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v
Chapter 1
HEAVY TAILS IN FINANCE FOR INDEPENDENT
OR MULTIFRACTAL PRICE INCREMENTS
BENOIT B. MANDELBROT
Sterling Professor of Mathematical Sciences, Yale University, New Haven, CT 065020-8283, USA
Contents
Abstract 4
1. Introduction: A path that led to model price by Brownian motion (Wiener or
fractional) of a multifractal trading time 5
1.1. From the law of Pareto to infinite moment "anomalies" that contradict the Gaussian "norm" 5
1.2. A scientific principle: scaling invariance in finance 6
1.3. Analysis alone versus statistical analysis followed by synthesis and graphic output 7
1.4. Actual implementation of scaling invariance by multifractal functions: it requires additional
assumptions that are convenient but not a matter of principle, for example, separability and
compounding 7
2. Background: the Bernoulli binomial measure and two random variants: shuffled
and canonical 8
2.1. Definition and construction of the Bernoulli binomial measure 8
2.2. The concept of canonical random cascade and the definition of the canonical binomial measure 9
2.3. Two forms of conservation: strict and on the average 9
2.4. The term "canonical" is motivated by statistical thermodynamics 10
2.5. In every variant of the binomial measure one can view all finite (positive or negative) powers
together, as forming a single "class of equivalence" 10
2.6. The full and folded forms of the address plane 11
2.7. Alternative parameters 11
3. Definition of the two-valued canonical multifractals 11
3.1. Construction of the two-valued canonical multifractal in the interval [0,1] 11
3.2. A second special two-valued canonical multifractal: the unifractal measure on the canonical
Cantor dust 12
3.3. Generalization of a useful new viewpoint: when considered together with their powers from
- to , all the TVCM parametrized by either p or 1- p form a single class of equivalence 12
3.4. The full and folded address planes 12
3.5. Background of the two-valued canonical measures in the historical development of multifractals
13
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
2 B.B. Mandelbrot
4. The limit random variable  = ([0,1]), its distribution and the star functional
equation 13
4.1. The identity EM = 1 implies that the limit measure has the "martingale" property, hence
the cascade defines a limit random variable  = ([0,1]) 13
4.2. Questions 14
4.3. Exact stochastic renormalizability and the "star functional equation" for  14
4.4. Metaphor for the probability of large values of , arising in the theory of discrete time
branching processes 14
4.5. To a large extent, the asymptotic measure  of a TVCM is large if, and only if, the pre-fractal
measure k([0,1]) has become large during the very first few stages of the generating cascade 15
5. The function (q): motivation and form of the graph 15
5.1. Motivation of (q) 15
5.2. A generalization of the role of : middle- and high-frequency contributions to microrandomness
15
5.3. The expected "partition function" Eq(dit) 16
5.4. Form of the (q) graph 17
5.5. Reducible and irreducible canonical multifractals 18
6. When u > 1, the moment Eq diverges if q exceeds a critical exponent qcrit
satisfying (q) = 0;  follows a power-law distribution of exponent qcrit 18
6.1. Divergent moments, power-law distributions and limits to the ability of moments to determine
a distribution 18
6.2. Discussion 19
6.3. An important apparent "anomaly": in a TVCM, the q-th moment of  may diverge 19
6.4. An important role of (q): if q > 1 the q-th moment of  is finite if, and only if, (q) > 0;
the same holds for (dt) whenever dt is a dyadic interval 19
6.5. Definition of qcrit; proof that in the case of TVCM qcrit is finite if, and only if, u > 1 20
6.6. The exponent qcrit can be considered as a macroscopic variable of the generating process 20
7. The quantity : the original Hölder exponent and beyond 21
7.1. The Bernoulli binomial case and two forms of the Hölder exponent: coarse-grained (or
coarse) and fine-grained 21
7.2. In the general TVCM measure,  = ~, and the link between "" and the Hölder exponent
breaks down; one consequence is that the "doubly anomalous" inequalities min < 0, hence
~ < 0, are not excluded 22
8. The full function f () and the function () 23
8.1. The Bernoulli binomial measure: definition and derivation of the box dimension function
f () 23
8.2. The "entropy ogive" function f (); the role of statistical thermodynamics in multifractals
and the contrast between equipartition and concentration 23
8.3. The Bernoulli binomial measure, continued: definition and derivation of a function () =
f () - 1 that originates as a rescaled logarithm of a probability 24
8.4. Generalization of () to the case of TVCM; the definition of f () as () + 1 is indirect
but significant because it allows the generalized f to be negative 24
8.5. Comments in terms of probability theory 25
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 3
8.6. Distinction between "center" and "tail" theorems in probability 26
8.7. The reason for the anomalous inequalities f () < 0 and  < 0 is that, by the definition of a
random variable (dt), the sample size is bounded and is prescribed intrinsically; the notion
of supersampling 26
8.8. Excluding the Bernoulli case p = 1/2, TVCM faces either one of two major "anomalies":
for p > -1/2, one has f (min) = 1 + log2 p > 0 and f (max) = 1 + log2(1 - p) < 0; for
p < 1/2, the opposite signs hold 27
8.9. The "minor anomalies" f (max) > 0 or f (min) > 0 lead to sample function with a clear
"ceiling" or "floor" 27
9. The fractal dimension D =  (1) = 2[-pulog2 u - (1 - p)v log2 v] and multifractal
concentration 27
9.1. In the Bernoulli binomial measures weak asymptotic negligibility holds but strong asymptotic
negligibility fails 28
9.2. For the Bernoulli or canonical binomials, the equation f () =  has one and only one solution;
that solution satisfies D > 0 and is the fractal dimension of the "carrier" of the measure 28
9.3. The notion of "multifractal concentration" 29
9.4. The case of TVCM with p < 1/2, allows D to be positive, negative, or zero 29
10. A noteworthy and unexpected separation of roles, between the "dimension
spectrum" and the total mass ; the former is ruled by the accessible  for
which f () > 0, the latter, by the inaccessible  for which f () < 0 30
10.1. Definitions of the "accessible ranges" of the variables: qs from q
min to q
max and s from

min to 
max; the accessible functions (q) and f () 30
10.2. A confrontation 30
10.3. The simplest cases where f () > 0 for all , as exemplified by the canonical binomial 31
10.4. The extreme case where f () < 0 and  < 0 both occur, as exemplified by TVCM when
u > 1 31
10.5. The intermediate case where min > 0 but f () < 0 for some values of  31
11. A broad form of the multifractal formalism that allows  < 0 and f () < 0 31
11.1. The broad "multifractal formalism" confirms the form of f () and allows f () < 0 for
some  32
11.2. The Legendre and inverse Legendre transforms and the thermodynamical analogy 32
Acknowledgments 32
References 32
4 B.B. Mandelbrot
Abstract
This chapter has two goals. Section 1 sketches the history of heavy tails in finance through
the author's three successive models of the variation of a financial price: mesofractal,
unifractal and multifractal. The heavy tails occur, respectively, in the marginal distribution
only (Mandelbrot, 1963), in the dependence only (Mandelbrot, 1965), or in both (Mandelbrot,
1997). These models increase in the scope of the "principle of scaling invariance",
which the author has used since 1957.
The mesofractal model is founded on the stable processes that date to Cauchy and Lévy.
The unifractal model uses the fractional Brownian motions introduced by the author. By
now, both are well-understood.
To the contrary, one of the key features of the multifractals (Mandelbrot, 1974a, b) remains
little known. Using the author's recent work, introduced for the first time in this
chapter, the exposition can be unusually brief and mathematically elementary, yet covering
all the key features of multifractality. It is restricted to very special but powerful cases:
(a) the Bernoulli binomial measure, which is classical but presented in a little-known fashion,
and (b) a new two-valued "canonical" measure. The latter generalizes Bernoulli and
provides an especially short path to negative dimensions, divergent moments, and divergent
(i.e., long range) dependence. All those features are now obtained as separately tunable aspects
of the same set of simple construction rules.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 5
My work in finance is well-documented in easily accessible sources, many of them reproduced
in Mandelbrot (1997 and also in 2001a, b, c, d). That work having expanded and
been commented upon by many authors, a survey of the literature is desirable, but this is
a task I cannot undertake now. However, it was a pleasure to yield to the entreaties of this
Handbook's editors by a text in which a new technical contribution is preceded by an introductory
sketch followed by a simple new presentation of an old feature that used to be
dismissed as "technical", but now moves to center stage.
The history of heavy tails in finance began in 1963. While acknowledging that the successive
increments of a financial price are interdependent, I assumed independence as a
first approximation and combined it with the principle of scaling invariance. This led to
(Lévy) stable distributions for the price changes. The tails are very heavy, in fact, powerlaw
distributed with an exponent  < 2.
The multifractal model advanced in Mandelbrot (1997) extends scale invariance to allow
for dependence. Readily controllable parameters generate tails that are as heavy as desired
and can be made to follow a power-law with an exponent in the range 1 <  < . This last
result, an essential one, involves a property of multifractals that was described in Mandelbrot
(1974a, b) but remains little known among users. The goal of the example described
after the introduction is to illustrate this property in a very simple form.
1. Introduction: A path that led to model price by Brownian motion (Wiener or
fractional) of a multifractal trading time
Given a financial price record P(t) and a time lag dt, define L(t,dt) = logP(t + dt) logP(t).
The 1900 dissertation of Louis Bachelier introduced Brownian motion as a model
of P(t). In later publications, however, Bachelier acknowledged that this is a very rough
first approximation: he recognized the presence of heavy tails and did not rule out dependence.
But until 1963, no one had proposed a model of the heavy tails' distribution.
1.1. From the law of Pareto to infinite moment "anomalies" that contradict the Gaussian
"norm"
All along, search for a model was inspired by a finding rooted in economics outside of
finance. Indeed, the distribution of personal incomes proposed in 1896 by Pareto involved
tails that are heavy in the sense of following a power-law distribution Pr{U > u} = u-.
However, almost nobody took this income distribution seriously. The strongest "conventional
wisdom" argument against Pareto was that the value  = 1.7 that he claimed leads
to the variance of U being infinite.
Infinite moments have been a perennial issue both before my work and (unfortunately)
ever since. Partly to avoid them, Pareto volunteered an exponential multiplier, resulting in
Pr{U > u} = u-
exp(-u).
6 B.B. Mandelbrot
Also, Herbert A. Simon expressed a universally held view when he asserted in 1953 that
infinite moments are (somehow) "improper". But in fact, the exponential multipliers are
not needed and infinite moments are perfectly proper and have important consequences. In
multifractal models, depending on specific features, variance can be either finite or infinite.
In fact, all moments can be finite, or they can be finite only up to a critical power qcrit that
may be 3, 4, or any other value needed to represent the data.
Beginning in the late 1950s, a general theme of my work has been that the uses of statistics
must be recognized as falling into at least two broad categories. In the "normal"
category, one can use the Gaussian distribution as a good approximation, so that the common
replacement of the term, "Gaussian", by "normal" is fully justified. To the contrary,
in the category one can call "abnormal" or "anomalous", the Gaussian is very misleading,
even as an approximation.
To underline this distinction, I have long suggested ­ to little effect up to now ­ that the
substance of the so-called ordinary central limit theorem would be better understood if it
is relabeled as the center limit theorem. Indeed, that theorem concerns the center of the
distribution, while the anomalies concern the tails. Following up on this vocabulary, the
generalized central limit theorem that yields Lévy stable limits would be better understood
if called a tail limit theorem. This distinction becomes essential in Section 8.5.
Be that as it may, I came to believe in the 1950s that the power-law distribution and
the associated infinite moments are key elements that distinguish economics from classical
physics. This distinction grew by being extended from independent to highly dependent
random variables. In 1997, it became ready to be phrased in terms of randomness and
variability falling in one of several distinct "states". The "mild" state prevails for classical
errors of observation and for sequences of near-Gaussian and near-independent quantities.
To the contrary, phenomena that present deep inequality necessarily belong to the "wild"
state of randomness.
1.2. A scientific principle: scaling invariance in finance
A second general theme of my work is the "principle" that financial records are invariant by
dilating or reducing the scales of time and price in ways suitably related to each other. There
is no need to believe that this principle is exactly valid, nor that its exact validity could ever
be tested empirically. However, a proper application of this principle has provided the
basis of models or scenarios that can be called good because they satisfy all the following
properties:
(a) they closely model reality,
(b) they are exceptionally parsimonious, being based on very few very general a priori
assumptions, and
(c) they are creative in the following sense: extensive and correct predictions arise as consequences
of a few assumptions; when those assumptions are changed the consequences
also change. By contrast, all too many financial models start with Brownian motion,
then build upon it by including in the input every one of the properties that one wishes
to see present in the output.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 7
1.3. Analysis alone versus statistical analysis followed by synthesis and graphic output
The topic of multifractal functions has grown into a well-developed analytic theory, making
it easy to apply the multifractal formalism blindly. But it is far harder to understand it and
draw consequences from its output. In particular, statistical techniques for handling multifractals
are conspicuous by their near-total absence. After they become actually available,
their applicability will have to be investigated carefully.
A chastening example is provided by the much simpler question of whether or not financial
series exhibit global (long range) dependence. My claim that they do was largely
based on R/S analysis which at this point relies heavily on graphical evidence. Lo (1991)
criticized this conclusion very severely as being subjective. Also, a certain alternative test
Lo described as "objective" led to a mixed pattern of "they do" and "they do not". This
pattern being practically impossible to interpret, Lo took the position that the simpler outcome
has not been shown wrong, hence one can assume that long range dependence is
absent.
Unfortunately, the "objective test" in question assumed the margins to be Gaussian.
Hence, Lo's experiment did not invalidate my conclusion, only showed that the test is
not robust and had repeatedly failed to recognize long range dependence.
The proper conclusion is that careful graphic evidence has not yet been superseded.
The first step is to attach special importance to models for which sample functions can be
generated.
1.4. Actual implementation of scaling invariance by multifractal functions: it requires
additional assumptions that are convenient but not a matter of principle, for
example, separability and compounding
By and large, an increase in the number and specificity in the assumptions leads to an
increase in the specificity of the results. It follows that generality may be an ideal unto
itself in mathematics, but in the sciences it competes with specificity, hence typically with
simplicity, familiarity, and intuition.
In the case of multifractal functions, two additional considerations should be heeded.
The so-called multifractal formalism (to be described below) is extremely important. But
it does not by itself specify a random function closely enough to allow analysis to be
followed by synthesis. Furthermore, multifractal functions are so new that it is best, in a
first stage, to be able to rely on existing knowledge while pursuing a concrete application.
For these and related reasons, my study of multifractals in finance has relied heavily on
two special cases.
One is implemented by the recursive "cartoons" investigated in Mandelbrot (1997) and
in much greater detail in Mandelbrot (2001c).
The other uses compounding.This process begins with a random function F() in which
the variable  is called an "intrinsic time". In the key context of financial prices,  is
called "trading time". The possible functions F() include all the functions that have been
previously used to model price variation. Foremost is the Wiener Brownian motion B(t)
8 B.B. Mandelbrot
postulated by Bachelier. The next simplest are the fractional Brownian motion BH (t) and
the Lévy stable "flight" L(t).
A separate step selects for the intrinsic trading time a scale invariant random functions
of the physical "clock time" t. Mandelbrot (1972) recommended for the function (t) the
integral of a multifractal measure. This choice was developed in Mandelbrot (1997) and
Mandelbrot, Calvet and Fisher (1997).
In summary, one begins with two statistically independent random functions F() and
(t), where (t) is non-decreasing. Then one creates the "compound" function F[(t)] =
(t). Choosing F() and (t) to be scale-invariant insures that (t) will be scale-invariant
as well. A limitation of compounding as defined thus far is that it demands independence
of F and , therefore restricts the scope of the compound function.
In a well-known special case called Bochner subordination, the increments of (t) are
independent. As shown in Mandelbrot and Taylor (1967), it follows that B[(t)] is a Lévy
stable process, i.e., the mesofractal model. This approach has become well-known. The
tails it creates are heavy and do follow a power law distribution but there are at least two
drawbacks. The exponent  is at most 2, a clearly unacceptable restriction in many cases,
and the increments are independent.
Compounding beyond subordination was introduced because it allows  to take any
value > 1 and the increments to exhibit long term dependence. All this is discussed elsewhere
(Mandelbrot, 1997 and more recent papers).
The goal of the remainder of this chapter is to use a specially designed simple case to
explain how multifractal measure suffices to create a power-law distribution. The idea is
that L(t,dt) = d(t) where  = BH [(t)]. Roughly, d(t) is |dBH |1/H . In the Wiener
Brownian case, H = 1/2 and d is the "local variance". This is how a price that fluctuates
up and down is reduced to a positive measure.
2. Background: the Bernoulli binomial measure and two random variants: shuffled
and canonical
The prototype of all multifractals is nonrandom: it is a Bernoulli binomial measure. Its
well-known properties are recalled in this section, then Section 3 introduces a random
"canonical" version. Also, all Bernoulli binomial measures being powers of one another,
a broader viewpoint considers them as forming a single "class of equivalence".
2.1. Definition and construction of the Bernoulli binomial measure
A multiplicative nonrandom cascade. A recursive construction of the Bernoulli binomial
measures involves an "initiator" and a "generator". The initiator is the interval [0,1] on
which a unit of mass is uniformly spread. This interval will recursively split into halves,
yielding dyadic intervals of length 2-k. The generator consists in a single parameter u,
variously called multiplier or mass. The first stage spreads mass over the halves of every
dyadic interval, with unequal proportions. Applied to [0,1], it leaves the mass u in [0,1/2]
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 9
and the mass v in [1/2,1]. The (k + 1)-th stage begins with dyadic intervals of length 2-k,
each split in two subintervals of length 2-k-1. A proportion equal to u goes to the left
subinterval and the proportion v, to the right.
After k stages, let 0 and 1 = 1 - 0 denote the relative frequencies of 0's and 1's in
the finite binary development t = 0.12 ...k. The "pre-binomial" measures in the dyadic
interval [dt] = [t,t + 2-k] takes the value
k(dt) = uk0 vk1 ,
which will be called "pre-multifractal". This measure is distributed uniformly over the
interval. For k  , this sequence of measures k(dt) has a limit (dt), which is the
Bernoulli binomial multifractal.
Shuffled binomial measure. The proportion equal to u now goes to either the left or
the right subinterval, with equal probabilities, and the remaining proportion v goes to the
remaining subinterval. This variant must be mentioned but is not interesting.
2.2. The concept of canonical random cascade and the definition of the canonical
binomial measure
Mandelbrot (1974a, b) took a major step beyond the preceding constructions.
The random multiplier M. In this generalization every recursive construction can be
described as follows. Given the mass m in a dyadic interval of length 2-k, the two subintervals
of length 2-k-1 are assigned the masses M1m and M2m, where M1 and M2 are
independent realizations of a random variable M called multiplier. This M is equal to u or
v with probabilities p = 1/2 and 1 - p = 1/2.
The Bernoulli and shuffled binomials both impose the constraint that M1 +M2 = 1. The
canonical binomial does not. It follows that the canonical mass in each interval of duration
2-k is multiplied in the next stage by the sum M1 + M2 of two independent realizations
of M. That sum is either 2u (with probability p2), or 1 (with probability 2(1 - p)p), or 2v
(with probability 1 - p2).
Writing p instead of 1/2 in the Bernoulli case and its variants complicates the notation
now, but will soon prove advantageous: the step to the TVCM will simply consist in
allowing 0 < p < 1.
2.3. Two forms of conservation: strict and on the average
Both the Bernoulli and shuffled binomials repeatedly redistribute mass, but within a dyadic
interval of duration 2-k, the mass remains exactly conserved in all stages beyond the k-th.
That is, the limit mass (t) in a dyadic interval satisfies k(dt) = (dt).
In a canonical binomial, to the contrary, the sum M1 + M2 is not identically 1, only its
expectation is 1. Therefore, canonical binomial construction preserve mass on the average,
but not exactly.
10 B.B. Mandelbrot
The random variable . In particular, the mass ([0,1]) is no longer equal to 1. It is a
basic random variable denoted by  and discussed in Section 4.
Within a dyadic interval dt of length 2-k, the cascade is simply a reduced-scale version
of the overall cascade. It transforms the mass k(dt) into a product of the form (dt) =
k(dt)(dt) where all the (dt) are independent realizations of the same variable .
2.4. The term "canonical" is motivated by statistical thermodynamics
As is well known, statistical thermodynamics finds it valuable to approximate large systems
as juxtapositions of parts, the "canonical ensembles", whose energy only depends on a
common temperature and not on the energies of the other parts. Microcanonical ensembles'
energies are constrained to add to a prescribed total energy. In the study of multifractals,
the use of this metaphor should not obscure the fact that the multiplication of canonical
factors introduces strong dependence among (dt) for different intervals dt.
2.5. In every variant of the binomial measure one can view all finite (positive or negative)
powers together, as forming a single "class of equivalence"
To any given real exponent g = 1 and multipliers u and v corresponds a multiplier Mg that
can take either of two values ug = ug with probability p, and vg = vg with probability
1 - p. The factor  is meant to insure pug + (1 - p)vg = 1/2. Therefore, [pug +
(1 - p)vg] = 1/2, that is,  = 1/[2EMg]. The expression 2EMg will be generalized and
encountered repeatedly especially through the expression
(q) = -log2 puq
+ (1 - p)vq
- 1 = -log2 2EMq
.
This is simply a notation at this point but will be justified in Section 5. It follows that
 = 2-(g), hence
ug = ug
2(g)
and vg = vg
2(g)
.
Assume u > v. As g ranges from 0 to , ug ranges from 1/2 to 1 and vg ranges from
1/2 to 0; the inequality ug > vg is preserved. To the contrary, as g ranges from 0 to ,
vg < ug. For example, g = -1 yields
ug =
1/u
1/u + 1/v
= v and vg =
1/v
1/v + 1/v
= u.
Thus, inversion leaves both the shuffled and the canonical binomial measures unchanged.
For the Bernoulli binomial, it only changes the direction of the time axis.
Altogether, every Bernoulli binomial measure can be obtained from any other as a reduced
positive or negative power. If one agrees to consider a measure and its reduced
powers as equivalent, there is only one Bernoulli binomial measure.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 11
In concrete terms relative to non-infinitesimal dyadic intervals, the sequences representing
log for different values of g are mutually affine. Each is obtained from the special
case g = 1 by a multiplication by g followed by a vertical translation.
2.6. The full and folded forms of the address plane
In anticipation of TVCM, the point of coordinates u and v will be called the address of a
binomial measure in a full address space. In that plane, the locus of the Bernoulli measures
is the interval defined by 0 < v, 0 < u, and u + v = 1.
The folded address space will be obtained by identifying the measures (u,v) and (v,u),
and representing both by one point. The locus of the Bernoulli measures becomes the
interval defined by the inequalities 0 < v < u and u + v = 1.
2.7. Alternative parameters
In its role as parameter added to p = 1/2, one can replace u by the ("informationtheoretical")
fractal dimension D = -ulog2 u - v log2 v which can be chosen at will in
this open interval ]0,1[. The value of D characterizes the "set that supports" the measure.
It received a new application in the new notion of multifractal concentration described in
Mandelbrot (2001c). More generally, the study of all multifractals, including the Bernoulli
binomial, is filled with fractal dimensions of many other sets. All are unquestionably positive.
One of the newest features of the TVCM will prove to be that they also allow negative
dimensions.
3. Definition of the two-valued canonical multifractals
3.1. Construction of the two-valued canonical multifractal in the interval [0,1]
The TVCM are called two-valued because, as with the Bernoulli binomial, the multiplier M
can only take 2 possible values u and v. The novelties are that p need not be 1/2, the
multipliers u and v are not bounded by 1, and the inequality u + v = 1 is acceptable.
For u + v = 1, the total mass cannot be preserved exactly. Preservation on the average
requires
EM = pu + (1 - p)v =
1
2
,
hence 0 < p = (1/2 - v)/(u - v) < 1.
The construction of TVCM is based upon a recursive subdivision of the interval [0,1]
into equal intervals. The point of departure is, once again, a uniformly spread unit mass.
The first stage splits [0,1] into two parts of equal lengths. On each, mass is poured uniformly,
with the respective densities M1 and M2 that are independent copies of M. The
second stage continues similarly with the interval [0,1/2] and [1/2,1].
12 B.B. Mandelbrot
3.2. A second special two-valued canonical multifractal: the unifractal measure on the
canonical Cantor dust
The identity EM = 1/2 is also satisfied by u = 1/2p and v = 0. In this case, let the lengths
and number of non-empty dyadic cells after k stages be denoted by t = 2-k and Nk. The
random variable Nk follows a simple birth and death process leading to the following
alternative.
When p > 1/2, ENk = (EN1)k = (2p)k = (dt)log(2p). To be able to write ENk =
(dt)-D, it suffices to introduce the exponent D = -log(2p). It satisfies D > 0 and defines
a fractal dimension.
When p < 1/2, to the contrary, the number of non-empty cells almost surely vanishes
asymptotically. At the same time, the formal fractal dimension D = -log(2p) satisfies
D < 0.
3.3. Generalization of a useful new viewpoint: when considered together with their
powers from - to , all the TVCM parametrized by either p or 1 - p form a
single class of equivalence
To take the key case, the multiplier M-1 takes the values
u-1 =
1/u
2(p/u + (1 - p)/v)
=
v
2(v + u) - 1
and v-1 =
u
2(v + u) - 1
.
It follows that pu-1 + (1 - p)v-1 = 1/2 and u-1/v-1 = v/u. In the full address plane,
the relations imply the following: (a) the point (u-1,v-1) lies on the extension beyond
(1/2,1/2) of the interval from (u,v) to (1/2,1/2) and (b) the slopes of the intervals from 0
to (u,v) and from 0 to (u-1,v-1) are inverse of one another. It suffices to fold the full phase
diagram along the diagonal to achieve v > u. The point (u-1,v-1) will be the intersection
of the interval corresponding to the probability 1 -p and of the interval joining 0 to (u,v).
3.4. The full and folded address planes
In the full address plane, the locus of all the points (u,v) with fixed p has the equation
pu + (1 - p)v = 1/2. This is the negatively sloped interval joining the points (0,1/2p)
and ([1/2(1 - p)],0). When (u,v) and (v,u) are identified, the locus becomes the same
interval plus the negatively sloped interval from [0,1/2(1 - p)] to (1/2p,0).
In the folded address plane, the locus is made of two shorter intervals from (1,1) to both
(1/2p,0) and ([1/2(1 - p)],0). In the special case u + v = 1 corresponding to p = 1/2,
the two shorter intervals coincide.
Those two intervals correspond to TVCM in the same class of equivalence. Starting
from an arbitrary point on either interval, positive moments correspond to points to the
same interval and negative moments, to points of the other. Moments for g > 1 correspond
to points to the left on the same interval; moments for 0 < g < 1, to points to the right on
the same interval; negative moments to points on the other interval.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 13
For p = 1/2, the class of equivalence of p includes a measure that corresponds to u = 1
and v = [1/2 - min(p,1 - p)]/[max(p,1 - p)]. This novel and convenient universal
point of reference requires p = 1/2. In terms to be explained below, it corresponds to
min = -logu = 0.
3.5. Background of the two-valued canonical measures in the historical development of
multifractals
The construction of TVCM is new but takes a well-defined place among the three main
approaches to the development of a theory of multifractals.
General mathematical theories came late and have the drawback that they are accessible
to few non-mathematicians and many are less general than they seem.
The heuristic presentation in Frisch and Parisi (1985) and Halsey et al. (1986) came
after Mandelbrot (1974a, b) but before most of the mathematics. Most importantly for
this paper's purpose, those presentations fail to include significantly random constructions,
hence cannot yield measures following the power law distribution.
Both the mathematical and the heuristic approaches seek generality and only later consider
the special cases. To the contrary, a third approach, the first historically, began in
Mandelbrot (1974a, b) with the careful investigation of a variety of special random multiplicative
measures. I believe that each feature of the general theory continues to be best
understood when introduced through a special case that is as general as needed, but no
more. The general theory is understood very easily when it comes last.
In pedagogical terms, the "third way" associates with each distinct feature of multifractals
a special construction, often one that consists of generalizing the binomial multifractal
in a new direction. TVCM is part of a continuation of that effective approach; it could have
been investigated much earlier if a clear need had been perceived.
4. The limit random variable  = ([0,1]), its distribution and the star functional
equation
4.1. The identity EM = 1 implies that the limit measure has the "martingale" property,
hence the cascade defines a limit random variable  = ([0,1])
We cannot deal with martingales here, but positive martingales are mathematically attractive
because they converge (almost surely) to a limit. But the situation is complicated because
the limit depends on the sign of D = 2[-pulog2 u - (1 - p)v log2 v].
Under the condition D > 0, which is discussed in Section 9, what seemed obvious is
confirmed: Pr{ > 0} > 0, conservation on the average continues to hold as k  , and
 is either non-random, or is random and satisfies the identity E = 1.
But if D < 0, one finds that  = 0 almost surely and conservation on the average holds
for finite k but fails as k  . The possibility that  = 0 arose in mathematical esoterica
and seemed bizarre, but is unavoidably introduced into concrete science.
14 B.B. Mandelbrot
4.2. Questions
(A) Which feature of the generating process dominates the tail distribution of ? It is
shown in Section 6 to be the sign of max(u,v) - 1.
(B) Which feature of the generating process allows  to have a high probability of being
either very large or very small? Section 6 will show that the criterion is that the
function (q) becomes negative for large enough q.
(C) Divide [0,1] into 2k intervals of length 2-k. Which feature of the generating process
determines the relative distribution of the overall  among those small intervals? This
relative distribution motivated the introduction of the functions f () and (), and is
discussed in Section 8.
(D) Are the features discussed under (B) and (C) interdependent? Section 10 will address
this issue and show that, even when  has a high probability of being large, its value
does not affect the distribution under (C).
4.3. Exact stochastic renormalizability and the "star functional equation" for 
Once again, the masses in [0,1/2] and [1/2,1] take, respectively, the forms M11 and
M22, where M1 and M2 are two independent realizations of the random variable M and
1, and 2 are two independent realizations of the random variable . Adding the two
parts yields
  1M1 + 2M2.
This identity in distribution, now called the "star equation", combines with E = 1 to
determine . It was introduced in Mandelbrot (1974a, b) and has since then been investigated
by several authors, for example by Durrett and Liggett (1983). A large bibliography
is found in Liu (2002).
In the special case where M is non-random, the star equation reduces to the equation
due to Cauchy whose solutions have become well-known: they are the Cauchy­Lévy stable
distributions.
4.4. Metaphor for the probability of large values of , arising in the theory of discrete
time branching processes
A growth process begins at t = 0 with a single cell. Then, at every integer instant of time,
every cell splits into a random non-negative number of N1 cells. At time k, one deals with
a clone of Nk cells. All those random splittings are statistically independent and identically
distributed. The normalized clone size, defined as Nk/ENk
1 has an expectation equal to 1.
The sequence of normalized sizes is a positive martingale, hence (as already mentioned)
converges to a limit random variable.
When EN > 1, that limit does not reduce to 0 and is random for a very intuitive reason.
As long as clone size is small, its growth very much depends on chance, therefore
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 15
the normalized clone size is very variable. However, after a small number of splittings, a
law of large numbers comes into force, the effects of chances become negligible, and the
clone grows near-exponentially. That is, the randomness in the relative number of family
members can be very large but acts very early.
4.5. To a large extent, the asymptotic measure  of a TVCM is large if, and only if, the
pre-fractal measure k([0,1]) has become large during the very first few stages of
the generating cascade
Such behavior is suggested by the analogy to a branching process, and analysis shows that
such is indeed the case. After the first stage, the measures 1([0,1/2]) and 1([1/2,1])
are both equal to u2 with probability p2, uv with probability 2p(1 - p), and v2 with
probability (1 - p)2. Extensive simulations were carried out for large k in "batches", and
the largest, medium, and smallest measure was recorded for each batch. Invariably, the
largest (resp., smallest)  started from a high (resp., low) overall level.
5. The function (q): motivation and form of the graph
So far (q) was nothing but a notation. It is important as it is the special form taken
for TVCM by a function that was first defined for an arbitrary multiplier in Mandelbrot
(1974a, b). (Actually, the little appreciated Figure 1 of that original paper did not include
q < 0 and worked with -(q), but the opposite sign came to be generally adopted.)
5.1. Motivation of (q)
After k cascade stages, consider an arbitrary dyadic interval of duration dt = 2-k. For
the k-approximant TVCM measure k(dt) the q-th power has an expected value equal to
[puq + (1 - p)vq]k = {EMq}k. Its logarithm of base 2 is
log2 puq
+ (1 - p)vq k
= k log2 puq
+ (1 - p)vq
= log2(dt) (q) + 1 .
Hence
E
q
k (dt) = (dt)(q)+1
.
5.2. A generalization of the role of : middle- and high-frequency contributions to
microrandomness
Exactly the same cascade transforms the measure in dt from k(dt) to (dt) and the
measure in [0,1] from 1 to . Hence, one can write
(dt) = k(dt)(dt).
16 B.B. Mandelbrot
Fig. 1. The full phase diagram of TVCM with coordinates u and v. The isolines of the quantity p are straight
intervals from (1/{2(1 - p)},0) to (0,1/{2p}). The values p and 1 - p are equivalent and the corresponding
isolines are symmetric with respect to the main bisector u = v. The acceptable part of the plane excludes the
points (u,v) such that either max(u,v) < 1/2 or min(u,v) > 1/2. Hence, the relevant part of this diagram is
made of two infinite halfstrips reducible to one another by folding along the bisector. The folded phase diagram
of TVCM corresponds to v < 0.5 < u. It shows the following curves. The isolines of 1 - p and p are straight
intervals that start at the point (1,1) and end at the points (1/{2p},0) and (1/{2(1 - p)},0). The isolines of D
start on the interval 1/2 < u < 1 of the u-axis and continue to the point (,0). The isolines of qcrit start at the
point (1,0) and continue to the point (,0). The Bernoulli binomial measure corresponds to p = 1/2 and the
canonical Cantor measure corresponds to the half line v = 0, u > 1/2.
In this product, frequencies of wavelength > dt, to be described as "low", contribute
k([0,1]), and frequencies of wavelength < dt, to be described as "high", contribute .
5.3. The expected "partition function" Eq(dit)
Section 6 will show that Eq need not be finite. But if it is, the limit measure (dt) =
k(dt)(dt) satisfies
Eq
(dt) = (dt)(q)+1
Eq
.
The interval [0,1] subdivides into 1/dt intervals dit of common length dt. The sum of
the q-th moments over those intervals takes the form
E(dt) = Eq
(dit) = (dt)(q)
Eq
.
Estimation of (q) from a sample. It is affected by the prefactor  insofar as one must
estimate both (q) and logEq.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 17
5.4. Form of the (q) graph
Due to conservation on the average, EM = pu + (1 - p)v = 1/2, hence (1) =
-log2[1/2] - 1 = 0. An additional universal value is (0) = -log2(1) - 1 = -1. For
other values of q, (q) is a cap-convex continuous function satisfying (q) < -1 for
q < 0.
For TVCM, a more special property is that (q) is asymptotically linear: assuming
u > v, and letting q  :
(q)  -log2 p - 1 - q logu and (-q)  -log2(1 - p) - 1 + q logv.
The sign of u - 1 affects the sign of logu, a fact that will be very important in Section 6.
Moving as little as possible beyond these properties. The very special tau function of the
TVCM is simple but Figure 2 suffices to bring out every one of the delicate possibilities first
reported in Mandelbrot (1974a), where -(q) is plotted in that little appreciated Figure 1.
Other features of  that deserve to be mentioned. Direct proofs are tedious and the short
proofs require the multifractal formalism that will only be described in Section 11.
Fig. 2. The function (q) for p = 3/4 and varying g. By arbitrary choice, the value g = 1 is assigned u = 1, from
which follows that g = -1 is assigned to the case v = 1. Behavior of (q) for the value g > 0: as q  -, the
graph of (q) is asymptotically tangent to  = -q log2 v, as q  , the graph of (q) is asymptotically tangent
to  = -q log2 u. Those properties are widely believed to describe the main facts about (q). But for TVCM they
do not. Thus, (q) is also tangent to  = q
max and  = q
min. Beyond those points of tangency, f becomes < 0.
For g > 1, that is, for u > 1, (q) has a maximum. Values of q beyond this maximum correspond to min < 0.
Because of the capconvexity of (q), the equation (q) = 0 may, in addition to the "universal" value q = 1,
have a root qcrit > 1. For u > 2.5, one deals with a very different phenomenon also first described in Mandelbrot
(1974a, b). One finds that the construction of TVCM leads to a measure that degenerates to 0.
18 B.B. Mandelbrot
The quantity D(q) = (q)/(q - 1). This popular expression is often called a "generalized
dimension", a term too vague to mean anything. D(q) is obtained by extending the
line from (q,) to (1,0) to its intercept with the line q = 0. It plays the role of a critical
embedding codimension for the existence of a finite q-th moment. This topic cannot be
discussed here but is treated in Mandelbrot (2003).
The ratio (q)/q and the "accessible" values of q. Increase q from - to 0 then to
+. In the Bernoulli case, (q)/q increases from max to , jumps down to - for
q = 0, then increases again from - to min. For TVCM with p = 1/2, the behavior
is very different. For example, let p < 1/2. As q increases from 1 to , (q) increases
from 0 to a maximum 
max, then decreases. In a way explored in Section 10, the values of
 > 
max are not "accessible".
5.5. Reducible and irreducible canonical multifractals
Once again, being "canonical" implies conservation on the average. When there exists a
microcanonical (conservative) variant having the same function f (), a canonical measure
can be called "reducible". The canonical binomial is reducible because its f () is
shared by the Bernoulli binomial. Another example introduced in Mandelbrot (1989b) is
the "Erice" measure, in which the multiplier M is uniformly distributed on [0,1]. But the
TVCM with p = 1/2 is not reducible.
In the interval [0,1] subdivided in the base b = 2, reducibility demands a multiplier M
whose distribution is symmetric with respect to M = 1/2. Since u > 0, this implies u < 1.
6. When u > 1, the moment Eq diverges if q exceeds a critical exponent qcrit
satisfying (q) = 0;  follows a power-law distribution of exponent qcrit
6.1. Divergent moments, power-law distributions and limits to the ability of moments to
determine a distribution
This section injects a concern that might have been voiced in Sections 4 and 5. The canonical
binomial and many other examples satisfy the following properties, which everyone
takes for granted and no one seems to think about: (a)  = 1, Eq < , (b) (q) > 0 for
all q > 0, and (c) (q)/q increases monotonically as q  .
Many presentations of fractals take those properties for granted in all cases. In fact, as
this section will show, the TVCM with u > 1 lead to the "anomalous" divergence Eq =
 and the "inconceivable" inequality (q) < 0 for qcrit < q < . Also, the monotonicity
of (q)/q fails for all TVCM with p = 1/2.
Since Pareto in 1897, infinite moments have been known to characterize the power-law
distributions of the form Pr{X > x} = x-qcrit . But in the case of TVCM and other canonical
multifractals, the complicating factor L(x) is absent. One finds that when u > 1, the overall
measure  follows a power law of exponent qcrit determined by (q).
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 19
6.2. Discussion
The power-law "anomalies" have very concrete consequences deduced in Mandelbrot
(1997) and discussed, for example, in Mandelbrot (2001c).
But does all this make sense? After all, (q) and Eq are given by simple formulas and
are finite for all parameters. The fact that those values cannot actually be observed raises a
question. Are high moments lost by being unobservable? In fact, they are "latent" but can
be made "actual" by a process is indeed provided by the process of "embedding" studied
elsewhere.
An additional comment is useful. The fact that high moments are non-observable does
not express a deficiency of TVCM but a limitation of the notion of moment. Features
ordinarily expressed by moments must be expressed by other means.
6.3. An important apparent "anomaly": in a TVCM, the q-th moment of  may diverge
Let us elaborate. From long past experience, physicists' and statisticians' natural impulse
is to define and manipulate moments without envisioning or voicing the possibility of their
being infinite. This lack of concern cannot extend to multifractals. The distribution of the
TVCM within a dyadic interval introduces an additional critical exponent qcrit that satisfies
qcrit > 1. When 1 < qcrit < , which is a stronger requirement that D > 0, the q-th
moment of (dt) diverges for q > qcrit.
A stronger result holds: the TVCM cascade generates a measure whose distribution follows
the power law of exponent qcrit.
Comment. The heuristic approach to non-random multifractals fails to extend to random
ones, in particular, it fails to allow qcrit < . This makes it incomplete from the viewpoint
of finance and several other important applications.
The finite qcrit has been around since Mandelbrot (1974a, b) (where it is denoted by )
and triggered a substantial literature in mathematics. But it is linked with events so extraordinarily
unlikely as to appear incapable of having any perceptible effect on the generated
measure. The applications continue to neglect it, perhaps because it is ill-understood.
A central goal of TVCM is to make this concept well-understood and widely adopted.
6.4. An important role of (q): if q > 1 the q-th moment of  is finite if, and only if,
(q) > 0; the same holds for (dt) whenever dt is a dyadic interval
By definition, after k levels of iteration, the following symbolic equality relates independent
realizations of M and . That is, it does not link random variables but distributions
k [0,1] = Mk-1 [0,1] + Mk-1 [0,1] .
Conservation on the average is expressed by the identity Ek-1([0,1]) = 1. In addition,
we have the following recursion relative to the second moment.
E2
[0,1] = 2EM2
E2
k-1 [0,1] + 2EM2
Ek-1 [0,1]
2
.
20 B.B. Mandelbrot
The second term to the right reduces to 1/2. Now let k  . The necessary and sufficient
condition for the variance of k([0,1]) to converge to a finite limit is
2 EM2
< 1 in other words (2) = -log2 EM2
- 1 > 0.
When such is the case, Kahane and Peyrire (1976) gave a mathematically rigorous
proof that there exists a limit measure ([0,1]) satisfying the formal expression
E2
[0,1] =
1
2(1 - 2(2))
.
Higher integer moments satisfy analogous recursion relations. That is, knowing that all
moments of order up to q - 1 are finite, the moment of order q is finite if and only if
(q) > 0.
The moments of non-integer order q are more delicate to handle, but they too are finite
if, and only if, (q) > 0.
6.5. Definition of qcrit; proof that in the case of TVCM qcrit is finite if, and only if, u > 1
Section 5.4 noted that the graph of (q) is always cap-convex and for large q > 0,
(q)  -log2 puq
+ -1  -log2 p - 1 - q log2 u.
The dependence of (q) on q is ruled by the sign of u - 1, as follows.
* The case when u < 1, hence min > 0. In this case, (q) is monotone increasing and
(q) > 0 for q > 1. This behavior is exemplified by the Bernoulli binomial.
* The case when u > 1, hence min < 0. In this case, one has (q) < 0 for large q. In addition
to the root q = 1, the equation (q) = 1 has a second root that is denoted by qcrit.
Comment. In terms of the function f () graphed on Figure 3, the values 1 and qcrit are
the slopes of the two tangents drawn to f () from the origin (0,0).
Within the class of equivalence of any p and 1 - p; the parameter g can be "tuned" so
that qcrit begins by being > 1 then converges to 1; if so, it is seen that D converges to 0.
* Therefore, the conditions qcrit = 1 and D = 0 describe the same "anomaly".
In Figure 1, isolines of qcrit are drawn for qcrit = 1,2,3, and 4. When q = 1 is the only
root, it is convenient to say that qcrit = . This isoset qcrit =  is made of the half-line
{v = 1/2 and u > 1/2} and of the square {0 < v < 1/2,1/2 < u < 1}.
6.6. The exponent qcrit can be considered as a macroscopic variable of the generating
process
Any set of two parameters that fully describes a TVCM can be called "microscopic". All
the quantities that are directly observable and can be called macroscopic are functions of
those two parameters.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 21
Fig. 3. The functions f () for p = 3/4 and varying g. All those graphs are linked by horizontal reductions or
dilations followed by translation and further self-affinity. It is widely anticipated that f () > 0 holds in all cases,
but for the TVCM this anticipation fails, as shown in this figure. For g > 0 (resp., g < 0) the left endpoint of f ()
(resp., the right endpoint) satisfies f () < 0 and the other endpoint, f () > 0.
For the general canonical multifractal, a full specification requires a far larger number
of microscopic quantities but the same number of macroscopic ones. Some of the latter
characterize each sample, but others, for example qcrit, characterize the population.
7. The quantity : the original Hölder exponent and beyond
The multiplicative cascades ­ common to the Bernoulli and canonical binomials and
TVCM ­ involve successive multiplications. An immediate consequence is that both the
basic (dt) and its probability are most intrinsically viewed through their logarithms.
A less obvious fact is that a normalizing factor 1/log(dt) is appropriate in each case.
An even less obvious fact is that the normalizations log/logdt and logP/logdt are of
far broader usefulness in the study of multifractals. The exact extend of their domain of
usefulness is beyond the goal of this chapter, but we keep some special cases that can be
treated fully by elementary arguments.
7.1. The Bernoulli binomial case and two forms of the Hölder exponent: coarse-grained
(or coarse) and fine-grained
Recall that due to conservation, the measure in an interval of length dt = 2-k is the same
after k stages and in the limit, namely, (dt) = k(dt). As a result, the coarse-grained
Hölder exponent can be defined in either of two ways,
(dt) =
log(dt)
log(dt)
and
~(dt) =
logk(dt)
log(dt)
.
22 B.B. Mandelbrot
The distinction is empty in the Bernoulli case but prove prove essential for the TVCM.
In terms of the relative frequencies 0 and 1 defined in Section 2.1,
(dt) = ~(dt) = (0,1) = -0 log2 u - 1 log2 v
= -0(log2 u - log2 v) - logv.
Since u > v, one has 0 < min = -log2 u  = ~ max = -log2 v < . In particular,
 > 0, hence ~ > 0. As dt  0, so does (dt), and a formal inversion of the definition
of  yields
(dt) = (dt)
.
This inversion reveals an old mathematical pedigree. Redefine 0 and 1 from denoting
the finite frequencies of 0 and 1 in an interval, into denoting the limit frequencies at an
instant t. The instant t is the limit of an infinite sequence of approximating intervals of
duration 2-k. The function ([0,t]) is non-differentiable because limdt0 (dt)/dt is not
defined and cannot serve to define the local density of  at the instant dt.
The need for alternative measures of roughness of a singularity expression first arose
around 1870 in mathematical esoterica due to L. Hölder. In fractal/multifractal geometry
this expression merged with a very concrete exponent due to H.E. Hurst and is continually
being generalized. It follows that for the Bernoulli binomial measure, it is legitimate to
interpret the coarse s as finite-difference surrogates of the local (infinitesimal) Hölder
exponents.
7.2. In the general TVCM measure,  = ~, and the link between "" and the Hölder
exponent breaks down; one consequence is that the "doubly anomalous"
inequalities min < 0, hence ~ < 0, are not excluded
A Hölder (Hurst) exponent is necessarily positive. Hence negative ~s cannot be interpreted
as Hölder exponents. Let us describe the heuristic argument that leads to this paradox and
then show that ~ < 0 is a serious "anomaly": it shows that the link between "some kind
of " and the Hölder exponent requires a searching look. The resolution of the paradox is
very subtle and is associated with the finite qcrit introduced in Section 6.5.
Once again, except in the Bernoulli case,  = 1 and (dt) = k(dt)(dt), hence
(dt) = ~(dt) +
log(dt)
logdt.
In the limit dt  0 the factor log = /log(dt) tends to 0, hence it seems that  = ~.
Assume u > 1, hence min < 0 and consider an interval where ~(dt) < 0. The formal
equality
"k(dt) = (dt)~
"
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 23
seems to hold and to imply that "the" mass in an interval increases as the interval length
 0. On casual inspection, this is absurd. On careful inspection, it is not ­ simply because
the variable dt = 2-k and the function k(dt) both depend on k. For example, consider the
point t for which 0 = 1. Around this point, one has k = uk-1 > k-1. This inequality
is not paradoxical.
Furthermore, Section 8 shows that the theory of the multiplicative measures introduces
~ intrinsically and inevitably and allows ~ < 0.
Those seemingly contradictory properties will be reexamined in Section 9. Values of
(dt) will be seen to have a positive probability but one so minute that they can never be
observed in the way  > 0 are observed. But they affect the distribution of the variable 
examined in Section 4, therefore are observed indirectly.
8. The full function f () and the function ()
8.1. The Bernoulli binomial measure: definition and derivation of the box dimension
function f ()
The number of intervals of denumerator 2-k leading to 0 and 1 is N(k,0,1) =
k!/(k0)!(k1)!, and dt is the reduction ratio r from [0,1] to an interval of duration dt.
Therefore, the expression
f (k,0,1) = -
logN(k,0,1)
log(dt)
= -
log[k!/(k0)!(k1)!]
log(dt)
is of the form f (k,0,1) = -logN/logr. Fractal geometry calls this the "box similarity
dimension" of a set. This is one of several forms taken by fractal dimension. More
precisely, since the boxes belong to a grid, it is a grid fractal dimension.
The dimension function f (). For large k, the leading term in the Stirling approximation
of the factorial yields
lim
k
f (k,0,1) = f (0,1) = -0 log2 0 - 1 log2 1.
8.2. The "entropy ogive" function f (); the role of statistical thermodynamics in
multifractals and the contrast between equipartition and concentration
Eliminate 0 and 1 between the functions f and  = -0 logu - 1 logv. This yields in
parametric form a function, f (). Note that 0 f () min{,1}. Equality to the right is
achieved when 0 = u. The value  where f =  is very important and will be discussed
in Section 9. In terms of the reduced variable 0 = ( - min)/(max - min), the function
f () becomes the "ogive"
~f (0) = -0 log2 0 - (1 - 0)log2(1 - 0).
24 B.B. Mandelbrot
This ~f (0) can be called a universal function. The f () corresponding to fixed p and
varying g are affine transforms of ~f (0), therefore of one another. The ogive function ~f
first arose in thermodynamics as an entropy and in 1948 (with Shannon) entered communication
theory as an information. Its occurrence here is the first of several roles the
formalism of thermodynamics plays in the theory of multifractals.
An essential but paradoxical feature. Equilibrium thermodynamics is a study of various
forms of near-equality, for example postulates the equipartition of states on a surface in
phase space or of energy among modes. In sharp contrast, multifractals are characterized
by extreme inequality between the measures in different intervals of common duration dt.
Upon more careful examination, the paradox dissolves by being turned around: the main
tools of thermodynamics can handle phenomena well beyond their original scope.
8.3. The Bernoulli binomial measure, continued: definition and derivation of a function
() = f () - 1 that originates as a rescaled logarithm of a probability
The function f () never fully specifies the measure. For example, it does not distinguish
between the Bernoulli, shuffled and canonical binomials. The function f () can be generalized
by being deduced from a function () = f ()-1 that will now be defined. Instead
of dimensions, that deduction relies on probabilities. In the Bernoulli case, the derivation
of  is a minute variant of the argument in Section 8.1, but, contrary to the definition of f ,
the definition of  easily extends to TVCM and other random multifractals.
In the Bernoulli binomial case, the probability of hitting an interval leading to 0 and 1
is simply P(k,0,) = N(k,0,1)2-k = k!/(k0)!(k1)!2-k. Consider the expression
(k,0,1) = -
log[P(k,0,1)]
log(dt)
,
which is a rescaled but not averaged form of entropy. For large k, Stirling yields
lim
k
(k,0,1) = (0,1) = -0 log2 0 - 1 log2 1 - 1
= f () - 1.
8.4. Generalization of () to the case of TVCM; the definition of f () as () + 1 is
indirect but significant because it allows the generalized f to be negative
Comparing the arguments in Sections 8.1 and 8.2 link the concepts of fractal dimension
and of minus log (probability). However, when f () is reported through f () = ()+1,
the latter is not a mysterious "spectrum of singularities". It is simply the peculiar but proper
way a probability distribution must be handled in the case of multifractal measures. Moreover,
there is a major a priori difference exploited in Section 10. Minus log (probability)
is not subjected to any bound. To the contrary, every one of the traditional definitions of
fractal dimension (including Hausdorff­Besicovitch or Minkowski­Bouligand) necessarily
yields a positive value.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 25
The point is that the dimension argument in Section 8.1 does not carry over to TVCM,
but the probability argument does carry over as follows. The probability of hitting an interval
leading to 0 and 1 now changes to P(k,0,1) = p(0k)!/(k0)!(k1)! One can
now form the expression
(k,0,1) = -
log[P(k,0,1)]
log(dt)
.
Stirling now yields
(0,1) = lim
k
(k,0,1)
= {-0 log2 0 - 1 log2 1} + 0 log2 p + 1 log2(1 - p) .
In this sum of two terms marked by braces, we know that the first one transforms (by
horizontal stretching and translation) into the entropy ogive. The second is a linear function
of , namely 0[log2 p - log2(1 - p)] + log2(1 - p). It transforms the entropy ogive by
an affinity in which the line joining the two support endpoints changes from horizontal to
inclined. The overall affinity solely depends on p, but 0 depends explicitly on u and v.
This affinity extends to all values of p. Another property familiar from the binomial
extends to all values of p. For all u and v, the graphs of (), hence of f () have a
vertical slope for q = .
Alternatively, (0,1) = -0 log2[0/p] - 1 log2[1/(1 - p)].
8.5. Comments in terms of probability theory
Roughly speaking, the measure  is a product of random variables, while the limit
theorems of probability theory are concerned with sums. The definition of  as
log(dt)/log(dt) replaces a product of random variables M by a weighted sum of random
variables of the form logM. Let us now go through this argument step by step in
greater rigor and generality. One needs a cumbersome restatement of k(dt).
The low frequency factor of k(dt) and the random variable Hlow. Consider once again
a dyadic cell of length 2-k that starts at t = 0.12 ...k. The first k stages of the cascade
can be called of low frequency because they involve multipliers that are constant over
dyadic intervals of length dt = 2-k or longer. These stages yield
k(dt) = M(1)M(1,2)M(1,...,k) = M.
We transform k(dt) into the low frequency random variable
Hlow =
log[k(dt)]
log(dt)
=
1
k
-log2 M(1) - log2 M(1,2) -  .
26 B.B. Mandelbrot
We saw in Section 4.5 that the first few values of M largely determine the distribution
of . But the last expression involves an operation of averaging in which the first terms
contributing to (dt) are asymptotically washed out.
8.6. Distinction between "center" and "tail" theorems in probability
The quantity ~k(dt) = 0 log2 u - 1 log2 v is the average of a sum of variables -logM;
but why is its distribution is not Gaussian and the graph of () is an entropy ogive rather
than a parabola? Why is this so? The law of large numbers tells us that ~k(dt) almost
surely converges to its expectation which tells us very little. A tempting heuristic argument
continues as follows. The central limit theorem is believed to ensure that for small
dt, Hlow(dt) becomes Gaussian, therefore the graph of logp(dt) should be expected to be
a parabola. This being granted, why is it that the Stirling approximation yields an entropy
ogive ­ not a parabola?
In fact, there is no paradox of any kind. While the central limit theorem is indeed central
to probability theory, all it asserts in this context is that, asymptotically, the Gaussian rules
the center of the distribution, its "bell". Renormalizations reduce this center to the immediate
neighborhood of the top of the () graph and the central limit theorem is correct in
asserting that the top of the entropy ogive is locally parabolic. But in the present context
this information is of little significance. We need instead an alternative that is only concerned
with the tail behavior which it ought to blow up. For this and many other reasons,
it would be an excellent idea to speak of center, not central limit theorem. The tail limit
theorem is due to H. Cramer and asserts that the tail consisting in the bulk of the graph is
not a parabola but an entropy ogive.
8.7. The reason for the anomalous inequalities f () < 0 and  < 0 is that, by the
definition of a random variable (dt), the sample size is bounded and is prescribed
intrinsically; the notion of supersampling
The inequality () < -1 characterizes events whose probability is extraordinarily small.
The finding that this inequality plays a significant role was not anticipated, remains difficult
to understand and appreciate, and demands comment.
The common response is that even extremely low probability events are captured if one
simply takes a sufficiently long sample of independent values. But this is impossible, even
if one forgets that, in the present uncommon context, the values are extremely far from
being statistically independent. Indeed, the choice the duration dt = 2-k has two effects.
Not only does it fix the distribution of (dt), but it also sets the sample size at the value
N = 1/dt = 2k. Roughly speaking, a sample of size N can only reveal values having a
probability greater than 1/N, which means () > -1.
In summary, it is true that decreasing dt to 2-k-1 increases the sample size. But it
also changes the distribution and does so in such a way that the bound  = -1 remains
untouched.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 27
This bound excludes u items of information that correspond to f () < 0 (for example,
the value of qcrit when finite). Those items remain hidden and latent in the sense that they
cannot be inferred from one sample of values of (dt). Ways of revealing those values, supersampling
and embedding, are examined in Mandelbrot (1989b, 1995) and forthcoming
Mandelbrot (2003).
Figure 3 shows, for p = 3/4, how the graph of f () depends on g.
8.8. Excluding the Bernoulli case p = 1/2, TVCM faces either one of two major
"anomalies": for p > -1/2, one has f (min) = 1 + log2 p > 0 and
f (max) = 1 + log2(1 - p) < 0; for p < 1/2, the opposite signs hold
The fact that the values of (min) = f (min) - 1 and (max) = f (max) - 1 are logarithms
of probabilities confirms and extends the definition of p() = f () - 1 as a limit
rescaled probability. Here, those endpoint values of f () are independent of g and the
affinity that deduces them from the entropy ogive (with ends on the horizontal axis) characterizes
the class of equivalence of p and 1-p. If, and only if, p = 1/2 and u+v = 1, that
is, in the familiar Bernoulli binomial case, one has (min) = (max) = log2(1/2) = -1
hence f (min) = f (max) = 0. When u + v = 1, one of the endpoints satisfies f > 0 and
the other satisfies f < 0. Sections 8.9 and 10 shall examine the sharply differing consequences
of those inequalities.
8.9. The "minor anomalies" f (max) > 0 or f (min) > 0 lead to sample function with a
clear "ceiling" or "floor"
Suppose that f (min) = 0 and f (max) = 0, as is the case for p = 1/2. Then, using terms
often applied to the printed page ­ but after it has been turned 90 to the side ­ the sample
functions are "non-justified" or "ragged" for both high and low values. That is, the values
tend to be unequal; one is clearly larger than all others, a second is clearly the second
largest, etc.
To the contrary, TVCM with p = 1/2 yield either f (max) > 0 or f (min) > 0. Sample
functions have a conspicuous "ceiling" (resp., a "floor"). That is, a largest (resp., smallest)
value is attained repeatedly for values of t belonging to a set of positive dimension. To
use the printers' vocabulary, when one side is "ragged" the other is "justified". On visual
inspection of the data, the ceiling is always visible; the floor merges with the time axis,
except when one plots log[(dt)].
9. The fractal dimension D =  (1) = 2[-pulog2 u - (1 - p)v log2 v] and
multifractal concentration
The function f () satisfies f () , with equality f () =  when  = D =  (1). From
the value of  = D follows one of the most important properties of multifractals. Mandelbrot
(2001d) proposed to call it "multifractal concentration". This section will first examine
its opposite, which is asymptotic negligibility.
28 B.B. Mandelbrot
9.1. In the Bernoulli binomial measures weak asymptotic negligibility holds but strong
asymptotic negligibility fails
Recall that during construction, the total binomial measure of [0,1] remains constant and
equal to 1. But the first few stages of construction make its distribution become very unequal
and a few values that stand out as sharp spikes. After k stages, the maximum measure
is uk, which is far larger than the minimum measure vk. From the relations
2-k
= dt, 2k
= N, -log2 u = min < 1, and - log2 v = min > 1,
it follows that
uk
= b(-logb u)(-k)
= (dt)min = N-min .
In words: even the maximum uk tends to 0. This is a weak form of asymptotic negligibility
following a power-law.
The preceding result holds for every multifractal for which there is an min > 0 that
plays the same role as in the binomial case. (In more general multifractals the same role is
held by some 
min > max{min,0}.)
Similarly, the total contribution of any fixed number of largest spikes is asymptotically
negligible.
9.2. For the Bernoulli or canonical binomials, the equation f () =  has one and only
one solution; that solution satisfies D > 0 and is the fractal dimension of the
"carrier" of the measure
We now proceed to the total contribution of a number of spikes that is no longer fixed but
increases with N. In the simplest of all possible worlds, many spikes would have been more
or less equal to the largest, and the sum of all the other spikes would have been negligible.
If so, the sum of Nmin spikes would have been of the order of Nmin N-min = 1.
While the world is actually more complicated there is an element of orderliness. The
equality 0 = u is achieved for  = f () = -ulogu - v logv = D. For finite but large k,
it follows that
(k,0,1)  2-k
= 2-kD
and N(k10,1)  2kf ()
= 2kD
.
Hence,
(k10, 1)N(k101) is approximately equal to 1.
Actually, this product is necessarily 1 but the difference tends to 0 as k  . That
is, an increasingly overwhelming bulk of the measure tends to "concentrate" in the cells
where  = D. The remainder is small, but in the theory of multifractals even very small
remainders are extremely significant for some purposes.
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 29
9.3. The notion of "multifractal concentration"
A key feature of multifractals is a subtle interaction between number and size that is elaborated
upon in Mandelbrot (2001d). Section 9.2 showed that the contributions that are large
are too few to matter. The small contributions are very numerous, but so extremely small
that their total contribution is negligible as well. The bulk of the measure is found in a
rather inconspicuous intermediate range one can call "mass carrying". Since D > min, the
ND spikes of size N-D are far smaller than the largest one. Separately, each is asymptotically
negligible. But their number ND is exactly large enough to insure that their total
contribution is nearly equal to the overall measure 1. When a sample is plotted, this range
does not stand out but it makes a perfect match between size and frequency.
Practically, the number of visible peaks is so small compared to ND that a combination
of the peaks and the intermediate range is still of the order of ND. The combined range
has the advantage of simplicity, since it includes the ND largest values. Note that the peaks
tend to be located in the midst of stretches of values of intermediate size.
9.4. The case of TVCM with p < 1/2, allows D to be positive, negative, or zero
Using the alternative expression for f () given in Section 8.4, the identity f () =  demands
the equality of the two expressions
f () = -0 log2
0
p
- 1 log2
1
1 - p
and  = -0 log2 u - 1 log2 v.
The solution is, obviously, 0 = pu and 1 = (1 -p)v. The sum 1 +1 is 1, as it must.
Hence, D = -pulog2 u - (1 - p)v log2 v, as announced. The novelty is that TVCM allow
D > 0, D = 0, and D < 0.
Familiar role of D under the inequality D > 0. Mandelbrot (1974a, b) obtained the
following criterion, which has become widely known and includes the TVCM case. When
positive, D is the fractal dimension of the "set that supports" the measure. Figure 1 shows
isolines of D for D = 0,1/4,1/2, and 3/4. The isoline for D = 1 is made of the interval
{u = 1, 0 < v < 1} and the half-line {v = 1, u 1}. The key result is that, contrary to the
Bernoulli binomial case, the half line 1 < q <  subdivides into up to three subranges of
values.
Largely unfamiliar consequence of the inequality D < 0. For all non-random multifractals,
 (1) > 0. A casual acquaintance with multifractals takes for granted that this is not
changed by randomness. But Mandelbrot (1974a, b) also allows for an alternative possibility,
which has so far remained little known. The example of TVCM shows that, in a
canonical case, the formally evaluated D can be negative. In the example of TVCM, D is
negative when the point (u,v) falls in a domain to the bottom right of the folded phase
diagram in Figure 1. The consequences of D < 0 are drastic: the multifractal reduces to 0
almost surely and is called degenerate.
A classical "pathological limit" as metaphor. This limit behavior of the distribution of
 seems incompatible with the fact that E = 1 by definition. But in fact, no contradiction
30 B.B. Mandelbrot
is observed. A convincing idea of the distribution is provided for each p, by the behavior
of the g   limit of the weights ug2(g) and vg2(g). This recalls a classical counterexample
of analysis, namely, the behavior for k   of the variable Pk defined as follows:
Pk = k with the probability 1/k and Pk = 0 with the probability 1 - 1/k. For finite k, one
has EPk = 1. But in the limit k  , P = 0, hence EP = 0, so that in the limit the
expectation drops discontinuously from 1 to 0. In practice, the preasymptotic measure is
extremely small with a high probability and huge with a tiny probability.
The condition D = 0. It defines the threshold of degeneracy.
10. A noteworthy and unexpected separation of roles, between the "dimension
spectrum" and the total mass ; the former is ruled by the accessible  for
which f () > 0, the latter, by the inaccessible  for which f () < 0
Brought together, Sections 4, 7, 8, and 9 imply, in plain words, that what you do not necessarily
see may affect you significantly. This section serves to underline that the notion
of canonical multifractal is very subtle and deserves to be well-understood and further
discussed.
10.1. Definitions of the "accessible ranges" of the variables: qs from q
min to q
max and s
from 
min to 
max; the accessible functions (q) and f ()
Mandelbrot (1995) worked to introduce to the function f () = max{0,f ()}. That is,
* In the interval [
min,
max] where f () > 0, f () = f ();
* When f () 0, f () = 0.
The graph of f () is identical to that of f () except that the "tails" with f < 0 are
truncated so that f  > 0. In terms of (q), the equality f () = 0 corresponds to lines that
are tangent to the graph of (q) and also go through (0,0). In the most general case, those
lines' slopes are 
min and 
max and the points of contact are denoted by q
max (satisfying
>0) and q
min (satisfying <0). Therefore, the function f () corresponds to the following
truncated function (q).
* When q < q
min, (q) = 
maxq;
* When q
min < q < q
max, (q) = (q);
* When q > q
max, (q) = 
minq.
In other words, the graph of  is identical to that of  except that beyond q
max or q
min
it follows the tangents that go through the origins. Therefore it is straight.
For the TVCM, one has either 
max = max with q
min = -, or 
min = min with
q
max = .
10.2. A confrontation
Section 4 noted that the largest values of ([0,1]) are generated when a sample cascade
begins with a few large values. Section 7 noted that the value of ([0,1]) ­ irrespective of
Ch. 1: Heavy Tails in Finance for Independent or Multifractal Price Increments 31
size ­ ceases, for k  , to have any impact on . Section 8 noted that, again for k  ,
values of  such that f () < 0 have a vanishing probability of being observed. Section 9.1
followed up by defining the accessible function f (). Section 9 returned to large values of
([0,1]) and noted their association with qcrit < . The values of  they involve satisfy
 < 0, hence a fortiori f () < 0. Those values do not occur in multifractal decomposition,
yet they are extremely important.
10.3. The simplest cases where f () > 0 for all , as exemplified by the canonical
binomial
Here, the large values of  are ruled by the left-most part of the graph of f (). That is, the
same graph controls those large values and the distribution of ([0,1]) among the 1/dt
intervals of length dt.
10.4. The extreme case where f () < 0 and  < 0 both occur, as exemplified by TVCM
when u > 1
Due to the inequality f () < , the graph of f () never intersects the quadrant where
 < 0 and f > 0. The key unexpected fact is that the portions of f () within other quadrants
play more or less separate roles. In the TVCM case, those quadrants are parts of one
(analytically simple) function. But in general they are nearly independent of each other.
The function f () was defined as having a graph that lies in the non-anomalousquadrant
 > 0 and f > 0. This f determines completely the multifractal decomposition of our
TVCM measure, in particular, the dimension D and the exponents q
min, q
max, 
min and

max.
To the contrary, qcrit is entirely determined by the doubly anomalous left tail located in
the quadrant characterized by f () < 0 and  < 0. A priori, it was quite unexpected that
this quadrant should exist and play any role, least of all a central role, in the theory of
multifractals. But in fact, qcrit has a major effect on the distribution, hence the value of the
total measure in an interval.
10.5. The intermediate case where min > 0 but f () < 0 for some values of 
When p < 1/2, but u < 1 so that qcrit =  and all moments are finite, large values of
 have a much lower probability than when u > 1. As always, however, their probability
distribution continues to be determined by the left tail of the probability graph where f < 0.
11. A broad form of the multifractal formalism that allows  < 0 and f () < 0
The collection of rules that relate (q) to f () is called "multifractal formalism". TVCM
was specifically designed to understand multifractals directly, thus avoiding all formalism.
32 B.B. Mandelbrot
However, general random multifractals more than TVCM demand their own broad multifractal
formalism. Once again, the most widely known form of the multifractal formalism
does not allow randomness and yields f () > 0, but the broad formalism first introduced
in Mandelbrot (1974a, b) concerns a generalized function for which f () < 0 is allowed.
11.1. The broad "multifractal formalism" confirms the form of f () and allows
f () < 0 for some 
Through a point on the graph of coordinates q and (q), draw the tangent to that graph.
Under wide conditions, the tangenťs slope is (q) and its intercept by the ordinate axis is
-f (q). Thus
(q) =
d(q)
dq
and - f (q) = (q) - q
d(q)
dq
.
Through the quantities (q) and f (q), a function f () is defined by using q as parame-
ter.
The slope f () is the inverse of the function (q). The tangent of slope f () intersects
the line  = 0 at the point of ordinate -(q). The D(q) tangenťs equation being
-(q) + q, its intersection with the bisector satisfies the condition - + q = , hence
D = (q)/(q - 1). This is the critical embedding dimension discussed in Section 5.4.
11.2. The Legendre and inverse Legendre transforms and the thermodynamical analogy
The transforms that replace q and (q) by  and f (), or conversely, are due to Legendre.
They play a central role in thermodynamics, as does already the argument that yielded
f () and () in the original formalism introduced in Mandelbrot (1974a, b).
Acknowledgments
The figures in this chapter were prepared by Evgenyi Vilenkin, Yale class of 2003.
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Mandelbrot, B.B., Taylor, H.M., 1967. On the distribution of stock price differences. Operations Research 15,
1057­1062.
Obukhov, A.M., 1962. Some specific features of atmospheric turbulence. Journal of Fluid Mechanics 13, 77­81.
Telser, L. (Ed.), 2000. Classic Futures: Lessons from the Past for the Electronic Age. Risk Books, London.
Chapter 2
FINANCIAL RISK AND HEAVY TAILS
BRENDAN O. BRADLEY and MURAD S. TAQQU
Department of Mathematics and Statistics, Boston University, 111 Cummington Street, Boston, MA 02215, USA
e-mail: bbradley@bu.edu, murad@bu.edu
web: http://math.bu.edu/people/murad
Contents
Abstract 36
1. Introduction 37
2. Historical perspective 38
2.1. Risk and utility 38
2.2. Markowitz mean­variance portfolio theory 39
2.3. CAPM and APT 40
2.4. Empirical evidence 43
3. Value at risk 45
3.1. Computation of VaR 47
3.1.1. Historical simulation VaR 47
3.1.2. Parametric VaR 48
3.1.3. Monte Carlo VaR 50
3.2. Parameter estimation 51
3.2.1. Historical volatility 51
3.2.2. ARCH/GARCH volatilities 53
3.2.3. Implied volatilities 57
3.2.4. Extreme value theory 58
4. Risk measures 58
4.1. Coherent risk measures 58
4.2. Expected shortfall 59
5. Portfolios and dependence 61
5.1. Copulas 61
5.2. Measures of dependence 66
5.2.1. Linear correlation 67
5.2.2. Rank correlation 67
5.2.3. Comonotonicity 68
5.2.4. Tail dependence 70
5.3. Elliptical distributions 72
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
36 B.O. Bradley and M.S. Taqqu
6. Univariate extreme value theory 77
6.1. Limit law for maxima 78
6.2. Block maxima method 80
6.3. Using the block maxima method for stress testing 82
6.4. Peaks over threshold method 83
6.4.1. Semiparametric approach 83
6.4.2. Fully parametric approach 85
6.4.3. Numerical illustration 89
6.4.4. A GARCH-EVT model for risk 91
7. Stable Paretian models 94
7.1. Stable portfolio theory 95
7.2. Stable asset pricing 98
Acknowledgments 100
References 101
Abstract
It is of great importance for those in charge of managing risk to understand how financial
asset returns are distributed. Practitioners often assume for convenience that the distribution
is normal. Since the 1960s, however, empirical evidence has led many to reject this
assumption in favor of various heavy-tailed alternatives. In a heavy-tailed distribution the
likelihood that one encounters significant deviations from the mean is much greater than
in the case of the normal distribution. It is now commonly accepted that financial asset
returns are, in fact, heavy-tailed. The goal of this survey is to examine how these heavy
tails affect several aspects of financial portfolio theory and risk management. We describe
some of the methods that one can use to deal with heavy tails and we illustrate them using
the NASDAQ composite index.
Ch. 2: Financial Risk and Heavy Tails 37
1. Introduction
Financial theory has long recognized the interaction of risk and reward. The seminal work
of Markowitz (1952) made explicit the trade-off of risk and reward in the context of a portfolio
of financial assets. Others such as Sharpe (1964), Lintner (1965), and Ross (1976),
have used equilibrium arguments to develop asset pricing models such as the capital asset
pricing model (CAPM) and the arbitrage pricing theory (APT), relating the expected return
of an asset to other risk factors. A common theme of these models is the assumption of normally
distributed returns. Even the classic Black and Scholes option pricing theory (Black
and Scholes, 1973) assumes that the return distribution of the underlying asset is normal.
The problem with these models is that they do not always comport with the empirical evidence.
Financial asset returns often possess distributions with tails heavier than those of
the normal distribution. As early as 1963, Mandelbrot (1963) recognized the heavy-tailed,
highly peaked nature of certain financial time series. Since that time many models have
been proposed to model heavy-tailed returns of financial assets.
The implication that returns of financial assets have a heavy-tailed distribution may be
profound to a risk manager in a financial institution. For example, 3 events may occur
with a much larger probability when the return distribution is heavy-tailed than when it
is normal. Quantile based measures of risk, such as value at risk, may also be drastically
different if calculated for a heavy-tailed distribution. This is especially true for the highest
quantiles of the distribution associated with very rare but very damaging adverse market
movements.
This chapter serves as a review of the literature. In Section 2, we examine financial
risk from an historical perspective. We review risk in the context of the mean­variance
portfolio theory, CAPM and the APT, and briefly discuss the validity of their assumption
of normality. Section 3 introduces the popular risk measure called value at risk (VaR).
The computation of VaR often involves estimating a scale parameter of a distribution. This
scale parameter is usually the volatility of the underlying asset. It is sometimes regarded as
constant, but it can also be made to depend on the previous observations as in the popular
class of ARCH/GARCH models.
In Section 4, we discuss the validity of several risk measures by reviewing a proposed
set of properties suggested by Artzner, Delbean, Eber and Heath (1999) that any sensible
risk measure should satisfy. Measures satisfying these properties are said to be coherent.
The popular measure VaR is, in general, not coherent, but the expected shortfall measure
is. The expected shortfall, in addition to being coherent, gives information on the expected
size of a large loss. Such information is of great interest to the risk manager.
In Section 5, we return to risk, portfolios and dependence. Copulas are introduced as a
tool for specifying the dependence structure of a multivariate distribution separately from
the univariate marginal distributions. Different measures of dependence are discussed including
rank correlations and tail dependence. Since the use of linear correlation in finance
is ubiquitous, we introduce the class of elliptical distributions. Linear correlation is shown
to be the canonical measure of dependence for this class of multivariate distributions and
the standard tools of risk management and portfolio theory apply.
38 B.O. Bradley and M.S. Taqqu
Since the risk manager is concerned with extreme market movements we introduce extreme
value theory (EVT) in Section 6. We review the fundamentals of EVT and argue
that it shows great promise in quantifying risk associated with heavy-tailed distributions.
Lastly, in Section 7, we examine the use of stable distributions in finance. We reformulate
the mean­variance portfolio theory of Markowitz and the CAPM in the context of the
multivariate stable distribution.
2. Historical perspective
2.1. Risk and utility
Perhaps the most cherished tenet of modern day financial theory is the trade-off between
risk and return. This, however, was not always the case, as Bernstein's (1996) narrative
on risk indicates. In fact, investment decisions used to be based primarily on expected return.
The higher the expected return, the better the investment. Risk considerations were
involved in the investment decision process, but only in a qualitative way, stocks are more
risky than bonds, for example. Thus any investor considering only the expected payoff EX
of a game (investment) would, in practice, be willing to pay a fee equal to EX for the right
to play.
The practice of basing investment decisions solely on expected return is problematic,
however. Consider the game known today as the Saint Petersburg Paradox, introduced in
1728 by Nicholas Bernoulli. The game involves flipping a fair coin and receiving a payoff
of 2n-1 roubles1 if the first head appears on the nth toss of the coin. The longer tails
appears, the larger the payoff. While in this game the expected payoff is infinite, no one
would be willing to wager an infinite sum to play, hence the paradox. Investment decisions
cannot be made on the basis of expected return alone.
Daniel Bernoulli, Nicholas' cousin, proposed a solution to the paradox ten years later.
He believed that, instead of trying to maximize their expected wealth, investors want to
maximize their expected utility of wealth. The notion of utility is now widespread in economics.2
A utility function U :R  R indicates how desirable is a quantity of wealth W.
One generally agrees that the utility function U should have the following properties:
(1) U is continuous and differentiable over some domain D.
(2) U (W) > 0 for all W  D, meaning investors prefer more wealth to less.
(3) U (W) < 0 for all W  D, meaning investors are risk averse. Each additional dollar of
wealth adds less to the investors utility when wealth is large than when wealth is small.
In other words, U is smooth and concave over D. An investor can use his utility function
to express his level of risk aversion.
1 In fact, it was ducats (Bernstein, 1996).
2 For introductions to utility theory see for example Ingersoll (1987) or Huang and Litzenberger (1988).
Ch. 2: Financial Risk and Heavy Tails 39
2.2. Markowitz mean­variance portfolio theory
In 1952, while a graduate student at the University of Chicago, Harry Markowitz (1952)
produced his seminal work on portfolio theory connecting risk and reward. He defined
the reward of the portfolio as the expected return and the risk as its standard deviation
or variance.3 Since the expectation operator is linear, the portfolio's expected return is
simply given by the weighted sum of the individual assets' expected returns. The variance
operator, however, is not linear. This means that the risk of a portfolio, as measured by the
variance, is not equal to the weighted sum of risks of the individual assets. This provides a
way to quantify the benefits of diversification.
We briefly describe Markowitz' theory in its classical setting where we assume that the
assets distribution is multivariate normal. We will relax this assumption in the sequel. For
example, in Section 5.3, we will suppose that the distribution is elliptical and, in Section
7.1, that it is an infinite variance stable distribution.
Consider a universe with n risky assets with random rates of return X = (X1,...,Xn),
with mean  = (1,...,n), covariance matrix  and portfolio weights w = (w1,...,
wn). If X is assumed to have a multivariate normal distribution X  N(,), then the return
distribution of the portfolio Xp = wTX is also normally distributed, Xp  N(p,2
p)
where p = wT and 2
p = wTw. The problem is to find the portfolio of minimum variance
that achieves a minimum level a of expected return:
min
w
wT
w
such that wT
 a, (1)
eT
w = 1.
Here e = (1,...,1) and T denotes a transpose. The last condition in (1),
eT
w =
n
i=1
wi = 1,
indicates that the portfolio is fully invested. Additional restrictions are usually added on the
weights4 and the problem is generally solved through quadratic programming. By varying
the minimum level a of expected return, a set of portfolios Xp is chosen, each of which is
optimal in the sense that an investor cannot achieve a greater expected return, p = EXp,
without increasing his risk, p. The set of optimal portfolios corresponds to a convex curve
(p,EXp) called the efficient frontier. Any rational investor making decisions based only
on the mean and variance of the distribution of returns of a portfolio would only choose
3 In practice, one minimizes the variance, but it is convenient to view risk as measured by the standard deviation.
4 For example, wi 0, in other words no short selling. Without the additional constraints, the problem can be
solved as a system of linear equations.
40 B.O. Bradley and M.S. Taqqu
Fig. 1. The efficient frontier (p,p). In the case when only risky assets R are available, the frontier traces out
a convex curve in risk-return space. The inclusion of a risk-free asset r, has a profound effect on the efficient set.
In this case, all efficient portfolios will consist of linear combinations of r and some risky portfolio R, where
(R,R) lies on the efficient frontier.
to own portfolios on this efficient frontier. The specific portfolio he chooses depends on
his level of risk aversion.5 If the universe of assets also includes a risk-free asset which
the investor may borrow and lend without constraint, then the optimal portfolio is a linear
combination of the risk-free asset r and a certain risky portfolio XR on the efficient frontier.
As shown in Figure 1, this line is tangent to the convex risky asset efficient frontier
at the point (R,EXR). The risky portfolio therefore maximizes the slope of this linear
combination,
max
w
E(XR) - r
XR
. (2)
Again, the specific weights given to the risk-free and risky assets depend on the individual
investors level of risk aversion.
2.3. CAPM and APT
The mean­variance portfolio theory of Markowitz describes the construction of an optimal
portfolio, in the mean­variance sense, for an individual investor. It requires only estimates
5 One can reconcile maximizing expected utility with the mean­variance portfolio theory of Markowitz, but
one has to assume either a quadratic utility function or that returns are multivariate normal or, more generally,
elliptical. (Elliptical distributions are introduced in Section 5.3.) For example, if returns are multivariate normal
and if Xp1 and Xp2 are the returns of two linear portfolios with the same expected return, then for all utility
functions U with properties listed in Section 2.1,
EU(Xp1 ) EU(Xp2 ) if and only if 2
p1
2
p2
.
See for example Ingersoll (1987).
Ch. 2: Financial Risk and Heavy Tails 41
for each asset mean return, and the covariance between assets.6 If all investors act in a way
consistent with Markowitz' theory, then under additional assumptions, one will be able to
learn something about the trade-off between risk and return in a market in equilibrium.7
This is what the CAPM does.
The capital asset pricing model (CAPM) is an equilibrium pricing model [see Sharpe
(1964) and Lintner (1965)] which relates the expected return of an asset to the risk-free
return, to the markeťs expected return and to the covariance between the market and the
asset. In addition to assuming that market participants use the mean­variance framework,
the model makes two additional major assumptions. First, the market is assumed frictionless.
This means that securities are infinitely divisible, there exist no transaction costs, no
taxes, and there are no trading restrictions. Second, the investors beliefs are homogeneous.
This means investors agree on mean returns and covariances for all assets in the market.
The efficient frontier in Figure 1 depended on the investors' belief. Under the CAPM
assumptions, since all investors assume the same expected return and covariances for all
assets in the market, they all have the same (risky) efficient frontier. However, the individual
investors choice of the optimal risky portfolio still depends on the investors own
level of risk aversion. Additionally, with the inclusion of a risk-free asset, we saw that the
investors portfolios become dramatically more simple. Each investor can own only two assets:
the risk-free asset and an optimal risky portfolio, with the relative weights depending
on the investors appetite for risk. But since each investor holds the same optimal portfolio
of risky assets, and since the market is assumed to be in equilibrium, this optimal risky
portfolio must be the market portfolio. Thus Figure 1 applies with R = M, where M denotes
the market portfolio. M consists of all risky assets held in proportion to their overall
market capitalization. Letting XM denote the return on the market portfolio, Xi denote the
return of asset i, and r denote the risk-free return, the CAPM establishes the following
relationship:
E(Xi - r) = iE(XM - r), (3)
where
i =
Cov(Xi,XM)
VarXM
. (4)
The CAPM thus relates in a linear way the expected premium EXi -r of holding the risky
asset i over the risk-free asset to the expected premium EXM - r of holding the market
portfolio over the risk-free asset. The constant of proportionality is the asseťs beta. The
coefficient i is a measure of asset i's sensitivity to the market portfolio. The expected
6 For a universe of n assets it is necessary to compute n(n-1)/2+n covariances. This means that if the universe
under consideration consists of n = 1000 assets, it is necessary to estimate over 500000 covariances.
7 By market equilibrium, we mean a market place where security prices are set so that supply equals demand.
42 B.O. Bradley and M.S. Taqqu
premium for asset i is greater than that of the market if i > 1 and less if i < 1. But if
i > 1, then the risk will be greater. Indeed, if we assume that
Xi - r = i(XM - r) + i, (5)
where i is such that Ei = 0 and Cov(i,XM ) = 0, then we have (3) and
2
Xi
= 2
i 2
XM
+ 2
i
. (6)
Equation (5) is often known as a single factor model for asset returns. Notice from (6)
that the asseťs risk is the sum of two terms, the systematic or market risk 2
i 2
XM
and the
unsystematic or residual risk 2
i
. For a portfolio Xp with weights w = (w1,...,wn), one
gets similarly 2
Xp
= 2
p2
XM
+ 2
p
where p = n
i=1 wii. If one additionally assumes
that Cov(i,j ) = 0 for all i = j then the residual risk is
2
p
=
n
i=1
w2
i 2
i
. (7)
It is bounded by c/n for some constant c, if for example, wi = 1/n, and hence the portfolio's
residual risk can be greatly reduced by diversification. The investor, for example, is
only rewarded for bearing systematic or market risk, that is, he can expect a higher return
than the market only by holding a portfolio which is riskier (p > 1) than the market.
In the CAPM, all assets are exposed to a single common source of randomness, namely
the market. The arbitrage pricing theory (APT) model, due to Ross (1976), is a generalization
of the CAPM in which assets are exposed to a larger number of common sources of
randomness. The APT differs from the CAPM in that the mean­variance framework that
led to (5) is now replaced by the assumption of a multifactor model
Xi = i + i1f1 +  + ikfk + i (8)
for generating security returns. All assets are exposed to the k sources of randomness
fj , j = 1,...,k, called factors. Additionally, each asset i is exposed to its own specific
source of randomness i. The equilibrium argument used in the CAPM led to the central
result (3). In the APT, the equilibrium assumption takes a slightly different form, namely,
one assumes that the market is free of arbitrage. The major result of the APT then relates
the expected premium of asset i to its exposure ij to factor j, and to each factor premium
j , j = 1,...,k. Specifically
EXi = r + i11 +  + ikk, (9)
where j , j = 1,...,k, is the expected premium investors demand for bearing the risk of
factor j. Notice that the factor premiums j are the same for each security, and it is the
Ch. 2: Financial Risk and Heavy Tails 43
Fig. 2. Left: Empirical probability density function (pdf) for NASDAQ standardized returns (solid) versus the
normal distribution (dot­dash) over the period February 1971 to February 2001. Right: Corresponding quantile­quantile
(QQ) plot with quantiles of the normal distribution on the abscissa and empirical quantiles on the
ordinate. Returns are expressed as a %.
exposure ij to each factor that depends on the security. Additionally if k = 1 in (8) and if
we assume the existence of a risk-free asset r, f1 = XM and that i are uncorrelated with
each other and the market, then 1 = E(XM - r) and we get back the CAPM.
2.4. Empirical evidence
Markowitz's mean­variance portfolio theory, as well as the CAPM and APT models, rely
either explicitly or implicitly on the assumption of normally distributed asset returns.8
Today, with long histories of price/return data available for a great many financial assets,
it is easy to see that this assumption is inadequate. Empirical evidence suggests that asset
returns have distributions which are heavier-tailed than the normal distribution. Figure 2
illustrates this for the NASDAQ.9 The quantile­quantile (QQ) plot10 shows clearly that the
distribution tails of the NASDAQ are heavier than the tails of the normal distribution. As
early as 1963, Mandelbrot (1963) and Fama (1965) rejected the assumption of normality
for other heavier-tailed distributions. In his 1963 paper, Mandelbrot not only confirmed the
poor fit of the normal distribution, but proposed the model which is known today as the
stable model for asset returns.
8 As noted before, the multivariate normal assumption is consistent with maximizing expected utility.
9 The daily NASDAQ time series, the corresponding returns and their maxima and minima are displayed in
Figure 16. The time series starts in February 1971 and ends February 2001 (actually from February 08, 1971 to
January 26, 2001). The corresponding empirical statistics can be found in Table 1.
10 A quantile­quantile (QQ) plot is a graphical check to see if two distributions are of the same type. Two random
variables X and Y are said to be of the same type if their distributions are the same up to a change in location and
scale. That is X
d
= aY + b for some a  R+, b  R. Since the QQ plot plots quantiles of two distributions, if they
are of the same type, the plot should be linear. In this case we are checking whether the empirical distribution of
NASDAQ standardized returns and the hypothesized normal distribution are of the same type.
44 B.O. Bradley and M.S. Taqqu
Table 1
Empirical statistics for daily returns (as %) of several financial assets: the S&P 500 index, the USD/British pound
exchange rate, the Thai Baht/USD exchange rate and the NASDAQ composite index
Asset Period Mean Std. dev. Skewness Kurtosis11 Min Max
S&P 500 01/51-03/2001 0.033 0.870 -1.61 43.9 -22.9 8.71
USD/GBP 02/1985-02/2001 0.006 0.677 0.043 3.40 -4.13 4.59
TB/USD 02/85-03/2001 0.011 0.663 4.22 158 -8.57 17.8
NASDAQ 02/1971-02/2001 0.044 1.08 -0.523 15.5 -12.0 13.3
Fig. 3. Ratio of tail probabilities P(T > k)/P(X > k) plotted in units of k. Here T  t4 and X is normal, both
with variance 2. T is more likely to take large values than X.
Recall that if the normal distribution is valid, then about 95% of the observations would
lie within two standard deviations of the mean, and about 99% would lie within three
standard deviations of the mean. In financial time series, large returns (both positive and
negative) occur far too often to be compatible with the normal distribution assumption.
The distribution of the financial return series are characterized not only by heavy tails, but
also by a high peakedness at the center. In the Econometric terminology, they are said to
be leptokurtotic.
To the risk manager trying to guard against large losses, the deviation from normality
cannot be neglected. Suppose for example that daily returns are distributed as a stable
distribution with 4 degrees of freedom (denoted t4) and a variance given by 2. Since this
distribution has a much heavier tail than a normal distribution with the same variance,
as one moves farther out into the tail of the distribution, rare events occur much more
frequently. Figure 3 shows how much more likely rare events occur under the t4 assumption
than under the normal, when rare is defined in terms of standard deviations.
11 In this chapter we use as definition of kurtosis
K(X) =
E(X - X)4
(VarX)2
- 3,
so that the normal distribution has a kurtosis of zero. Heavy tails, therefore, will lead to positive kurtosis.
Ch. 2: Financial Risk and Heavy Tails 45
3. Value at risk
In the early 1990s, a number of financial institutions (J.P. Morgan, Bankers Trust, ...)
proposed a new risk measure to quantify by a single number the firms aggregate exposure
to market risk. This measure, commonly known today as value at risk (VaR), is now used to
measure not only market risk but other forms of risk to which the firm is exposed, such as
credit, operational, liquidity, and legal risk. VaR is defined as the loss of a financial position
over a time horizon  that would be exceeded with small probability 1 - , that is,
P(Loss > VaR) 1 - . (10)
The confidence level  is typically a large number12 between 0.95 and 1.
To define VaR precisely, let X be the random variable whose cumulative distribution
function FX describes the negative profit and loss distribution (P&L) of the risky financial
position at the specified horizon time . Negative values of X correspond now to profits and
positive values of X correspond to losses. This is a useful convention in risk management
since there is then no ambiguity when discussing large losses (large values of X correspond
to large losses).
Formally, value at risk is a quantile of the probability distribution FX, that is roughly,
the x corresponding to a given value of 0 <  = FX(x) < 1.
Definition 3.1. Let X be the random variable whose cumulative distribution function FX
describes the negative profit and loss distribution (P&L) of the risky financial position
at the specified horizon time  (so that losses are positive). Then, for a confidence level
0 <  < 1,
VaR(X) = inf x | FX(x)  . (11)
We set, avoiding technicalities
VaR(X) = F-1
X (),
where F-1
X denotes the inverse function of FX
13 (see Figure 4). Hence the value VaR(X)
over the horizon time  would be exceeded on the average 100(1 - ) times every 100
time periods.
12 In statistics,  and 1 -  are usually interchanged because , in statistics, denotes typically the Type 1 hypothesis
testing error and is chosen small. The corresponding confidence level is then 1 - .
13 This is strictly correct when FX is strictly increasing and continuous. Otherwise, one needs to use the generalized
inverse of FX, denoted F
X , and defined as
F
X () = inf{x | FX(x) }, 0 <  < 1.
The definition (11) of VaR(X) is then VaR(X) = F
X (). Thus, if FX(x) =  for x0 x x1, then
VaR(X) = F
X () = x0.
46 B.O. Bradley and M.S. Taqqu
Fig. 4. VaR(X) for different cumulative distributions functions (cdfs) of the loss distribution X. The cdf on the
right corresponds to an asset with discontinuous payoff, for example a binary option. See Definition 3.1.
Because of its intuitive appeal and simplicity, it is no surprise that VaR has become the
de facto standard risk measure used around the world today. For example, today VaR is
frequently used by regulators to determine minimum capital adequacy requirements. In
1995, the Basle Committee on Banking Supervision14 suggested that banks be allowed
to use their own internal VaR models for the purpose of determining minimum capital
reserves. The internal models approach of the Basle Committee is a ten day VaR at the
 = 99% confidence level multiplied by a safety factor of at least 3. Thus if VaR = 1M,
the institution is required to have at least 3M in reserve in a safe account.
The safety factor of three is an effort by regulators to ensure the solvency of their institutions.
It has also been argued, see Stahl (1997) or Danielsson et al. (1998), that the safety
factor of three comes from the heavy-tailed nature of the return distribution. Since most
VaR calculations are based on the simplifying assumption that the distribution of returns
are normal,15 how bad does this assumption effect VaR? Assume that the Profit and Loss
(P&L) distribution is symmetric and has finite variance 2. Then regardless of the actual
distribution, if X represents the random loss over the specified horizon time with mean
zero, Chebyshev's inequality gives
P[X > c]
1
2c2
.
So if we are interested in VaR bounds for  = 0.99, setting 1/2c2 = 0.01 gives c = 7.071,
and this implies VaRmax
=0.99(X) = 7.071. If the VaR calculation were done under the assumption
of normality (Gaussian distribution) then VaRGa
=0.99(X) = 2.326, and so if
the true distribution is indeed heavy-tailed with finite variance then the correction for
VaR=0.99 of three is reasonable, since 3 × 2.326 = 6.978.
14 See Basle Committee on Banking Supervision (1995a, 1995b). Basle is a city in Switzerland. In French, Basle
is Bâle, in German, it is Basel. Basle is the old name for the city. The accent in Bâle stands for the s that has been
dropped from Basle.
15 See for example the RiskMetrics manual (RiskMetrics, 1996).
Ch. 2: Financial Risk and Heavy Tails 47
3.1. Computation of VaR
Before we discuss how VaR(X) is computed, we need to say a few words about X. Typically
X represents the risk of some aggregated position which is influenced by many
underlying risk factors Y1,...,Yd ,
X = f (Y1,...,Yd). (12)
The functional form of the dependence of X on the factors Y1,...,Yd is usually never
known exactly, but it may be approximated in several standard ways depending on the
nature of the position. For example, f is linear in the case of a portfolio of straight equity
positions. The function f is non-linear, for example, if the portfolio contains a call option
on an equity since the value of the call changes non-linearly with respect to a change in
the underlying asset. The usual procedure is to approximate the change in the calls value
with respect to its underlying by the options delta. For small changes in the underlying
such an approximation is reasonable. However for large changes in the underlying, the
approximation can be quite bad. In an effort to improve the approximation, a second order
term is sometimes added, the options gamma. This second order approximation is referred
to as the delta­gamma approximation.
In practice, the VaR of a risky position X is calculated in one of three ways: through
historical simulation, through a parametric model, or through some sort of Monte Carlo
simulation. Each way involves assumptions and approximations and it is the responsibility
of the user to be aware of them. The risk manager who blindly performs the model calculations
does so at his or her peril. For a full treatment of the commonly used procedures for
the calculation of VaR, see Jorion (2001), Dowd (1998) or Wilson (1998). See Duffie and
Pan (1997) for a discussion of heavy tails and VaR calculations. We now describe the three
ways of calculating VaR.
3.1.1. Historical simulation VaR
The historical simulation model uses the historical returns of assets currently held in the
portfolio in order to calculate VaR.16 First, returns over the horizon time  are constructed
for each asset in the portfolio using historical price information. Then portfolio returns
are computed using the current weight distribution of assets as though the portfolio had
been held during the whole historical period which is being sampled. The VaR is then
read from the historical sample by using the order statistics. For example, if 1000 time
periods are sampled, then 1000 portfolio returns are calculated, one for each time period.
Let X(1)
p X(2)
p  X(1000)
p be the order statistics of these returns, where losses are
positive. Then VaR=0.95(Xp) = X(50)
p . The size of the sample is chosen by the user, but
may be constrained by the available data for some of the assets currently held.
16 Over a fixed time horizon, VaR may be reported in units of rate of return (%) or of currency (profit and loss)
since these are essentially the same, up to multiplication by the initial wealth/value.
48 B.O. Bradley and M.S. Taqqu
The model is simple to implement and has several advantages. Since it is based on historical
prices it allows for a non-linear dependence between assets in the portfolio and underlying
risk factors. Also since it uses historical returns it allows for the presence of heavy
tails without making assumptions on the probability distributions of returns of the assets in
the portfolio. There is therefore no model risk. In addition, there is no need to worry about
the dependence structure of assets within the portfolio since it is already reflected in the
price and return data.
The drawbacks are typical of models involving historical data. There may not be enough
data available and there may be no reason to believe that the future will look like the
past. For example, if the user would like to compute VaR for regulatory requirements, then
 = 10 days. With about 260 business days, there are only 26 such observations in each
year, four years worth of data are required to get about 100 historical simulations. This
is the absolute minimum necessary to calculate VaR with  = 0.99, since with 100 data
points, there is but a single observation in the tail. If one or several of the assets in the
portfolio have insufficient histories then adjustments must be made. For example, some
practitioners bootstrap from the shorter return histories in order to take advantage of the
longer histories on other assets.
When working only with historical data it is important to realize that we are assuming
that the future will look like the past. If this assumption is likely to be unrealistic, the VaR
estimate may be dangerously off the mark. For instance, if the sample period or window
is devoid of large price changes, then our historical VaR will be low. But it will be large
if there were large price fluctuations during the sample period. As large price fluctuations
leave the sample window, the VaR will change accordingly. This yields a highly variable
estimate and one which does not take into account the current financial climate. The deficiencies
of historical simulation notwithstanding, its ease of use makes it the most popular
method for VaR calculations.
3.1.2. Parametric VaR
The parametric VaR model assumes that the returns possess a specific distribution, usually
normal. The parameters of the distribution are estimated using either historical data or
forward looking option data.
Example 3.1. Assume that over the desired time horizon  the (negative) return distribution
of a portfolio is given by FX  N( ,2
 ). Then the value at risk of portfolio X for
horizon  and confidence level  > 0.5 is given by
VaR(X) = inf x | FX(x)  = F-1
X () =  +  -1
(),
where -1() is the  quantile of the standard normal distribution.
More generally, if the (negative) return distribution of X is any FX with finite mean 
and finite variance 2
 , then
VaR(X) =  +  q, (13)
Ch. 2: Financial Risk and Heavy Tails 49
where q is the  quantile of the standardized version of X. In other words, q = F-1
X
()
where X = (X -  )/ .
If the VaR is computed under the assumption that returns are light-tailed, say normal,
when in fact they are heavy tailed, say t (Student-t distribution with  degrees of freedom),
the risk may be seriously underestimated for high confidence levels. This is because
for large , F-1
normal() F-1
t
(), so that the value of x that achieves Fnormal(x) =  is
smaller than the value of x that achieves Ft (x) = . It is thus very important that the
return distribution be modelled well. A wide variety of parametric distributions can be
considered.
Within the portfolio context, the most easily implemented parametric model is the so
called delta-normal method, where the joint distribution of the risk factor returns is multivariate
normal and the returns of the portfolio are assumed to be a linear function of
the returns of the underlying risk factors. In this case the portfolio returns are themselves
normally distributed.
Example 3.2. Take a portfolio of equities whose (negative) returns are given by Xp =
w1X1 +  + wnXn where wi is the weight given to asset i and Xi is the assets (negative)
return over the horizon in question. Assume (X1,...,Xn)  N(0,). Then, for
  (0.5,1),
VaR(Xp) = -1
() wTw =
--
VaR
T
--
VaR,
where
--
VaR = (VaR(w1X1),...,VaR(wnXn)) is the vector of the individual weighted
asset VaRs and  is the asset return correlation matrix. See Dowd (1998) for details.
When the number of assets is large, the central limit theorem is often invoked in defense
of the normal model. Even if the individual asset returns are non-normal, the central limit
theorem tells us that the weighted sum of many assets should be approximately normal.
This argument may be disposed of in various ways. Consider, for example, the empirical
distribution of daily returns of a large diversified index such as the NASDAQ, which is
clearly heavy-tailed (see Figure 2). From a probabilistic point of view it is not at all obvious
that the assumptions of the central limit theorem are satisfied. For example, if the returns
do not have finite variance, there may be convergence to the class of stable distributions.
The class of stable distributions (also known as -stable or stable Paretian) may be
defined in a variety of ways. More will be said about them in Section 7. We define, at
this stage, a stable distribution as the only possible limiting distribution of appropriately
normalized sums of independent random variables.
Definition 3.2. The random variable X has a stable distribution if there exists a sequences
of i.i.d. random variables {Yi} and constants {an}  R and {bn}  R+ such that
Y1 +  + Yn
bn
- an
d
- X as n  . (14)
50 B.O. Bradley and M.S. Taqqu
The stable distribution of X in (14) is characterized by four parameters (,,,) and
we write X  S(,,). The parameter   (0,2] is called the index of stability or the tail
exponent and controls the decay in the tails of the distribution. The remaining parameters
, ,  control scale, skewness, and location respectively. If the Yi have finite variance (the
case in the usual CLT) then  = 2 and the distribution of X is Gaussian. For all   (0,2)
the distribution is non-Gaussian stable and possess heavy tails.
Example 3.3. Properties of weekly returns of the Nikkei 225 Index over a 12 year period
are examined in Mittnik, Rachev and Paolella (1998). The authors fit the return distribution
using a number of parametric distributions, including the normal, Student-t and stable. According
to various measures of goodness of fit, the partially asymmetric Weibull, Student-t
and the asymmetric stable provide the best fit. The fit by the normal is shown to be relatively
poor. The stable distribution, in addition, fits best the tail quantiles of the empirical
distribution, which is a result most relevant to the calculation of VaR.
The central limit theorem typically assumes independence. Although it has extensions
to allow for mild dependence, this dependence must be sufficiently weak. In fact, for a
given number of assets, the greater the dependence, the worse the normal approximation.
This affects the speed of the convergence. Since a VaR calculation involves the tails of the
distribution, it is most important that the approximation hold in the tails. However, even
when the conditions for the central limit theorem hold, the convergence in the tail is known
to be very slow. The normal approximation may then only be valid in the central part of the
distribution. In this case, the return distribution may be better approximated by a heaviertailed
distribution such as the Student-t or hyperbolic whose use in finance is becoming
more common.
The hyperbolic distribution is a subclass of the class of generalized hyperbolic distributions.
The generalized hyperbolic distributions were introduced in 1977 by BarndorffNielsen
(1977) in order to explain empirical findings in geology. Today these distributions
are becoming popular in finance, and in particular in risk management. Two subclasses, the
hyperbolic and the inverse Gaussian, are most commonly used. Both these subclasses may
be shown to be mixtures of Gaussians. As such, they possess heavier tails than the normal
distribution but not as heavy as the stable distribution. For an introduction to generalized
hyperbolic distributions in finance, see for example Eberlein and Keller (1995), Eberlein
and Prause (2002) or Shiryaev (1999).
3.1.3. Monte Carlo VaR
Monte Carlo procedures are perhaps the most flexible methods for computing VaR. The risk
manager specifies a model for the underlying risk factors, which incorporates somehow
their dependence. For example, the risk factors in (12) may be described by the stochastic
differential equation
dY
(i)
t = Y
(i)
t 
(i)
t dt + 
(i)
t dW
(i)
t , (15)
Ch. 2: Financial Risk and Heavy Tails 51
for i = 1,...,d, where Wt = (W(1)
t ,...,W(d)
t ) is a multivariate Wiener process. Once parameters
of the model are estimated, for example by using historical data, or option implied
estimates, the risk factors paths are then computer generated, thousands of paths for each
risk factor. Each set of simulated paths for the risk factors yields a portfolio path and the
portfolio is priced accordingly. Each computed price of the portfolio represents a point on
the portfolio's return distribution. After many such points are obtained the portfolio's VaR
may then be read off the simulated distribution.
This method has the advantage of being extremely versatile. It allows for heavy tails,
non-linear payoffs and a great many other user specifications. Within the Monte Carlo
framework, risk managers may use their own pricing models to determine non-linear payoffs
under many different scenarios for the underlying risk factors. The method has also
the advantage of allowing for time varying parameters within the risk factor processes. See
for example Broadie and Glasserman (1998).
There are two major drawbacks to Monte Carlo methods. First, they are computationally
very expensive. Thousands of simulations of the risk factors may have to be carried out for
results to be trusted. For a portfolio with a large number of assets this procedure may
quickly become unmanageable, since each asset within the portfolio must be valued using
these simulations. Second, the method is prone to model risk. The risk factors and the
pricing models of assets with non-linear payoffs may both be mis-specified. And, as is the
case of the parametric VaR, there is the risk of mis-specifying the model parameters.
3.2. Parameter estimation
The parametric and Monte Carlo VaR methods require parameters to be estimated. When
one is interested in short time horizons, the primary goal is to estimate the volatility and
covariance/correlation.17 We outline some of the common estimation techniques here.
3.2.1. Historical volatility
There are two different approaches to modelling volatility and covariance using only historical
data. The more common approach gives constant weights to each data point. It
assumes that volatility and covariance are constant over time. The other approach attempts
to address the fact that volatility and covariance are time dependent by giving more weight
to the more recent data points in the sample window.
First assume that variances and covariances do not to change over time. Take a large
window of length n in which historical data on the risk factors is available. Let Yi,tk be the
return of factor i at time period tk. The variance of factor i and covariance of factors i and
j are then computed by giving equal weights to each data point in the past. The n-period
estimates at time T for the variance and covariance
^2
i =
1
n - 1
T -1
t=T -n
(Yi,t - ^Yi )2
, where ^Yi =
1
n
T -1
t=T -n
Yi,t , (16)
17 For example, over short time horizons, the mean return is usually assumed to be zero.
52 B.O. Bradley and M.S. Taqqu
and
^i,j =
1
n - 1
T -1
t=T -n
(Yi,t - ^Yi )(Yj,t - ^Yj ) (17)
respectively.18 Since equal weight is given to each data point in the sample, the estimated
volatility and covariance change only slowly. If one keeps the window length fixed, the
estimated values will rise or fall as new large returns enter the sample period and old large
returns leave it. This means that even a single extreme return will affect the estimates in
the same way, whether it occurred at time T -1 or time T -n. The estimated variance and
covariance, therefore, are greatly influenced by the choice of the window size n.
Another stylized fact of financial time series, however, is that volatility itself is volatile.
With this in mind, another historical estimate of variance and covariance uses a weighting
scheme which gives more weight to more recent observations. The corresponding estimates
of variance and covariance are
^2
i (T ) =
T -1
t=T -n
t (Yi,t - ^Yi )2
,
^i,j (T ) =
T -1
t=T -n
t (Yi,t - ^Yi )(Yj,t - ^Yj ),
where the weights t , T -1
t=T -n t = 1, are chosen to reflect current volatility conditions. In
particular, more weight is given to recent observations: 1 > T -1 > T -2 >  > T -n >
0. The model using exponentially decreasing weights, such as that used by RiskMetrics, is
probably the most popular. In RiskMetrics, the volatility estimator is given by
^i(T ) = (1 - )
n
t=1
t-1(Yi,T -t - ^Yi )2, (18)
where the decay factor  is chosen to best match a large group of assets.19 The covariance
estimate is similar. RiskMetrics chooses  = 0.94 in the case of daily returns.
18 The normalization constant n - 1 gives an unbiased estimate. It is sometimes replaced by n in order to correspond
to the maximum likelihood estimate.
19 In this estimate it is assumed that the decay parameter  and window length n are such that the approximation
n
t=1
t-1 =
1
1 - 
is valid.
Ch. 2: Financial Risk and Heavy Tails 53
The choice (18) allows the forecast of the next periods volatility given the current information,
and hence to make parametric VaR calculations given the current information. To
see this, assume that the time T (negative) return distribution XT is being modelled by
XT
d
= T ZT , (19)
where Zt , t  Z, is an innovation process, that is a sequence of i.i.d. mean zero and unit
variance random variables. Letting Ft denote the filtration20 we have
2
T +1|FT
= (1 - )

t=0
t
X2
i,T -t
= (1 - )X2
T + (1 - )(X2
T -1 + X2
T -2 + 2
X2
T -3 + )
= (1 - )X2
T + 2
T |FT -1
.
This allows us to make our VaR calculation depend on the conditional return distribution
FXT +1|FT
. If VaRT +1
 (X) denotes the estimated value at risk for X at confidence level 
for the period T + 1 at time T , then, by (19),
VaRT +1
 (X) = T +1|FT
q,
where q is the  quantile of the innovation process Zt+1. In RiskMetrics Z is N(0,1), in
which case the return process Xt is conditionally normal.21
The modelling of the volatility using exponential weights and the assumption of conditional
normality has two major effects. First, the volatility estimator, which is now truly
time varying, attempts to account for the local volatility conditions by giving more weight
to the most recent observations. It also has a second less obvious, but no less profound
effect on the calculation of VaR. Even though the conditional return distribution may be
assumed to be normal (thin-tailed) within the VaR calculation, the unconditional return
distribution will typically have heavier tails than the normal. This result is not surprising
since we may think of our time t return as being sampled from a normal distribution with
changing variance. This means that our unconditional distribution is more likely to fit the
empirical returns and thus to provide a better estimate of the true VaR.
3.2.2. ARCH/GARCH volatilities
The ARCH/GARCH class of conditional volatility models were first proposed by Engle
(1982) and Bollerslev (1986) respectively. We will again assume that the (negative) return
20 Conditioning over FT means conditioning over all the observations X1,... ,XT .
21 RiskMetrics allows the assumption of conditional normality to be relaxed in favor of heavier-tailed conditional
distributions. For example the conditional distribution of returns may be mixture of normals or a generalized error
distribution, that is, a double sided exponential.
54 B.O. Bradley and M.S. Taqqu
process to be modelled is of the form (19) where Zt are i.i.d. mean zero, unit variance
random variables representing the innovations of the return process. In the GARCH(p,q)
model,22 the conditional variance is given by
2
t = 0 +
p
i=1
iX2
t-i +
q
j=1
j 2
t-j .
In its most common form, Zt  N(0,1), so that the returns are conditionally normal.
Just as in the exponentially weighted model for volatility (see Section 3.1.1), the GARCH
model with a conditionally normal return distribution can lead to heavy tails in the unconditional
return distribution. In the case of the GARCH(1,1) model
Xt = tZt, where Zt  N(0,1) i.i.d.,
2
t = 0 + 1X2
t-1 + 12
t-1,
it is straightforward to show that under certain conditions23 the unconditional centered
kurtosis is given by
K =
EX4
t
(EX2
t )2
- 3 =
62
1
1 - 2
1 - 211 - 32
1
,
which for most financial return series will be greater than zero. For example, in the case
of a stationary ARCH(1) model, Xt = 0 + 1X2
t-1Zt , with 0 > 0 and 1  (0,2 e ),
where  is Euler's constant,24 Embrechts, Klüppelberg and Mikosch (1997) show that the
unconditional distribution is formally heavy-tailed, that is
P(X > x)  cx,
x  , (20)
where /2 > 0 is the unique solution to the equation h(u) = (21)u


(u + 1
2 ) = 1.
The ARCH/GARCH models allow for both volatility clustering (periods of large volatility)
and for heavy tails. The GARCH(1,1) estimated volatility process t for the NASDAQ
is displayed in Figure 5. The assumption of conditional normality can be checked, for ex-
22 The ARCH(p) model first proposed by Engle is equivalent to the GARCH(p,0) model later proposed by
Bollerslev. The advantage of the GARCH model over the ARCH model is that it requires fewer parameters to be
estimated, because AR models (ARCH) of high order are often less parsimonious than ARMA models (GARCH)
of lower order.
23 These conditions are 1 + 1 < 1 to guarantee stationarity, and 32
1 + 211 + 2
1 < 1 for K > 0. Both are
generally met in financial time series.
24 Euler's constant  is given by  = limn( n
k=1
1
k - lnn) and is approximately   0.577.
Ch. 2: Financial Risk and Heavy Tails 55
Fig. 5. GARCH(1,1) volatilities t for NASDAQ.
Fig. 6. Quantile­quantile (QQ) plot of the conditionally normal GARCH(1,1) standardized ex post innovations
for NASDAQ with the N (0,1) distribution.
ample, by examining a QQ plot of the ex post innovations, that is Zt = Xt /^t . Figure 6
displays the QQ plot of Zt in the traditional, conditionally normal GARCH(1,1) model for
the NASDAQ. The fit of the GARCH(1,1) conditionally normal model in the lower tail is
poor, showing the lower tail of Zt is heavier than the normal distribution.
If the distribution of the historical innovations Zt-n,...,Zt is heavier-tailed than
the normal, one can modify the model to allow a heavy-tailed conditional distribution
FXt+1|Ft .25 In Panorska, Mittnik and Rachev (1995) and Mittnik, Paolella and Rachev
(1997), returns on the Nikkei index are modelled using an ARMA-GARCH model of the
form
Xt = a0 +
r
i=1
aiXt-i + t +
s
j=1
bj t-j (21)
(contrast with (19)), where t = t Zt , with Zt an i.i.d. location zero, unit scale heavytailed
random variable. The conditional distribution of the return series FXt |Ft-1
is given
25 For example the GARCH module in the statistical software package SPlus allows for three different nonGaussian
conditional distributions. As long as the user can estimate the GARCH parameters, usually through
maximum likelihood, there are virtually no limits to the choice of the conditional distribution.
56 B.O. Bradley and M.S. Taqqu
by the distribution type of Zt . The ARMA structure in (21) is used to model the conditional
mean E(Xt |Ft-1) of the return series Xt . The GARCH structure is imposed on the scale
parameter26 t through
2
t = 0 +
p
i=1
i2
t-i +
q
j=1
j 2
t-j .
Several choices for the distribution of Zt are tested. In the case where Zt are realizations
from a stable distribution, the GARCH model used is
t = 0 +
p
i=1
i|t-i| +
q
j=1
j t-j ,
and the index of stability exponent  for the stable distribution is constrained to be greater
than one.
Using several goodness of fit measures, the authors find that it is better to model the conditional
distribution of returns for the Nikkei than the unconditional distribution, since the
unconditional distribution cannot capture the observed temporal dependencies of the return
series.27 Within the tested models for Zt , the partially asymmetric Weibull, the Student-t,
and the asymmetric stable all outperform the normal. In order to perform reliable value
at risk calculations one must model the tail of the distribution Zt particularly well. The
Anderson­Darling (AD) statistic can be used to measure goodness of fit in the tails. Letting
Femp(x) and Fhyp(x) denote the empirical and hypothesized parametric distributions
respectively, the AD statistic
AD = sup
xR
|Femp(x) - Fhyp(x)|
Fhyp(x)(1 - Fhyp(x))
gives more weight to the tails of the distribution. Using this statistic, as well as others,
the authors propose the asymmetric stable distribution as the best of the tested models for
performing VaR calculations at high quantiles.
The class of ARCH/GARCH models have become increasingly popular for computing
VaR. The modelling of the conditional distribution has two immediate benefits. First, it
allows for the predicted volatility (or scaling) to use local information, i.e., it allows for
volatility clustering. Second, since volatility is allowed to be volatile, the unconditional
distribution will typically not be thin-tailed. This is true, as we have seen, even when the
conditional distribution is normal.
26 In their model t is to be interpreted as a scale parameter, not necessarily a volatility, since for some of the
distributional choices for Zt , the variance may not exist.
27 The type of the conditional distribution is that of Zt , the unconditional distribution is that of Xt .
Ch. 2: Financial Risk and Heavy Tails 57
There now exist many generalizations of the class of ARCH/GARCH models. Models
such as EGARCH, HGARCH, AGARCH, and others, all attempt to use the local volatility
structure to better predict future volatility while trying to account for other observed
phenomenon. See Bollerslev, Chou and Kroner (1992) for a review. The time series of
returns {Xt}tZ in (19) is generally assumed to be stationary. In a recent paper, Mikosch
and St˘aric˘a (2000) show that this assumption is not supported, at least globally, by the
S&P 500 from 1953 to 1990 and the DEM/USD foreign exchange rate from 1975 to 1982.
The authors show that when using a GARCH model the parameters must be updated to
account for changes of structure (changes in the unconditional variance) of the time series.
A method for detecting these changes is also proposed. Additionally, they show that the
long range dependence behavior associated with the absolute return series, another of the
so called stylized facts of financial time series, may only be an artifact of structural changes
in the series, that is, to non-stationarity.
Stochastic volatility models are not limited to the class of ARCH/GARCH models and
their generalizations. Other models may involve additional sources of randomness. For
example, the model of Hull and White (1987)
dYt = Yt + tYt dW
(1)
t ,
dVt = Vt + Vt dW(2)
t ,
where 2
t = Vt and (W(1)
t ,W(2)
t ) is a bivariate Wiener process, introduces a second source
of randomness through the volatility. The two sources of randomness W
(1)
t and W
(2)
t need
not be uncorrelated. Again, the introduction of a stochastic scaling generally leads to an
unconditional return distribution which is leptokurtotic. See Shiryaev (1999), for an introduction
to stochastic volatility models in discrete and continuous time.
3.2.3. Implied volatilities
The parametric VaR calculation requires a forecast of the volatility. All of the models examined
so far have used historical data. One may prefer to use a forward looking data
set instead of historical data in the forecast of volatility, for example options data, which
provide the market estimate of future volatility. To do so, one could use the implied volatility
derived from the Black­Scholes model. In this model, European call options prices
Ct = C(St ,K,r,,T - t) are an increasing function of the volatility . The stock price St
at time t, the strike price K, the interest rate r and the time to expiration T - t are known
at time t. Since  is the only unknown parameter/variable, we may then use the observed
market price Ct to solve for . This estimate of  is commonly called the (Black­Scholes)
implied volatility. The Black­Scholes model, however is imperfect. While  should be
constant, one typically observes that  depends on the time to expiration T - t and on the
strike price K. For fixed T - t, the implied volatility  = (T - t,K) as a function of
the strike price K is often convex, a phenomenon known as the volatility smile. To obtain
volatility estimates it is common to use at-the-money options, where St = K, since they
are the most actively traded and hence are thought to provide the most accurate estimates.
58 B.O. Bradley and M.S. Taqqu
3.2.4. Extreme value theory
Since VaR calculations are only concerned with the tails of a probability distribution, techniques
from Extreme Value Theory (EVT) may be particularly effective. Proponents of
EVT have made compelling arguments for its use in calculating VaR and for risk management
in general. We will discuss EVT in Section 6.
4. Risk measures
We have considered two different measures of risk: standard deviation and value at risk.
Standard deviation, used by Markowitz and others, is still commonly used in portfolio
theory today. The second measure, VaR, is the standard measure used today by regulators
and investment banks. We detailed some of the computational issues surrounding these
measures but have not discussed their validity.
It is easy to criticize standard deviation and value at risk. Even in Markowitz's pioneering
work on portfolio theory, the shortcomings of standard deviation as a risk measure
were recognized. In Markowitz (1959), an entire chapter is devoted to semi-variance28 as
a potential alternative. In Artzner et al. (1997), for example, measures based on standard
deviation are criticized based on their inability to describe rare events and VaR is criticized
because of its inability to aggregate risks in a logical manner. In two now famous papers
(Artzner et al., 1997, 1999) on financial risk, the authors propose a set of properties any
reasonable risk measure should satisfy. Any risk measure which satisfies these properties
is called coherent. We shall now introduce these properties and indicate why the risk measures
described above are not coherent.
4.1. Coherent risk measures
Suppose that the financial position of an investor will lead at time T to a loss X,29 which is
a random variable. Let G be the set of all such X. A risk measure  is defined as a mapping
from G to R. Intuitively, for a given potential loss X in the future we may think of (X) as
the minimum amount of cash that we need to invest prudently today (in a reference instrument)
to be allowed to take the position X.30 A risk measure  may be coherent or not.
Definition 4.1. Given a reference instrument with return r, possibly random, a risk measure
 satisfying the following four axioms is said to be coherent:
28 In order to put the accent on (negative) returns above the mean, semi-variance is defined as
X = E[(X - EX)1{X>EX}]2.
29 Losses are positive and profits negative. This is at odds with the authors' original notation.
30 The authors refer to X as risk and axiomatically define acceptance sets which are sets of acceptable risks, and
proceed to define measures of risk as describing the risks proximity to the acceptance set.
Ch. 2: Financial Risk and Heavy Tails 59
Translation invariance. For all X  G and all   R, we have (X + r) = (X) + .
This means that adding the amount  to the position, and investing it prudently, reduces
the overall risk of the position by .
Subadditivity. For all X1 and X2  G, (X1 + X2) (X1) + (X2). Hence a merger
does not create extra risk. This is the basis for diversification.
Positive homogeneity. For all  0 and all X  G, (X) = (X). This requires that
the risk scales with the size of a position. If the size of a position renders it illiquid, then
this should be considered when modelling the future net worth.
Monotonicity. For all X and Y  G with X Y, we have (X) (Y). If the future net
loss X is greater, then X is more risky.
The term coherent measure of risk has found its way into the risk management vernacular.
It is defined, for example, in the second edition of Philippe Jorion's Value at Risk
(Jorion, 2001).
Note that the axioms of translation invariance and monotonicity rule out standard deviation
as a coherent measure of risk. Indeed, since X+r = X, translation invariance fails,
and since  also penalizes the investor for large profits as well as large losses, monotonicity
fails as well. Consider, for example, two portfolios X and Y which are identical except
for the free lottery ticket held in Y. We have X Y, since there is no down-side to the free
ticket and therefore the potential losses in Y are smaller than in X. Nevertheless, the standard
deviation measure assigns to Y a higher risk, hence monotonicity fails. Markowitz's
alternative risk measure semi-variance is not coherent either because it is not subadditive.
4.2. Expected shortfall
VaR is not a coherent measure of risk because it fails to be subadditive in general. One can
indeed easily construct scenarios [see Albanese (1997)] where for two positions X and Y
it is true that
VaR(X + Y) > VaR(X) + VaR(Y).
This is contrary to the risk managers feelings, that the overall risk of different trading desks
is bounded by the sum of their individual risks. In short, VaR fails to aggregate risks in a
logical manner. In addition, VaR tells us nothing about the size of the loss that exceeds it.
Two distributions may have the same VaR yet be dramatically different in the tail.
Hence neither the standard deviation nor VaR are coherent. On the other hand, the expected
shortfall, also called tail conditional expectation, is a coherent risk measure. Intuitively,
the expected shortfall addresses the question: given that we will have a bad day,
how bad do we expect it to be? It is a more conservative measure than VaR and looks at the
average of all losses that exceed VaR. Formally, the expected shortfall for risk X and high
confidence level  is defined as follows:
60 B.O. Bradley and M.S. Taqqu
Definition 4.2. Let X be the random variable whose distribution function FX describes the
negative profit and loss distribution (P&L) of the risky financial position at the specified
horizon time  (thus losses are positive). Then the expected shortfall for X is
S(X) = E X|X > VaR(X) . (22)
Suppose, for example, that a portfolio's risk is to be calculated through simulation. If
1000 simulations are run, then for  = 0.95, the portfolios VaR would be the smallest
of the 50 largest losses. The corresponding expected shortfall would be estimated by the
numerical average of these 50 largest losses. Expected shortfall, therefore, tells us something
about the expected size of a loss exceeding VaR. It is subadditive, coherent and puts
fewer restrictions on the distribution of X, requiring only a finite first moment to be well
defined. Additionally, it may be reconciled with the idea of maximizing expected utility.
Levy and Kroll (1978) show that for all utility functions U with the properties described in
Section 2.1 and all random variables X and Y (representing losses) that
EU(-X) EU(-Y)  S(X) S(Y) for all   (0,1).
Expected shortfall can be used in portfolio theory as a replacement of the standard deviation
if the distribution of X is normal, or more generally, elliptical. As we will see in
Section 5.3, in this case any positive homogeneous translation invariant risk measure will
yield the same optimal linear portfolio for the same level of expected return.
Unlike standard deviation, expected shortfall, as defined in (22), does not measure deviation
from the mean. Bertsimas, Lauprete and Samarov (2000) define shortfall31 as
s(X) = E X|X > VaR(X) - EX. (23)
The subtraction of the mean makes it more similar to the standard deviation X =
E(X - EX)2 and again, as far as portfolio theory is concerned, in the case of elliptical
distributions, one obtains the same optimal portfolio for the same level of expected
return if one uses s to measure risk. In fact, it can be shown that for a linear portfolio
Xp = w1X1 +  + wnXn of multivariate normally distributed returns X  N(,), that
s(Xp) =
(-1())
1 - 
p,
where (x) and (x) are respectively, the pdf and cdf of a standard normal random variable
evaluated at x. In other words,
arg min
Aw=b
wT
w = arg min
Aw=b
s wT
X ,
31 We still assume losses are positive. This is at odds with the authors notation.
Ch. 2: Financial Risk and Heavy Tails 61
for all   (0,1), where Aw = b is any set of linear constraints, including constraints that
do not require all portfolios to have the same mean. Note, however, that s is not coherent
since it violates the axioms of translation invariance and monotonicity.
5. Portfolios and dependence
The measure of dependence most popular in the financial community is linear correlation.32
Its popularity may be traced back to Markowitz' mean variance portfolio theory
since, under the assumption of multivariate normality, the correlation is the canonical measure
of dependence. Outside of the world of multivariate normal distributions, correlation
as a measure of dependence may lead to misleading conclusions (see Section 5.2.1).33 The
linear correlation between two random variables X and Y, defined by
(X,Y) =
Cov(X,Y)
XY
, (24)
is a measure of linear dependence between X and Y. The word linear is used because
when variances are finite, (X,Y) = 1 if and only if Y is an affine transformation of X
almost surely, that is if Y = aX + b a.s. for some constants a  R \ {0}, and b  R. When
the distribution of returns X is multivariate normal, the dependence structure of the returns
is determined completely by the covariance matrix  or, equivalently, by the correlation
matrix . One has  = [][] where [] is a diagonal matrix with the standard deviations
j on the diagonal.
When returns are not multivariate normal, linear correlation may no longer be a meaningful
measure of dependence. To deal with potential alternatives, we will introduce the
concept of copulas, describe various measures of dependence and focus on elliptical distributions.
For additional details and proofs, see Embrechts, McNeil and Straumann (2001),
Lindskog (2000b), Nelsen (1999), Joe (1997) and Fang, Kotz and Ng (1990).
5.1. Copulas
When X = (X1,...,Xn)  N(,), the distribution of any linear portfolio of the Xj 's is
normal with known mean and variance. In the non-normal case, the joint distribution of X,
F(x1,...,xn) = P(X1 x1,...,Xn xn)
is not fully described by its mean and covariance. One would like, however, to describe the
joint distribution by specifying separately the marginal distributions, that is, the distribution
of the components X1,...,Xn, and the dependence structure. One can do this with copulas.
32 Also known as Pearson's correlation.
33 Linear correlation is actually the canonical measure of dependence for the class of elliptical distributions. This
class will be introduced shortly and may be thought of as an extension of multivariate normal distributions.
62 B.O. Bradley and M.S. Taqqu
Definition 5.1. An n-copula is any function C :[0,1]n  [0,1] satisfying the following
properties:
(1) For every u = (u1,...,un) in [0,1]n we have that C(u) = 0 if at least one component
uj = 0 and C(u) = uj if u = (1,...,1,uj ,1,...,1).
(2) For every a,b  [0,1]n such that a b
2
i1=1

2
in=1
(-1)i1++in
C(u1i1 ,...,unin ) 0, (25)
where uj1 = aj and uj2 = bj for j = 1,...,n.
Corollary 5.1 below provides a concrete way to construct copulas. It is based on the following
theorem due to Sklar [see Sklar (1996), Nelsen (1999)], which states that by using
copulas one can separate the dependence structure of the multivariate distribution from the
marginal behavior.
Theorem 5.1 (Sklar). Let F be an n-dimensional distribution function with marginals
Xj  Fj for j = 1,...,n. Then there exists an n-copula C :[0,1]n  [0,1] such that for
every x = (x1,...,xn)  Rn,
F(x1,...,xn) = C F1(x1),...,Fn(xn) . (26)
Furthermore, if the Fj are continuous then C is unique. Conversely, if C is an n-copula
and Fj are distribution functions, then F in (26) is an n-dimensional distribution function
with marginals Fj .
The function C is called the copula of the multivariate distribution of X. Assuming continuity
of the marginals Fj , j = 1,...,n, we see that the copula C of F is the joint distribution
of the uniform transformed variables Fj (Xj ),
C(u1,...,un) = F F-1
1 (u1),...,F-1
n (un) . (27)
Corollary 5.1. If the Fj are the cdfs of U(0,1) random variables, then xj = Fj (xj ),
0 < xj < 1, and (26) becomes F(x1,...,xn) = C(x1,...,xn). Therefore the copula C
may be thought of as the cumulative distribution function (cdf) of a random vector with
uniform marginals.
Copulas allow us to model the joint distribution of X in two natural steps. First, one
models the univariate marginals Xj . Second, one chooses a copula that characterizes the
dependence structure of the joint distribution. Any n-dimensional distribution function can
serve as a copula. The following examples relate familiar multivariate distributions to their
associated copulas and marginals.
Ch. 2: Financial Risk and Heavy Tails 63
Example 5.1. Suppose X1,...,Xn are independent then
F(x1,...,xn) = P(X1 x1,...,Xn xn)
= P(X1 x1)P(Xn xn)
= F1(x1)Fn(xn).
Hence, in the case of independence, C(u1,...,un) = u1 un for all (u1,...,un) 
[0,1]n.
Example 5.2. Suppose (X1,...,Xn) is multivariate standard normal with linear correlation
matrix . Let (z) = P(Z z) for Z  N(0,1). Then
F(x1,...,xn) = P(X1 x1,...,Xn xn)
= P F1(X1) F1(x1),...,Fn(Xn) Fn(xn)
= CGa
 (x1),...,(xn) ,
where
CGa
 (u1,...,un) =
1

||(2)n
-1(u1)
-

-1(un)
e-
1
2 sT-1s
ds (28)
is called the multivariate Gaussian copula.
Example 5.3. Suppose (X1,...,Xn) is multivariate t with  degrees of freedom and linear
correlation matrix .34 Let t(x) = P(T x) where T  t. Then
F(x1,...,xn) = P(X1 x1,...,Xn xn)
= P F1(X1) F1(x1),...,Fn(Xn) Fn(xn)
= Ct
 t(x1),...,t(xn) ,
where
Ct
 (u1,...,un) =
(+n
2 )
(
2 )

||()n
t-1
 (u1)
-

t-1
 (un)
-
1 +
sT-1s

-(+n)/2
ds
(29)
is called the multivariate t copula.
34 Its cdf is given by (29) where the upper limits t-1
 (u1),... ,t-1
 (un) are replaced by x1,... ,xn respectively.
A multivariate t is easy to generate. Generate a multivariate normal with covariance matrix  and divide it by
2
 / where 2
 is an independent chi-squared random variable with  degrees of freedom.
64 B.O. Bradley and M.S. Taqqu
In Examples 5.2 and 5.3, || denotes the determinant of the matrix . In these examples,
the copulas were introduced through the joint distribution, but it is important to remember
that the copula characterizes the dependence structure of the multivariate distribution
through (26). The Gaussian and t copulas (28) and (29) exist separately from their associated
multivariate distributions.
Example 5.4. The bivariate Gumbel copula CGu
 is given by
CGu
 (u1,u2) = exp - (-lnu1)1/
+ (-lnu2)1/ 
, (30)
where 0 <  1 is a parameter controlling the dependence,   0+ implies perfect dependence
(see Section 5.2.3), and  = 1 implies independence.
Example 5.5. The bivariate Clayton copula CCl
 is given by
CCl
 (u1,u2) = u
-
1 + u
-
2 - 1
-1/
, (31)
where 0 <  <  is a parameter controlling the dependence,   0+ implies independence,
and    implies perfect dependence. This copula family is sometimes referred
to as the Kimeldorf and Sampson family.
Both the Gumbel and Clayton copulas are strict Archimedean copulas. Archimedean copulas
are defined as follows. Let  :[0,1]  [0,) with (0) =  and (1) = 0 be a continuous,
convex, strictly decreasing function. The transformation -1 maintains the uniform
1-dimensional distribution since -1(u) = u, u  [0,1]. To obtain a 2-dimensional
distribution function use instead of -1(u), u  [0,1], the function -1((u) + (v)),
u,v  [0,1].
Definition 5.2. A strict Archimedean copula with generator  is of the form
C(u,v) = -1
(u) + (v) , u,v  [0,1]. (32)
Example 5.6. The function (t) = (-lnt)1/,0 <  1, generates the bivariate Gumbel
copula CGu
 (see Example 5.4).
Example 5.7. The function (t) = (t- - 1)/, > 0, generates the bivariate Clayton
copula CCl
 (see Example 5.5).
Example 5.8. The function (t) = -ln((e-t - 1)/(e- - 1)),   R \ {0}, generates the
bivariate Frank copula
CFr
 (u,v) = -
1

ln 1 +
(e-u - 1)(e-v - 1)
e- - 1
[see Frank (1979)].
Ch. 2: Financial Risk and Heavy Tails 65
If (0) < , then the term strict in Definition 5.2 is dropped and -1(s) in (32) is replaced
by the pseudo-inverse [-1](s) which equals -1(s) if 0 s (0) and is zero otherwise.
Example 5.9. The function (t) = 1 - t, t  [0,1], satisfies (0) = 1 and hence
[-1](t) = max(1 - t,0). It generates the non-strict Archimedean copula
C(u,v) = max(u + v - 1,0).
The class of Archimedean copulas has many nice properties, including various simple
multivariate extensions. For more on Archimedean copulas see Lindskog (2000b), Nelsen
(1999), Joe (1997) and Embrechts, Lindskog and McNeil (2001).
Figure 7 illustrates how the choice of a copula can affect the joint distribution. Each
figure shows contours of constant density of a bivariate distribution (X,Y) with standard
normal marginals and linear correlations   0.7. The differences in the distributions is
due to the choice of the copula. [For an introduction on the choice of a copula, see Frees
and Valdez (1998).]
Fig. 7. Contours of constant density for different bivariate distributions with standard normal marginals. All have
roughly the same linear correlation, and differ only in their copula. Clockwise from upper left: Gaussian, t2,
Gumbel, Clayton. See Examples 5.2­5.5 for the copula definitions.
66 B.O. Bradley and M.S. Taqqu
The following theorem provides a bound for the joint cdf.
Theorem 5.2 (Fréchet). Let F be the joint cdf of distribution with univariate marginals
F1,...,Fn. Then for all x  Rn
max{0,F1(x1) +  + Fn(xn) - (n - 1)}
CL(F1(x1),...,Fn(xn))
F(x1,...,xn)
C(F1(x1),...,Fn(xn))
min{F1(x1),...,Fn(xn)}
CU (F1(x1),...,Fn(xn))
.
The function CU (u1 ...,un) is a copula for all n 2, but the function CL(u1,...,un) is
a copula for n = 2 only. If n = 2, the copulas CL and CU are the bivariate cdf's of the
random vectors (U,1 - U) and (U,U) respectively, where U  U(0,1).
Another important property of copulas is their invariance under an increasing transformation
of the marginals.
Theorem 5.3. Let X1,...,Xn be continuous random variables with copula C. Let
1,...,n be strictly increasing transformations. Then the random vector (1(X1),...,
n(Xn)) has the same copula C as (X1,...,Xn).
5.2. Measures of dependence
As already mentioned, linear correlation is the only measure of dependence involved in
the mean­variance portfolio theory. This theory assumes, either implicitly or explicitly,
that returns are multivariate normal. This assumption seems implausible today given the
many complex financial products in the marketplace and the empirical evidence against
normality. Without the restrictive assumption of normality, is linear correlation still an
appropriate measure of dependence?
Linear correlation is often used in the financial community to describe any form of
dependence. As illustrated in Embrechts, McNeil and Straumann (2001, 1999), linear correlation
is often a very misunderstood measure of dependence. Consider the following
example.
Example 5.10. Figure 8 represent 10000 simulations from bivariate distributions (X,Y)L
and (X,Y)R. In both cases X and Y have a standard normal distribution with (approximately)
the same linear correlation   0.7. Thus, on the basis of the marginal distributions
and linear correlation, the two distributions are indistinguishable. The two distributions are
however clearly different. If positive values represent losses, the distribution on the right is
clearly of greater concern to the risk manager since large losses in X and Y occur simultaneously.
The two distributions differ only in their copula.
In the figure on the left the dependence structure is given by the bivariate Gaussian
copula. Since the marginals are standard normal, this means that distribution is the bivariate
Ch. 2: Financial Risk and Heavy Tails 67
Fig. 8. Simulation of 10000 realizations from bivariate distributions both with standard normal marginals and
linear correlation of   0.7. The distribution on the left has a Gaussian copula, on the right a Gumbel copula.
Compare the shapes with those illustrated in Figure 7, where the population distribution is used.
standard normal distribution with the given correlation coefficient. The copula in the figure
on the right the Gumbel copula given in (30) with  = 1/2. Various values of  were tried
until the simulation sample linear correlation was   0.7.
We now briefly describe several measures of dependence which may be useful to the
risk manager. Again the reader in encouraged to look at the above references, especially
Embrechts, McNeil and Straumann (2001) for details.
5.2.1. Linear correlation
The linear correlation coefficient , defined in (24), is a commonly misused measure of
dependence. To illustrate the confusion involved in interpreting it, consider the following
classic example. Let X  N(,2) and let Y = X2. Then (X,Y) = 0, yet clearly X and
Y are dependent. Unless we are willing to make certain assumptions about the multivariate
distribution, linear correlation can therefore be a misleading measure of dependence. Since
the copula of a multivariate distribution describes its dependence structure we would like
to use measures of dependence which are copula-based. Linear correlation is not such a
measure.
5.2.2. Rank correlation
Two well-known rank correlation measures which are copula based and have better properties
than linear correlation are the Kendalĺs tau and Spearman's rho.
Definition 5.3. Let (X1,Y1) and (X2,Y2) be two independent copies of (X,Y). Then ,
denoted  , is given by
 (X,Y) = P (X1 - X2)(Y1 - Y2) > 0 - P (X1 - X2)(Y1 - Y2) < 0 .
68 B.O. Bradley and M.S. Taqqu
If the marginal distributions FX and FY of X and Y are continuous and if F is the
bivariate distribution function of (X,Y) with copula C, then  can be expressed in terms
of C as follows [see Embrechts, McNeil and Straumann (2001)]:
 (X,Y) = 4
1
0
1
0
C(u,v)dC(u,v) - 1.
Definition 5.4. Let X  FX and Y  FY . Spearman's correlation, denoted S, is the linear
correlation of FX(X) and FY (Y), that is,
S(X,Y) =  FX(X),FY (Y) .
Spearman's correlation can also be expressed in a form similar to Definition 5.3 [see Lindskog
(2000b)]. Let (X1,Y1),(X2,Y2) and (X3,Y3) be three independent copies of (X,Y).
Then
S(X,Y) = 3 P (X1 - X2)(Y1 - Y3) > 0 - P (X1 - X2)(Y1 - Y3) < 0 .
If the marginal distributions are continuous, S is related to the copula of the joint distribution
as follows:
S(X,Y) = 12
1
0
1
0
C(u,v)dudv - 3.
Whereas linear correlation is a measure of linear dependence, both Kendalĺs tau and
Spearman's rho are measures of monotonic dependence. Since they are copula based, they
are invariant under strictly increasing transformations.35 Indeed, if 1, 2 are strictly increasing
transformations, then
 1(X1),2(X2) =  (X1,X2),
S 1(X1),2(X2) = S(X1,X2), but
 1(X1),2(X2) = (X1,X2).
5.2.3. Comonotonicity
An additional important property of these rank correlations is their handling of perfect dependence.
By perfect dependencewe mean intuitively that X and Y are monotone functions
of the same source of randomness. Recall that in the bivariate case, the Fréchet bounds CL
and CU in Theorem 5.2 are themselves copulas. The following theorem shows that if the
copula is CL or CU then X and Y are perfectly dependent.
35 Recall that invariance under increasing transformations is a property of copulas.
Ch. 2: Financial Risk and Heavy Tails 69
Theorem 5.4 (Embrechts, McNeil and Straumann, 2001). Suppose that the copula C of
(X,Y) is either CL or CU . Then there exist monotone functions  and  and a random
variable Z such that
(X,Y)
d
= (Z),(Z) .
If C = CL then  and  are increasing and decreasing respectively. If C = CU , then both
 and  are increasing.
X and Y are said to be countermonotonic if they have copula CL. If they have copula
CU , they are said to be comonotonic. In fact, when FX and FY are continuous,
C = CL  Y = T (X) a.s., T = F-1
Y  (1 - FX) ,
C = CU  Y = T (X) a.s., T = F-1
Y  FX .
Kendalĺs tau and Spearman's rho handle perfect dependence in a reasonable manner.
Indeed,
Theorem 5.5 (Embrechts, McNeil and Straumann, 2001). Let (X,Y)  F with continuous
marginals and copula C. Then
 (X,Y) = -1  S(X,Y) = -1  C = CL
 X and Y are countermonotonic,
 (X,Y) = 1  S(X,Y) = 1  C = CU
 X and Y are comonotonic.
The following theorem due to Höffding and Fréchet deals with linear correlation. See
Embrechts, McNeil and Straumann (2001) for its proof.
Theorem 5.6. Let (X,Y) be a random vector with marginals non-degenerate FX and FY
and unspecified dependence structure. If X and Y have finite variance, then
(1) The set of possible linear correlations is a closed interval [min,max] with min < 0 <
max.
(2) The extremal linear correlation  = min is attained iff X and Y are countermonotonic;
 = max is attained iff X and Y are comonotonic.
(3) min = -1  X and -Y are of the same type;36 max = 1  X and Y are of the same
type.
The following example shows that linear correlation does not handle perfect dependence
in a reasonable manner.
36 Recall that two random variables are the same type if their distributions are the same up to a change in location
and scale.
70 B.O. Bradley and M.S. Taqqu
Fig. 9. Range of maximal and minimal linear correlation in Example 5.11. The x-axis is in units of . As 
increases, both the maximal and minimal linear correlations tend to zero.
Example 5.11 (Embrechts, McNeil and Straumann, 2001). Let X  Lognormal(0,1) and
Y  Lognormal(0,2) with  > 0. By Theorem 5.6,  = min and  = max when X
and Y are countermonotonic and comonotonic respectively. By Theorem 5.4, (X,Y)
d
=
((Z),(Z)), and in fact, (X,Y)
d
= (eZ,e-Z) when X and Y are countermonotonic
and (X,Y)
d
= (eZ,eZ) when X and Y are comonotonic, where Z  N(0,1). Hence
min = (eZ,e-Z) and max = (eZ,eZ) where Z  N(0,1). Using the properties of
the lognormal distribution, these maximal and minimal correlations can be evaluated explicitly
and one gets
min =
e- - 1
(e - 1)(e2
- 1)
, max =
e - 1
(e - 1)(e2
- 1)
.
As  increases, the maximal and minimal linear correlation both tend to zero even though
X and Y are monotonic functions of the same source of randomness. This is illustrated in
Figure 9.
5.2.4. Tail dependence
There is a saying in finance that in times of stress all correlations go to one.37 While
it shows that the financial community uses linear correlation to describe any measure of
dependence, it can also serve as motivation for the next measure of dependence, known as
tail dependence.
Bivariate tail dependence measures the amount of dependence in the upper and lower
quadrant tail of the distribution. This is of great interest to the risk manager trying to guard
against concurrent bad events in the tails.
37 See Cizeau, Potters and Bouchaud (2001) for example.
Ch. 2: Financial Risk and Heavy Tails 71
Definition 5.5. Let X  FX and Y  FY and observe that as   1-, F-1
X ()   and
F-1
Y ()  . The coefficient of upper tail dependence U is
U (X,Y) = lim
1P
Y > F-1
Y ()|X > F-1
X () (33)
provided the limit exists. If U = 0, then X and Y are said to asymptotically independent
in the upper tail. If U  (0,1], then X and Y are asymptotically dependent in the upper
tail. The coefficient of lower tail dependence L is similarly defined:
L(X,Y) = lim
0+
P Y < F-1
Y ()|X < F-1
X () .
Since
U (X,Y)
= lim
1-
1 - P(X F-1
X ()) - P(Y F-1
Y ()) + P(X F-1
X (),Y F-1
Y ())
1 - P(X F-1
X ())
,
U , as well as L, can be expressed in terms of copulas. Let (X,Y) have continuous distribution
F with copula C. It is easily seen that the coefficient of upper tail dependence U
can be expressed as
U (X,Y) = lim
1-
C(,)
1 - 
, (34)
where C(,) = 1 - 2 + C(,).38 Similarly,
L(X,Y) = lim
0+
C(,)

.
Example 5.12. Recall the simulation Example 5.10. In this example, both distributions
had the same marginal distributions with the same linear correlation. Yet the distributions
were clearly different in the upper tail. This difference came from the choice of copula and
may now be quantified by using the notion of upper tail dependence. In Figure 8 on the
left, F(x,y) = CGa
 ((x),(y)),  denotes the standard N(0,1) cdf and CGa
 is given by
38 If (U1,U2)T  C then
C(u1,u2) = P(U1 > u1,U2 > u2) = 1 - u1 - u2 + C(u1,u2).
72 B.O. Bradley and M.S. Taqqu
(28) that is, the distribution is a bivariate standard normal with linear correlation  = 0.7.
The coefficient of upper tail dependence can be calculated explicitly,39
U (X,Y) = 2 lim
x

x

1 - 

1 + 
= 0,
which is a general characteristic of Gaussian copulas. This means that if we go far enough
out into the tail then extreme events occur independently in X and Y. In the figure of the
right,
F(x,y) = CGu
 (x),(y) ,
with CGu
 given by (30), where the dependence parameter  was chosen to give (approximately)
the same linear correlation.40 In the case of the Gumbel copula a simple calculation
shows that for all 0 <  < 1, the coefficient of upper tail dependence is
U (X,Y) = 2 - 2
.
Hence, for the Gumbel copula, U = 0 for 0 <  < 1.
Suppose the risk manager tries to account for heavy tails of a distribution by simply
modelling the joint distribution as a multivariate t. He will not get U = 0 as in the case
of the multivariate normal distribution.
Example 5.13. If (X,Y)  t with any linear correlation   (-1,1) then it can be shown
(Embrechts, McNeil and Straumann, 2001) that
U (X,Y) = 2t+1
( + 1)(1 - )
1 + 
.
Hence for all   (-1,1) there is upper tail dependence of the bivariate t. The stronger
the linear correlation and the lower the degrees of freedom, the stronger the upper tail
dependence.
5.3. Elliptical distributions
There are distributions other than multivariate normal where linear correlation can be used
effectively. These are the spherical, or more generally, the elliptical distributions. Elliptical
distributions extend in a natural way the class of multivariate normal distributions. Linear
correlation (when it exists) will still be the canonical measure of dependence, yet elliptical
distributions can display heavy tails.
39 (x) = 1 - (x), and, below t(x) = 1 - t(x).
40 The dependence parameter  of the bivariate Gumbel copula is related to Kendalĺs tau by  = 1 - .
Ch. 2: Financial Risk and Heavy Tails 73
We shall define first the spherical distributions. These extend the class of standard multivariate
normal distributions with zero correlations (Fang, Kotz and Ng, 1990; Embrechts,
McNeil and Straumann, 2001).
Definition 5.6. The random vector X  Rn is said to be spherically distributed if
 X
d
= X   O(n),
where O(n) is the group of n × n orthogonal matrices.
In other words, the distribution of X is invariant under rotation of the coordinates. Here
are further characterizations.
Theorem 5.7. The random vector X  Rn has a spherical distribution iff its characteristic
function X satisfies one of the following equivalent conditions:
(1) X( Tt) = X(t)   O(n);
(2) There exists a function ():R+  R such that X(t) = (tTt), that is, X(t) =
( n
i=1 t2
i ), where t = (t1,...,tn). Alternatively, spherical distributions admit a stochastic
representation, namely, X  Rn has a spherical distribution iff there exists a
non-negative random variable R and random vector U independent of R and uniformly
distributed over the unit hypersphere Sn = {s  Rn | s = 1} such that
X
d
= RU. (35)
Example 5.14. Let X  N(0,In) then
X(t) = e-(1/2)(tTt)
= e-(1/2)( n
i=1 t2
i )
,
and so (u) = e-u/2. Additionally, R  2
n in the stochastic representation (35).
The function  is called the characteristic generator of the spherical distribution. We
write
X  Sn()
to indicate that X  Rn is spherically distributed with generator . Note that if X possesses
a density, then Theorem 5.7 requires that it is of the form
f (x) = g xT
x = g
n
i=1
x2
i
for some non-negative function g. The curves of constant density are spheroids in Rn.
74 B.O. Bradley and M.S. Taqqu
Table 2
Partial list of spherical distributions used in finance
Type pdf f (x) or ch.f.  (t)
Normal f (x) = c exp(-xTx/2)
t f (x) = c(1 + xTx/)-(+n)/2
Logistic f (x) = c exp(-xTx)/[1 + exp(-xTx)]2
Scale mixture f (x) = c 
0 t-n/2 exp(-xTx/2t)dG(t), G(t) a c.d.f.
Stable laws  (t) = exp{r(tTt)/2}, 0 <  2 and r > 0
Example 5.15. If X  Rn has a multivariate t distribution with zero correlation, then
f (x) =
(+n
2 )
(
2 )()n/2
1 +
xTx

-(+n)/2
.
X is therefore spherically distributed.
Table 2 gives a partial list of the spherical distributions used in finance.
Recall that if X  N(0,In), then Y =  + AX has a multivariate normal distribution
with mean  and covariance matrix  = AAT. Elliptical distributions are defined from
spherical distributions in a similar manner. They are affine transformations of spherical
distributions.
Definition 5.7. Let X  Rn,   Rn, and   Rn×n. Then X has an elliptical distribution
with parameters  and  if
X
d
=  + AY,
where Y  Sk(), and A  Rn×k,  = AAT, with rank() = k.
Since the characteristic function of X may be written
X(t) = eitT
 tT
t ,
we use the notation
X  En(,,).
In this representation only  is uniquely determined. Since both  and  are determined
up to a positive constant  may be chosen to be the covariance matrix if variances are finite
(which we assume here). An elliptically distributed random variable X  En(,,)
is thus described by its mean, covariance matrix and its characteristic generator. If X pos-
Ch. 2: Financial Risk and Heavy Tails 75
sesses a density, then it is of the form
f (x) = ||-1/2
g (x - )T
-1
(x - ) (36)
so that contours of constant density are ellipsoids in Rn.41
The following theorem describes some properties of linear combinations, marginal distributions
and conditional distributions of elliptical distributions.
Theorem 5.8 (Fang, Kotz and Ng, 1990). Let X  En(,,).
(1) If B  Rm×n and   Rm, then
 + BX  Em  + B,BBT
, .
Hence any linear combination of elliptically distributed variates is elliptical with the
same characteristic generator.
(2) Partition X, ,and  into
X =
X(1)
X(2) ,  =
(1)
(2) ,  =
11 12
21 22
,
where X(1)  Rm, (1)  Rm and 11  Rm×m, 0 < m < n. Then
X(1)
 Em (1)
,11, , X(2)
 En-m (2)
,22, .
Hence all marginals of an elliptical distribution are also elliptical with the same gen-
erator.
(3) Partition X, ,and  as above and assume that  is strictly positive definite. Then
X(1)
|X(2)
= x(2)
0  Em 1.2,11.2, ~ ,
where
1.2 = (1)
+ 12-1
22 x
(2)
0 - (2)
, 11.2 = 11 - 12-1
22 21.
Hence the conditional distribution of X(1) given X(2) is also elliptical, though with
different generator.42
41 For example if rank() = n and Y has density of the form g(yTy).
42 The form of the generator ~ can be related to  through the stochastic representation of an elliptically distributed
random vector in (35). See Fang, Kotz and Ng (1990) for details.
76 B.O. Bradley and M.S. Taqqu
The importance of the class of elliptical distributions to risk management can be seen in
the following theorem. It indicates that the standard approaches to risk management apply
to a linear portfolio with elliptically distributed risk factors.
Theorem 5.9 (Embrechts, McNeil and Straumann, 2001). Suppose X  En(,,) with
finite variances for all univariate marginals. Let
P = Z =
n
i=1
wiXi wi  R
be the set of all linear portfolios. Then:
(1) (Subadditivity of VaR.) For any two portfolios Z1,Z2  P and 0.5  < 1,
VaR(Z1 + Z2) VaR(Z1) + VaR(Z2).
(2) (Equivalence of variance and any other positive homogeneous risk measure.) Let  be
any real valued, positive homogeneous risk measure depending only on the distribution
of a random variable X. Then for Z1,Z2  P,
(Z1 - EZ1) (Z2 - EZ2)  2
Z1
2
Z2
.
(3) (Markowitz risk minimizing portfolio.) Let  be as in (2), but also translation invariant,
and let
E = Z =
n
i=1
wiXi wi  R,
n
i=1
wi = 1, EZ = r
be the subset of portfolios with the same expected return r. Then
argmin
ZE
(Z) = argmin
ZE
2
Z.
The theorem43 states that:
43 Because of the importance of Theorem 5.9 and because its proof is illuminating and straightforward we shall
sketch it. It is based on the observation that (Z1,Z2) is elliptical and so portfolios Z1, Z2 and Z1 + Z2 are all of
the same type. Let q, 1/2 <  < 1, denote the  quantile of the corresponding standardized distribution. Then
VaR(Z1) = EZ1 + Z1
q,
VaR(Z2) = EZ2 + Z2
q,
VaR(Z1 + Z2) = EZ1 + EZ2 + Z1+Z2
q
Ch. 2: Financial Risk and Heavy Tails 77
* For any linear portfolio of elliptical risk factors, VaR is a coherent measure of risk.
* If the risk factors are elliptical, the linear correlation is the canonical measure of depen-
dence.
* For elliptical risk factors, the Markowitz mean variance optimal portfolio, for a given
level of expected return, will be the same regardless of whether the risk measure is given
by the variance, VaR, expected shortfall or any other positive homogeneous, translation
invariant risk measure. Hence, all the usual techniques of portfolio theory and risk management
apply.
* It may be strange at first that the expected shortfall S(X), for example, which does
not involve subtraction of the mean (see (22)), can be used instead of the variance in
Markowitz' risk minimization portfolio theory. This is because one considers a set of
portfolios E, all of the same mean. Since S(X - EX) = S(X) - EX and since EX is
the same for all portfolios X in E, the term EX can be ignored.
Note that elliptical distributions are not required to be thin-tailed. The multivariate normal
is but one elliptical distribution. The risk manager may well feel that the risk factors under
consideration are better modelled using a heavy-tailed elliptical distribution.44 The usual
techniques then apply, but the risk of a linear portfolio will be greater than if the risk factors
were assumed multivariate normal.
6. Univariate extreme value theory
Managing extreme market risk is a goal of any financial institution or individual investor.
In an effort to guarantee solvency, financial regulators require most financial institutions to
maintain a minimum level of capital in reserve. The recommendation of the Basle Committee
(1995b) of a minimum capital reserve requirement based on VaR is an attempt to manage
extreme market risks. Recall that VaR is nothing more that a quantile of a probability
but Z1+Z2
Z1
+ Z2
and q > 0, proving (1). Next, note that there exists a > 0 such that Z1 - EZ1
d
=
a(Z2 -EZ2), so that a 1  2
1 2
2 . Since the risk measure  is assumed positive homogeneous and depends
only on the distribution of Z,
(Z1 - EZ1) = (a(Z2 - EZ2)) = a(Z2 - EZ2)
and hence
(Z1 - EZ1) (Z2 - EZ2)  a 1  2
Z1
2
Z2
(37)
which proves (2). Now consider only portfolios in E. Then (37) holds with EZ1 = EZ2 = r. However, using
translation invariance of , (Zj - r) = (Zj ) - r for j = 1,2. This gives
(Z1) (Z2)  2
Z1
2
Z2
proving (3).
44 In a recent paper, Lindskog (2000a) compares estimators for linear correlation showing that the standard
covariance estimator (17) performs poorly for heavy-tailed elliptical data. Several alternatives are proposed and
compared.
78 B.O. Bradley and M.S. Taqqu
distribution. The minimum capital reserve is then a multiple of this high quantile, usually
computed with  = 0.99. Therefore it is very important to attempt to model correctly the
tail of probability distribution of returns (profit and losses). The primary difficulty is that
we are trying to model events about which we know very little. By definition, these events
are rare. The model must allow for these rare but very damaging events. Extreme value
theory (EVT) approaches the modelling of these rare and damaging events in a statistically
sound way. Once the risks have been modelled they may be measured. We will use VaR
and Expected Shortfall to measure them.
Extreme value theory (EVT) has its roots in hydrology, where, for example, one needed
to compute how high a sea dyke had to be to guard against a 100 year storm. EVT has
recently found its way into the financial community. The reader interested in a solid
background may now consult various texts on EVT such as Embrechts, Klüppelberg and
Mikosch (1997), Reiss and Thomas (2001) and Beirlant, Teugels and Vynckier (1996).
For discussions of the use of EVT in risk management, see Embrechts (2000) and Diebold,
Schuermann and Stroughair (2000).
The modelling of extremes may be done in two different ways: modelling the maximum
of a collection of random variables, and modelling the largest values over some high
threshold. We start, for historical reasons, with the first method, called block maxima.
6.1. Limit law for maxima
The Fisher­Tippett theorem is one of two fundamental theorems in EVT. It does for the
maxima of i.i.d. random variables what the central limit theorem does for sums. It provides
the limit law for maxima.
Theorem 6.1 (Fisher­Tippett, 1928). Let (Xn) be a sequence of i.i.d. random variables
with distribution F. Let Mn = max(X1,...,Xn). If there exist norming constants cn > 0
and dn  R and some non-degenerate distribution function H such that
Mn - dn
cn
d
- H,
then H is one of the following three types:
Fr´echet (x) =
0, x 0,
exp -x,
x > 0,
 > 0,
Weibull (x) =
exp -(-x)
, x 0,
1, x > 0,
 > 0,
Gumbel (x) = exp -e-x
, x  R.
The distributions , , and  are called standard extreme value distributions. The expressions
given above are cumulative distribution functions. The Weibull is usually defined
Ch. 2: Financial Risk and Heavy Tails 79
Fig. 10. Densities of the generalized extreme value distribution H . Left: Weibull with  = -0.5. Middle: Gumbel
with  = 0. Right: Fréchet with  = 0.5.
as having support (0,) but, in the context of extreme value theory, it has support on
(-,0), as indicated in the theorem. These distributions are related:
X    lnX
   -
1
X
 .
A one-parameter representation of these distributions (due to Jenkinson and von Mises)
will be useful. The reparameterized version is called the generalized extreme value (GEV)
distribution.
H (x) =
exp -(1 + x)-1/
,  = 0,
exp -e-x
,  = 0,
where 1 + x > 0. The standard extreme value distributions , , and  follow by
taking  = -1 > 0,  = --1 < 0, and  = 0 respectively.45 There densities are sketched
in Figure 10. The parameter  is the shape parameter of H . Since for any random variable
X  FX and constants   R and  > 0, the distribution function of X =  + X is
given by FX(x) = FX((x - )/), we can add location and scale parameters to the above
parameterization, and consider
H,, (x) = H
x - 

.
If the Fisher­Tippett theorem holds, then we say that F is in the maximum domain of
attraction of H and write F  MDA(H). Most distributions in statistics are in MDA(H )
for some . If F  MDA(H ) and  = 0 or F  MDA(H ) and  < 0, then F is said to
be thin-tailed or short-tailed respectively. Thin-tailed distributions ( = 0) include the normal,
exponential, gamma and lognormal. Short-tailed distributions ( < 0) have a finite
45 Consider, for example, the Fréchet distribution where  = -1 > 0. Since the support of H is 1 + x > 0,
one has
H-1 (x) = exp{-(1 + -1x)-} = (1 + -1x)
for 1 + -1x > 0.
80 B.O. Bradley and M.S. Taqqu
right-hand end point and include the uniform and beta distributions. The heavy-tailed distributions,
those in the domain of attraction of the Fréchet distribution, F  MDA(H ), for
 > 0, are of particular interest in finance. They are characterized in the following theorem
due to Gnedenko.
Theorem 6.2 (Gnedenko, 1943). The distribution function F  MDA(H ) for  > 0 if and
only if F(x) = 1 - F(x) = x-1/ L(x) for some slowly varying function L.46
Distributions such as the Student-t, -stable and Pareto are in this class. Note that if X  F
with F  MDA(H ),  > 0 then all moments EX are infinite for  > 1/. Note also that
 < 1 corresponds to  > 1, where  is as in Theorem 6.1.
6.2. Block maxima method
We now explain the block maxima method, where one assumes in practice that the maximum
is distributed as H,, . The implementation of this method requires a great deal
of data. Let X1,X2,...,Xmn be daily (negative) returns and divide them into m adjacent
blocks of size n. Choose the block size n large enough so that our limiting theorem results
apply to M
(j)
n = max(X(j-1)n+1,...,X(j-1)n+n) for j = 1,...,m. Our data set must then
be long enough to allow for m blocks of length n. There are three parameters, ,  and ,
which need to be estimated, using for example maximum likelihood based on the extreme
value distribution. The value of m must be sufficiently large as well, to allow for a reasonable
confidence in the parameter estimation. This is the classic bias-variance trade-off
since for a finite data set, increasing the number of blocks m, which reduces the variance,
decreases the block size n, which increases the bias. Once the GEV model H,, is fit
using M(1)
n ,...,M(m)
n , we may estimate quantities of interest.
For example, assuming n = 261 trading days per year, we may want to find R261,k, the
daily loss we expect to be exceeded in one year every k years.47 If this loss is exceeded in
a given day, this day is viewed as an exceedance day and the year to which the day belongs
is regarded as an exceedance year. While an exceedance year has at least one exceedance
day, we are not concerned here with the total number of exceedance days in that year. This
would involve taking into consideration the propensity of extremes to form clusters. Since
we want M261 to be less than R261,k for k - 1 of k years, R261,k is the 1 - 1/k quantile of
M261:
R261,k = inf r P(M261 r) 1 -
1
k
. (38)
46 The function L is said to be slowly varying (at infinity) if
lim
x
L(tx)
L(x)
= 1, t > 0.
47 Note the obvious hydrological analogy: How high to build a sea dyke to guard against a k year storm.
Ch. 2: Financial Risk and Heavy Tails 81
If we assume that M261 has approximately the H,, distribution, the quantile R261,k is
given by
R261,k = H-1
,, 1 -
1
k
(39)
=  +


-ln 1 -
1
k
-
1 ,  = 0, (40)
since the inverse function of y = exp{-(1 + x)}-1/ is x = (1/)[(-lny)- - 1]. Confidence
intervals for R261,k may also be constructed using profile log-likelihood functions.
The idea is as follows. The GEV distribution H,, depends on three parameters.
Substitute R261,k for  using (40) and denote the reparameterized H as H,R261,k, after
some abuse of notation. Then obtain the log-likelihood L(,R261,k,|M1,...,Mm)
for our m observations from H,R261,k, . Take H0: R261,k = r as the null hypothesis
in an asymptotic likelihood ratio test and let 0 = (  R, R261,k = r,   R+) and
 = (  R, R261,k  R,   R+) be the constrained and unconstrained parameter spaces
respectively. Then under certain regularity conditions we have that
-2 sup
0
L(|M1,...,Mm) - sup

L(|M1,...,Mm)  2
1
as m   where  = (,R261,k,) and 2
1 is a chi-squared distribution with one
degree of freedom. Let L(^,r, ^) = sup0
L(|M1,...,Mm) and L(^,R261,k, ^) =
sup L(|M1,...,Mm) denote the constrained and unconstrained maximum log-likelihood
values respectively. The  confidence interval for R261,k is the set
r: L ^,r, ^ L ^,R261,k, ^ -
1
2
2
1 () ,
that is, the set r for which the null hypothesis cannot be rejected for level . See McNeil
(1998a) or Këllezi and Gilli (2000) for details.
Example 6.1. We have 7570 data points for the NASDAQ, which we subdivided into m =
31 blocks of roughly n = 261 trading days. (The last block, which corresponds to January
2001, has relatively few trading days, but was included because of the large fluctuations.)
Estimating the GEV distribution by maximum likelihood leads to ^ = 0.319, ^ = 2.80 and
^ = 1.38. The value of ^ corresponds to ^ = 1/^ = 3.14, which is in the expected range
for financial data. The GEV fit is not perfect (see Figure 11). Choosing k = 20 yields an
estimate of the twenty year return level R261,20 = 9.62%. Figure 12, which displays the
log-likelihood corresponding to the null-hypothesis that R261,20 = r, where r is displayed
on the abscissa, also provides the corresponding confidence interval.
82 B.O. Bradley and M.S. Taqqu
Fig. 11. The GEV distribution H^, ^,^
fitted using the 31 annual maxima of daily (negative, as %) NASDAQ
returns.
Fig. 12. The profile log-likelihood curve for the 20 year return level R261,20 for NASDAQ. The abscissa displays
return levels (as %) and the ordinate displays log-likelihoods. The point estimate R261,20 = 9.62% corresponds
to the location of the maximum and the asymmetric 95% confidence interval, computed using the profile
log-likelihood curve, is (6.79%,21.1%).
6.3. Using the block maxima method for stress testing
For the purpose of stress testing (worst case scenario), it is the high quantiles of the daily
return distribution F that we are interested in, not those of Mn. If the Xi  F have a
continuous distribution, we have
P(Mn Rn,k) = 1 -
1
k
.
If they are also i.i.d.,
P(Mn Rn,k) = P(X Rn,k)
n
,
where X  F, and hence
P(X Rn,k) = 1 -
1
k
1/n
. (41)
Ch. 2: Financial Risk and Heavy Tails 83
This means that Rn,k is the (1 - 1/k)1/n quantile of the marginal distribution F. Suppose
we would like to calculate VaR at very high quantiles for the purposes of stress testing. The
block size n has been fixed for the calibration of the model. This leaves the parameter k
for the Rn,k return level free. High  quantiles, x = F-1(), of F may then be computed
from (41) by choosing  = (1 - 1/k)1/n, that is k = 1/(1 - n). Hence
VaR(X) = Rn,k, where k =
1
1 - n
. (42)
For the NASDAQ data, our choice of k = 20, corresponds to
 = 0.9998 and VaR=0.9998(X) = R261,20 = 9.62%.
In practice  is given, and one chooses k = 1/(1 - n), then computes Rn,k using (40) and
thus one obtains VaR(X) = Rn,k.
We assumed independence but, in finance, this assumption is not realistic. At best, the
marginal distribution F can be viewed as stationary. For the extension of the Fisher­Tippett
theorem to stationary time series see Leadbetter, Lindgren and Rootzén (1983, 1997) and
McNeil (1998a). See McNeil (1998b) for a non-technical example pertaining to the block
maxima method and the market crash of 1987.
6.4. Peaks over threshold method
The more modern approach to modelling extreme events is to attempt to focus not only the
largest (maximum) events, but on all events greater than some large preset threshold. This
is referred to as peaks over threshold (POT) modelling. We will discuss two approaches
to POT modelling currently found in the literature. The first is a semi-parametric approach
based on a Hill type estimator of the tail index (Beirlant, Teugels and Vynckier, 1996;
Danielsson and de Vries, 1997, 2000; Mills, 1999). The second approach is a fully parametric
approach based on the generalized Pareto distribution (Embrechts, Klüppelberg and
Mikosch, 1997; McNeil and Saladin, 1997; Embrechts, Resnick and Samorodnitsky,1999).
6.4.1. Semiparametric approach
Recall that FX is in the maximum domain of attraction of the Fréchet distribution if and
only if FX(x) = x-L(x) for some slowly varying function L. Suppose FX is the distribution
function of a loss distribution over some time horizon, where we would like to
calculate a quantile based risk measure such as VaR. Assume for simplicity that the distribution
of large losses is of Pareto type
P(X > x) = cx,
 > 0, x > x0. (43)
The semiparametric approach uses a Hill type estimator for  and order statistics of historical
data to invert and solve for VaR.
84 B.O. Bradley and M.S. Taqqu
We first focus on VaR. Let X(1) X(2)  X(n) be the order statistics of an historical
sample of losses of size n, assumed i.i.d. with distribution FX. If X is of Pareto type in
the tail and X(k+1) is a high order statistic then for x > X(k+1),
FX(x)
FX(X(k+1))
=
x
X(k+1)
-
.
The empirical distribution function estimator FX(X(k+1)) = k/n suggests the following
estimator of FX in the upper tail,
FX(x) = 1 -
k
n
x
X(k+1)
-^
for x > X(k+1)
.
By inverting this relation, one can express x in terms of FX(x), so that fixing q =
FX(x) one gets48 x = VaRq(X). The value of q should be large, namely, q = FX(x) >
F(X(k+1)) = 1 - k/n. This yields
VaRq(X) = X(k+1) n
k
(1 - q)
-1/^
. (44)
We obtained an estimator for VaR but it depends on k through X(k+1), on the sample
size n and ^. To estimate , Hill (1975) proposed the following estimator ^(Hill) which is
also dependent on the order statistics and sample size:
^(Hill)
= ^(Hill)
k,n =
1
k
k
i=1
lnX(i)
- lnX(k+1)
-1
. (45)
The consistency and asymptotic normality properties of this ^(Hill) estimator are known
in the i.i.d. case and for certain stationary processes. There are however, many issues surrounding
Hill-type estimators, see for example Beirlant, Teugels and Vynckier (1996),
Embrechts, Klüppelberg and Mikosch (1997) and Drees, de Haan and Resnick (2000).
To obtain VaRq(X), one also needs to choose the threshold level X(k+1) or, equivalently,
k. Danielsson et al. (2001) provide an optimal choice for k by means of a two stage
bootstrap method. Even in this case, however, optimal means merely minimizing the asymptotic
mean squared error, which leaves the user uncertain as to how to proceed in the
finite sample case. Traditionally the choice of k is done visually by constructing a Hill plot.
The Hill plot {(k, ^
(Hill)
k,n ): k = 1,...,n-1} is a visual check for the optimal choice of k.
The choice of k and therefore of ^
(Hill)
k,n , is inferred from a stable region of the plot since
48 We write here VaRq and not VaR since now  represents the heavy-tail exponent.
Ch. 2: Financial Risk and Heavy Tails 85
Fig. 13. Hill plots for the NASDAQ data set. Left: The Hill plot {(k, ^
(Hill)
k,n ): k = 1,... ,n-1}. Right: The AltHill
plot{(, ^
(Hill)
n ,n
): 0  < 1}. The Hill plot is difficult to read, whereas the AltHill plot gives the user an estimate
of ^AltHill  3.
in the Pareto case, where (43) holds, ^(Hill)
n-1,n is the maximum likelihood estimator for . In
the more general case
1 - F(x)  xL(x),
x  ,  > 0, (46)
where L is a slowly varying function, the traditional Hill plot is often difficult to interpret.
Resnick and St˘aric˘a (1997) suggest an alternative plot, called an AltHill plot by plotting
{(, ^(Hill)
n ,n
): 0  < 1} where n denotes the smallest integer greater than or equal
to n . This plot has the advantage of stretching the left-hand side of the plot, which corresponds
to smaller values of k, often making the choice of k easier. See Figure 13 for
examples of the Hill and AltHill plots for the ordered negative returns X(j) for the NAS-
DAQ.
6.4.2. Fully parametric approach
The fully parametric approach uses the generalized Pareto distribution (GPD) and the second
fundamental theorem in EVT by Pickands, Balkema and de Haan. The GPD is a twoparameter
distribution
G,(x) =



1 - 1 +
x

-1/
,  = 0,
1 - exp -
x

,  = 0,
where an additional parameter  > 0 has been introduced. The support of G,(x) is x 0
for  0 and 0 x -/ for  < 0. The distribution is heavy-tailed when  > 0. GPD
distributions with  = 1 are displayed in Figure 14.
86 B.O. Bradley and M.S. Taqqu
Fig. 14. GPD distribution functions G,, all with  = 1. Left:  = -0.5, Middle:  = 0, Right:  = 0.5, which
corresponds to a location adjusted Pareto distribution with  = 2.
Definition 6.1. Let X  F with right-end-point xF = sup{x  R | F(x) < 1} . For
any high threshold u < xF define the excess distribution function
Fu(x) = P(X - u x|X > u) for 0 x < xF - u. (47)
The mean excess function of X is then
eX(u) = E(X - u|X > u). (48)
If X has exceeded the high level u, Fu(x) measures the probability that it did not exceed
it by more than x. Note that for 0 x < xF - u, we may express Fu(x) in terms of F,
Fu(x) =
F(u + x) - F(u)
1 - F(u)
,
and the mean excess function eX(u) may be expressed as a function of the excess distribution
Fu as
eX(u) =
xF -u
0
x dFu(x).
The following theorem relates Fu to a GPD through the maximum domain of attraction of
a GEV distribution. In fact, it completely characterizes the maximum domain of attraction
of H .
Theorem 6.3 (Pickands, 1975, Balkema and de Haan, 1974). Let X  F. Then for every
  R, X  MDA(H ) if and only if
lim
uxF
sup
0<x<xF -u
Fu(x) - G,(u)(x) = 0
for some positive function .
Ch. 2: Financial Risk and Heavy Tails 87
This theorem says that the excess distribution Fu may be replaced by the GPD distribution
G when u is very large. To see how it can be used, note that by (47) above, we may
write
F(x) = F (u)Fu(x - u) (49)
for x > u. Assuming that u is sufficiently large, we may then approximate Fu by G,(u)
and use the empirical estimator, for F(u),
F(u) =
Nu
n
, where Nu =
n
i=1
1{Xi>u}
and where n is the total number of observations. The upper tail of F(x) may then be
estimated by
F(x) = 1 - F = 1 -
Nu
n
1 + ^
x - u
^
-1/^
for all x > u. (50)
This way of doing things allows us to extrapolate beyond the available data which would
not be possible had we chosen an empirical estimator for F(x), x > u. We can therefore
deal with potentially catastrophic events which have not yet occurred.
The parameters  and  of the GPD G,(u) may be estimated by using, for example,
maximum likelihood once the threshold u has been chosen. The data points that are used
in the maximum likelihood estimation are Xi1 - u,...,Xik - u where Xi1 ,...,Xik are the
observations that exceed u. Again there is a bias-variance trade-off in the choice of u. To
choose a value for u, a graphical tool known as the mean excess plot (u,eX(u)) is often
used.
The mean excess plot relies on the following theorem for generalized Pareto distribu-
tions.
Theorem 6.4 (Embrechts, Klüppelberg and Mikosch, 1997). Suppose X has GPD distribution
with  < 1 and . Then, for u < xF ,
eX(u) =
 + u
1 - 
,  + u > 0.
The restriction  < 1 implies that the heavy-tailed distribution must have at least a finite
mean.
If the threshold u is large enough so that Fu is approximately G, then, by Theorem
6.4, the plot (u,e(u)) is linear in u. How then is one to pick u? The mean excess
plot is a graphical tool for examining the relationship between the possible threshold u and
88 B.O. Bradley and M.S. Taqqu
the mean excess function eX(u) and checking the values of u where there is linearity. In
practice it is not eX(u), but its sample version
^eX(u) =
n
i=1(Xi - u)+
n
i=1 1{Xi>u}
which is plotted against u. After using the mean excess plot to pick the upper threshold u
one obtains an estimator of the tail of the distribution by applying (50). For the NASDAQ
data, since linearity seems to start at relatively small values of u (Figure 15), we choose
u = 1.59 which corresponds to the 95% of the empirical NASDAQ return distribution.
To obtain VaR(X) for VaR(X) > u, one simply inverts the tail estimator (50), which
yields
VaR(X) = u +
^
^
n
Nu
(1 - )
-^
- 1 . (51)
Since expected shortfall is a risk measure with better technical properties than VaR we
would like to find an estimator for it which uses our GPD model of the tail. Recalling the
definitions of the expected shortfall (22) and the mean excess function (48) we have that
S(X) = VaR(X) + eX VaR(X) .
Since the excess distribution Fu is approximated by a GPD G,(u) with  < 1 then, applying
Theorem 6.4, we get for VaR(X) > u,
S(X) = VaR(X) +
 + (VaR(X) - u)
1 - 
=
 + VaR(X) - u
1 - 
.
This suggests the following estimator for expected shortfall,
S(X) =
^x
1 - ^
+
^ - ^u
1 - ^
, (52)
Fig. 15. Sample mean excess plot (u, ^eX(u)) for NASDAQ.
Ch. 2: Financial Risk and Heavy Tails 89
where ^x = VaR(X) may be obtained by using (51). As in the case of block maxima,
confidence intervals for VaR and S may be constructed using profile log-likelihood func-
tions.
6.4.3. Numerical illustration
To illustrate the usefulness of EVT in risk management, we consider the following example.
Let X1,...,Xn represent the daily negative returns of the NASDAQ index over most
of its history from February 1971 to February 2001, which gives a time series of n = 7570
data points.
The price and return series are displayed in Figure 16. Let X(1)  X(n) be the
corresponding order statistics. Suppose the risk manager wants to obtain value at risk and
expected shortfall estimates of the returns on the index at some high quantile. Assume that
{Xi}n
i=1 are i.i.d. so that Theorem 6.1 holds. Then, using Theorem 6.3, we model the tail
of the excess distribution Fu by a GPD G, and use (49) to model the distribution F(x)
Fig. 16. Time series of NASDAQ daily prices, (log) returns and annual maxima and minima daily returns given
as a percent for the period February 1971 (when it was created) to February 2001. If Pt is the price (level) at
time t, the returns are defined as 100ln(Pt /Pt-1) and expressed as %. The crash of 1987 is clearly visible. The
NASDAQ price level peaked in March of 2000.
90 B.O. Bradley and M.S. Taqqu
of the observations for all x > u. We use Theorem 6.4 and the sample mean excess plot,
Figure 15, to pick the high threshold u = 1.59%. This leaves us with k = 379 observations
from which we estimate the parameters of the GPD by maximum likelihood. The
estimates give  = 0.189 and  = 0.915. The model fit is checked by using a QQ plot
displayed in Figure 17. Accepting the model, we go on to calculate the value at risk and
expected shortfall for various high quantiles  by using (51) and (52). The results for the
NASDAQ are plotted in Figure 18 (solid lines). If one had assumed that the observations
were normally distributed (dashed lines), both the VaR and the expected shortfall would
have been significantly underestimated for high quantiles.
For example, at the  = 0.99 confidence level, VaR(X) = 6.59% under the normal
model versus VaR(X) = 8.19% for the GPD model. For the expected shortfall, the difference
is even more dramatic. For the normal model, S(X) = 7.09% versus S(X) = 10.8%
for the GPD model. This is to be expected, since under the assumption of normality it may
be shown (Embrechts, Klüppelberg and Mikosch, 1997) that
S
VaR
 1 as   1-
,
Fig. 17. For the NASDAQ return data (as %), there were 379 exceedances above the high threshold u = 1.59%.
These are fitted with a GPD distribution G^, ^
with ^ = 0.189 and ^ = 0.915. Left: The fitted GPD distribution
(dark curve) and the empirical one (dotted curve). Right: QQ-plot of sample quantiles versus the quantiles of the
fitted G^, ^
distribution.
Fig. 18. Risk estimates for NASDAQ in percent returns versus . Left: Value at risk VaR, for GPD (solid) and
normal (dashed). Right: Expected shortfall S, for GDP (solid) and normal (dashed). The parameters of the GPD
are fitted by maximum likelihood using 30 years of data. The sample mean and volatility of the normal distribution
are computed by (16) using the most recent year of daily observations.
Ch. 2: Financial Risk and Heavy Tails 91
whereas for the GPD model
S
VaR
-
1
1 - 
as   1-
.
These results indicate that for very high quantiles, the expected shortfall S and the value
at risk VaR are comparable under normality, but for the GPD with  < 1, S tends to be
larger than VaR.
6.4.4. A GARCH-EVT model for risk
In order to invoke Theorems 6.1 and 6.3 in the numerical illustration above it was necessary
to assume that the (negative) returns {Xt}tZ were i.i.d. However, from inspection of Figures
16 and 19, it is apparent that this assumption is unrealistic. The time series of returns
is characterized by periods of varying volatility, that is, the time series is heteroscedastic.
The heteroscadicity of the time series may cause problems for the estimation of the parameters
of the GPD model since we would expect the high threshold u to be violated more
often during periods of high volatility. Smith (2000) suggests using Bayesian techniques
to model time-varying GPD parameters. In this section, we review a model proposed by
McNeil and Frey (2000) which extends the EVT methodology to models of financial time
series that allow for stochastic volatility and apply this model to the NASDAQ data set.
Fig. 19. Sample auto correlation functions with lags on the abscissa and sample autocorrelation on the ordinate:
returns (top left), squared returns (bottom left), GARCH innovations (top right), squared GARCH innovations
(bottom right). The sample consists of 1000 daily returns for the NASDAQ ending February 2001. Horizontal
lines indicate the 95% confidence bands (1.96/

n) corresponding to Gaussian white noise.
92 B.O. Bradley and M.S. Taqqu
Recall from Section 3.2.2 that the standard GARCH(1,1) model is given by49
Xt = tZt, where Zt  FZ i.i.d., (53)
2
t = 0 + 1X2
t-1 + 12
t-1. (54)
Since the time t + 1 volatility t+1 is known at time t we have that
VaR(Xt+1|Ft) := inf x  R | FXt+1|Ft (x)  = t+1z, (55)
where z = F-1
Z (). The same argument shows that the conditional expected shortfall
S(Xt+1|Ft) := E Xt+1|Xt+1 > VaR(Xt+1|Ft),Ft = t+1E(Z|Z > z).
Traditionally the innovation distribution FZ is assumed normal. Figures 6 and 20 show
that this assumption may still underestimate the tails of the loss portion of the distribution.
McNeil and Frey propose a two step procedure to estimate VaR and expected shortfall
of the conditional distribution. First they use a GARCH(1,1) model for the volatility of
the (negative) return series {Xt}. This gives a series of model implied innovations Zt =
Xt /t . Second, EVT is used to model the tails of the distribution of these innovations.
This approach has the obvious benefit that the resulting innovations Zt are much closer to
satisfying the requirements of Theorems 6.1 and 6.3 than is the original series. We illustrate
the methodology with an example using the NASDAQ data.
(1) Let (xt-n+1,...,xt-1,xt) be n daily negative returns of the NASDAQ. We take50
n = 1000 and use pseudo-maximum-likelihood (PML) to estimate the model parameters
^ = ( ^0, ^1, ^1) in (54) under the assumption51 that FZ is normal in (53). The
parameter vector ^ depends on the true distribution of (Xt-n+1,...,Xt-1,Xt ), which
Fig. 20. QQ plots versus the normal for returns (left) and innovations (right) in Figure 19. Notice that the lower
(loss) tail of the innovations are still heavier than the normal distribution.
49 Since the NASDAQ series appears to have a zero conditional mean we do not set Xt = t + t Zt and model
the mean t , for example as an AR(1) process t = Xt-1.
50 We keep the sample size moderate in order to avoid the IGARCH effect, that is 1 + 1 = 1, corresponding to
non-stationarity. See Mikosch and St˘aric˘a (2000) for details.
51 The term pseudo refers to the fact that one is not maximizing the true likelihood.
Ch. 2: Financial Risk and Heavy Tails 93
is assumed stationary, and on the distribution FZ used to compute the likelihood function.52
When we assume FZ is normal we fit a model whose distributional assumptions
we do not believe. Under standard regularity conditions this is justified since ^ is a consistent
estimator of  (in fact, asymptotically normal) even if FZ is non-normal. See
Gouriéroux (1997) and references therein for details.
(2) The model innovations (zt-n+1,...,zt-1,zt ) = (xt-n+1/^t-n+1,...,xt-1/^t-1,
xt/^t ) are now calculated. If the model is tenable, these innovations should be i.i.d.
Figure 19 shows that while the i.i.d. assumption is not realistic for the series of returns,
it is defensible for the series of innovations.53 While the returns appear uncorrelated,
their squares clearly are not, and hence the returns are dependent. The GARCH innovations
and their squares appear uncorrelated. The i.i.d. assumption is therefore more
tenable.
(3) Examination of the QQ plot of the innovations in Figure 20 reveals that the loss tail is
heavier than that of the normal. Therefore the EVT tools of Section 6.4.2 are now applied
to the innovations (zt-n+1,...,zt-1,zt). Let z(1)  z(n) be the order statistics
of the innovation sample. We choose the threshold u = 1.79, again corresponding
to the 95% of the empirical distribution of innovations, which leaves k = 50 observations
(z(n-k+1),...,z(n)), from which to estimate the GPD parameters by maximum
likelihood. The estimates give  = 0.323 and  = 0.364.
Observe that  = 0.323 corresponds to a heavier tail than  = 0.189 which we found in
Section 6.4.3. We are fitting here, however, over a particularly volatile period of 1000 days
of the NASDAQ ending February 2001, whereas in Section 6.4.3, we considered nearly 30
Fig. 21. Backtest results for the GARCH-EVT methodology of McNeil and Frey. Under the assumption that the
model correctly estimates the conditional quantiles we expect violations 5% and 1% of the time for  = 0.95 and
 = 0.99 respectively. VaR for  = 0.95 and  = 0.99 are given by the solid and dotted lines respectively. We
obtain 5.8% violations of the  = 0.95 level and 1% violations of the  = 0.99 level.
52 The condition 1 + 1 < 1 is sufficient for stationarity of the GARCH model. We found ^0 = 0.080, ^1 =
0.181 and ^1 = 0.811. However, as indicated in the sequel, the GARCH model is constantly updated, and hence
is never used on an infinite horizon.
53 Ljung-Box tests also found no evidence against the i.i.d. assumption for the innovations.
94 B.O. Bradley and M.S. Taqqu
years worth of returns where for the majority of the time the NASDAQ was significantly
less volatile (see Figure 16).
Since the model is assumed stationary, we could, in principle, use the estimated GARCH
parameters to compute ^t+1|Ft using (54) for t beyond February 2001. Using z corresponding
to the GPD distribution G,, we would obtain, by using (55), VaR(Xt+1|Ft)
for t beyond February 2001. In practice, however, stationarity is not always assured and in
any case one wants to use the most recent data available in order to calibrate the model.
In order to backtest the methodology we use the most recent 500 days in our NASDAQ
data set. For each day, t + 1, in this data set we use the previous n = 1000 days (negative)
returns (Xt-n+1,...,Xt-1,Xt ) to calibrate the model and estimate VaR(Xt+1|Ft) for
 = 0.95 and  = 0.99 using the steps above. We compare VaR(Xt+1|Ft) with the actual
loss xt+1. A violation, at the  level, is said to occur whenever xt+1 > VaR(Xt+1|Ft).
Results for the period ending February 2001 are given in Figure 21.
7. Stable Paretian models
The works of Mandelbrot (1963) and Fama (1965) introduced the use of stable distributions
to finance. The excessively peaked and heavy-tailed nature of the return distribution
led the authors to reject the standard hypothesis of normally distributed returns in favor of
the stable distribution. Since this time, the stable distribution has been used to model both
the unconditional, and conditional return distributions. In addition, portfolio theories and
market equilibrium models have been constructed using it. For an in depth introduction to
the general properties of stable distributions see Samorodnitsky and Taqqu (1994) and the
upcoming text Nolan (2001). A major reference for applications in finance is Rachev and
Mittnik (2000).
In Definition 3.2, the stable distribution S(,,) is defined as the limiting distribution
of the sum of i.i.d. random variables. Like the normal distribution, stable distributions are
closed under addition, and are often defined by this property. Recall that if X1  N(1,2
1 )
and X2  N(2,2
2 ) are independent then X1 +X2  N(1 +2,2
1 +2
2 ). Similarly, for
stable random variables, if X1  S(1,1,1) and X2  S(2,2,2) are independent,
then X1 + X2  S(,,) where
 = 
1 + 
2
1/
,  =
1
1 + 2
2

1 + 
2
,  = 1 + 2.
It is in this sense that the stable distribution is a natural heavy-tailed alternative to the
normal distribution. However, a common criticism of the stable distribution is that their
tails are too heavy. One has P(X > x)  cx- as x  . For 0 <  < 2, this implies
that E|X|p <  if 0 < p < . In particular, EX2 = , that is, all non-Gaussian stable
distributions have infinite variance.
The stable distributions can be defined and parameterized in different ways. One way
to specify a stable distribution is through its characteristic function. This is helpful since
Ch. 2: Financial Risk and Heavy Tails 95
in general there exists no closed form for the probability density function,54 which historically,
has been an impediment to their widespread use. Today, however, there are efficient
computer programs to evaluate their densities using fast Fourier transform methods
(Rachev and Mittnik, 2000; Nolan, 2001).
Definition 7.1. A random variable X is said to have a stable distribution if there are parameters
  (0,2],   [0,),   [-1,1] and   R such that its characteristic function
has the following form:
X(t) =



exp -
|t|
1 - i(signt)tan

2
+ it for  = 1,
exp -|t| 1 + i
2

(signt)ln|t| + it for  = 1.
(56)
If both the skewness and location parameters  and  are zero, X is said to be symmetric
stable, which is denoted X  SS, and its characteristic function takes the simple form
X(t) = e-|t|
.
If X  SS, then it is characterized completely by its index of stability  and its scale
parameter . If  = 2, the Gaussian case, then the scale parameter is  = 1
2 Var(X).
7.1. Stable portfolio theory
In Section 2.2 we introduced the mean­variance portfolio theory of Markowitz. The model
assumed that the distribution of asset returns is multivariate normal, and provides efficient
portfolios, that is, portfolios with maximum expected return for a given level of risk,
where risk is measured by the variance of the portfolio. It is possible to extend the ideas
of portfolio theory to the case where asset returns have a multivariate stable distribution,
even though, variances are now infinite. We need first to define a stable random vector and
specify its characteristic function.
Definition 7.2. The random vector X = (X1,...,Xn) is said to be a stable random vector
in Rn if for any a,b > 0 there exists c > 0 and d  Rn such that
aX1 + bX2
d
= cX + d, (57)
where Xj , j = 1,2, are independent copies of X.
54 The exceptions to this rule are the distributions S2(,0,), S1(,0,),and S1/2(,1,) which correspond
to the Gaussian, Cauchy and Lévy distributions respectively.
96 B.O. Bradley and M.S. Taqqu
The constants in (57) are related by c = a +b, where   (0,2] is the index of stability.
Setting n = 1 in (57) yields one of the alternate definitions of a stable random variable
alluded to earlier. In the case of a stable random vector, the scale and skewness parameters
 and  are replaced by a finite measure X on the unit hyperspherein Rn. For convenience
here, let (,) denote the inner product so that (t,s) = n
i=1 tisi .55
Theorem 7.1. Let 0 <  < 2. Then X = (X1,...,Xn) is a stable random vector with
index of stability  if and only if there exists a finite measure X on the unit hypersphere
Sn = {s  Rn| s = 1} and a vector   Rn such that
 (t) =



exp -
Sn
(t,s)

1 - isign (t,s) tan

2
X(ds) + i(t,) ,  = 1,
exp -
Sn
(t,s) 1 + i
2

sign (t,s) ln (t,s) X(ds) + i(t,) ,  = 1.
(58)
The pair (X,) is unique.
The measure X is called the spectral measure of the stable random vector X and specifies
the dependence structure. If X is SS in Rn, then the characteristic function takes the
simple form
 (t) = exp -
Sn
(t,s)

X(ds) ,
where  is the unique symmetric spectral measure. The expression in (58) for the characteristic
function is also valid for the normal case  = 2. When  = 2, it reduces to
 2(t) = exp{- Sn
|(t,s)|2X(ds)} but in this case X is no longer unique. To get a feeling
for X, suppose X = (X1,X2) and that the distribution is Gaussian. Then
S2
(t,s)
2
(X1,X2)(ds) =
S2
(t1s1 + t2s2)
2
(X1,X2)(ds)
= t2
1 2
1 + 2t1t21,2 + t2
1 2
1 ,
where
2
i =
S2
s2
i (X1,X2)(ds), i = 1,2, and 1,2 =
S2
s1s2(X1,X2)(ds),
55 Previously we wrote tTs instead of (t,s).
Ch. 2: Financial Risk and Heavy Tails 97
and where integration over the circle S2 means integration on {s = (s1,s2) | s2
1 + s2
2 = 1}.
One recognizes the normal characteristic function with VarX1 = 22
1 , VarX2 = 22
2 and
Cov(X1,X2) = 21,2. Since different choices of (X1,X2) can yield the same values for 2
1 ,
2
2 and 1,2, the choice of X is not unique in the Gaussian case.
As in the case of a normal random vector, if X is multivariate stable with index of
stability 0 <  < 2, then all linear combinations of the components of X are stable with
the same . So, if X is a stable random vector in Rn, and w  Rn, we know that Y =
(w,X) = n
i=1 wiXi is S(Y ,Y ,Y ). Using the characteristic function (58), it can be
shown [see Samorodnitsky and Taqqu (1994), Example 2.3.4], that
Y =
Sn
(w,s)

X(ds)
1/
, (59)
Y =
Sn
|(w,s)| sign(w,s)X(ds)
Sn
|(w,s)|X(ds)
, (60)
Y =



(w,) for  = 1,
(w,) -
2
 Sn
(w,s)ln|(w,s)|X(ds) for  = 1.
(61)
In the mean­variance portfolio theory, the risk to be minimized for any level of expected
return is given by the portfolios' variance. If the asset returns are assumed multivariate
stable with index of stability 0 <  < 2 then the variance is infinite and cannot be used. In
the stable portfolio theory, it is assumed that 1 <  < 2, EX =  and that X -   SS.
Let w be the vector of weights for the risky portfolio Xp = (w,X). Given the relationship
between the scale parameter and the variance in the Gaussian case (that is, stable with
 = 2), it is natural to use the scale parameter Xp of the resulting stable distribution
instead of the standard deviation. It is given by (59). This brings us to the corresponding
stable portfolio problem:
min
w
Xp =
Sn
(w,s)

X(ds)
1/
such that (w,) a, (62)
(w,e) = 1.
The risk measure Xp = (w,X) is a convex function of w and the problem is generally
solved using sequential quadratic programming. See Belkacem (1997) and Rachev and
Mittnik (2000) and references therein for details of the procedure and on the estimation
of the index of stability, spectral measure and scale parameters. If a risk free asset is included
in the asset universe, then we end up with a maximization problem similar to (2) in
Section 2.2, but where the risk measure is the scale parameter Xp of the risky portfolio.
98 B.O. Bradley and M.S. Taqqu
7.2. Stable asset pricing
Since there exists a portfolio theory under the assumption of a multivariate stable distribution
of asset returns (1 <  < 2), it is natural to ask whether there exists an analogous
CAPM. The answer is positive, and it was first introduced by Fama (1970). For recent
descriptions of the stable CAPM see Belkacem, Lévy Véhel and Walter (1996) and, of
course, Rachev and Mittnik (2000).
The assumptions behind the stable CAPM are the same as in the Gaussian case in Section
2.3 with the assumption of joint normality of asset returns replaced by that of jointly
stable asset returns with index of stability   (1,2). That is, we assume EX =  and that
X -   SS. Recall from the traditional CAPM and Equations (3) and (4), that the expected
premium of holding the risky asset i over the riskless asset is proportional to the
expected premium of holding the market portfolio over the riskless asset. The constant of
proportionality was the risky assets beta given by (4). In the stable CAPM, we require
an alternative measure of dependence since covariances do not exist. Naturally, the scale
parameter  replaces the standard deviation.
The covariation is a natural alternative to the covariance in the stable case when
1 <  < 2. This measure possesses many, but not all, of the useful properties of covariance
in the Gaussian case. We define and present several of the properties of covariation.
Details may be found in Samorodnitsky and Taqqu (1994) and Rachev and Mittnik (2000).
Definition 7.3. Let X1 and X2 be jointly SS with 1 <  2 and let (X1,X2) be the
spectral measure of the random vector (X1,X2). The covariation of X1 on X2 is given by
[X1,X2] =
S2
s1s
-1
2 (X1,X2)(ds) (63)
where s p denotes the signed power s p = |s|p(signs).
In the Gaussian case  = 2 it reduces to
[X1,X2]2 =
1
2
Cov(X1,X2). (64)
Note, however, that whereas in the Gaussian case the dependence structure is fully characterized
by the covariance, in the stable one needs to use X, and the covariation does
not fully characterize the dependence structure. We now derive the stable CAPM under the
preceding assumptions, following Belkacem, Lévy Véhel and Walter (1996).
Consider a portfolio of a riskless asset with rate of return r and a risky asset Xi with
weights w and 1 - w respectively. The expected rate of return of the portfolio Xp = wr +
Ch. 2: Financial Risk and Heavy Tails 99
(1 - w)Xi is then EXp = wr + (1 - w)EXi , and its risk, as given by its scale parameter,
is p = (1 - w)i .56 The risk-return trade-off is then given by
EXp = r +
EXi - r
i
p (65)
after setting w = 1 - p/i . Under the assumptions of CAPM, investors have homogeneous
beliefs, that is, they all agree on the multivariate stable parameters. This means that
all investors hold the market portfolio (as in Section 2.3) as their risky asset and the riskreturn
trade-off (65) becomes
EXp = r +
EXM - r
M
p, (66)
where XM and M are the rate of return and scale parameter respectively of the market.
Now consider the suboptimal portfolio Xp = wXi + (1 - w)XM obtained by adding to
the market portfolio a certain position in asset i (the portfolio is optimal if w = 0). Since
X -   SS we know that Xi - i and XM - M are jointly SS. By properties of
symmetric stable random vectors this means that Xp  S(p,0,p), where the scale and
location parameters are given by (59) and (61), that is

p =
S2
ws1 + (1 - w)s2

(Xi,Xp)(ds1,ds2), (67)
p = EXp = wi + (1 - w)M , (68)
respectively. Differentiating with respect to w gives
p
w
= i - M, (69)
p
w
=
1
-1
p

p
w
=
1
-1
p S2
(s1 - s2) ws1 + (1 - w)s2
-1
(Xi,Xp)(ds1,ds2). (70)
So evaluating (69) and (70) at w = 0 and using Definition 7.3 we get
p
p w=0
=
p
w
p
w w=0
=
-1
M (i - M)
[Xi,XM ] - 
M
, (71)
56 Note that if X  S(,,) then aX + b  S(|a|,sign(a),a + b) if 1 <  < 2.
100 B.O. Bradley and M.S. Taqqu
Fig. 22. The stable efficient frontier. The portfolio Xp = wXi + (1 - w)XM is suboptimal, and hence must be
dominated by the efficient frontier.
since at w = 0 the portfolio Xp becomes XM and p becomes M . Moreover, in market
equilibrium the trade-off between risk and return is given by (66), so that the slope
p/p at w = 0 is given by (M - r)/M (see Figure 22). Hence
M - r
M
=
-1
M (i - M)
[Xi,XM] - 
M
. (72)
This may be rewritten in the familiar CAPM form (3) as
E(Xi - r) = iE(XM - r),
where now, in the stable case,
i =
[Xi,XM]

M
. (73)
Note that if we assume Gaussian returns, then X -   SS with  = 2, and by using
(64), we recover
i =
Cov(Xi,XM )
Var(XM)
,
that is, the traditional CAPM result.
Acknowledgments
We would like to thank Paul Embrechts and Filip Lindskog for many valuable comments
which led to an improved exposition of this material. This research was partially supported
by the NSF Grant ANI-9805623 at Boston University.
Ch. 2: Financial Risk and Heavy Tails 101
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Chapter 3
MODELING FINANCIAL DATA WITH STABLE DISTRIBUTIONS
JOHN P. NOLAN
Department of Mathematics and Statistics, American University
Contents
Abstract 106
1. Basic facts about stable distributions 107
2. Appropriateness of stable models 111
3. Computation, simulation, estimation and diagnostics 113
4. Applications to financial data 114
5. Multivariate stable distributions 116
6. Multivariate computation, simulation, estimation and diagnostics 121
7. Multivariate application 124
8. Classes of multivariate stable distributions 126
9. Operator stable distributions 128
10. Discussion 128
References 129
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
106 J.P. Nolan
Abstract
Stable distributions are a class of probability distributions that allow heavy tails and skewness.
In addition to theoretical reasons for using stable laws, they are a rich family that
can accurately model different kinds of financial data. We review the basic facts, describe
programs that make it practical to use stable distributions, and give examples of these distributions
in finance. A non-technical introduction to multivariate stable laws is also given.
Ch. 3: Modeling Financial Data 107
1. Basic facts about stable distributions
Stable distributions are a class of probability laws that have intriguing theoretical and practical
properties. Their applications to financial modeling comes from the fact that they
generalize the normal (Gaussian) distribution and allow heavy tails and skewness, which
are frequently seen in financial data. In this chapter, we focus on the basic definition and
properties of stable laws, and show how they can be used in practice. We give no proofs;
interested readers can find these in Zolotarev (1986), Samorodnitsky and Taqqu (1994),
Janicki and Weron (1994), Uchaikin and Zolotarev (1999), Rachev and Mittnik (2000) and
Nolan (2003).
The defining characteristic, and reason for the term stable, is that they retain their shape
(up to scale and shift) under addition: if X,X1,X2,...,Xn are independent, identically
distributed stable random variables, then for every n
X1 + X2 +  + Xn
d
= cnX + dn (1)
for some constants cn > 0 and dn. The symbol
d
= means equality in distribution, i.e., the
right- and left-hand sides have the same distribution. The law is called strictly stable if dn =
0 for all n. Some authors use the term sum stable to emphasize the stability under addition
and to distinguish it from other concepts, e.g., max-stable, min-stable, etc. The normal
distributions satisfy this property: the sum of normals is normal. Likewise the Cauchy laws
and the Lévy laws (see below) satisfy this property. The class of all laws that satisfy (1)
is described by four parameters, which we call (,,,), see Figure 1 for some density
graphs. In general, there are no closed form formulas for stable densities f and cumulative
distribution functions F, but there are now reliable computer programs for working with
these laws.
The parameter  is called the index of the law or the index of stability or characteristic
exponent and must be in the range 0 <  2. The constant cn in (1) must be of the form
n1/. The parameter  is called the skewness of the law, and must be in the range -1
 1. If  = 0, the distribution is symmetric, if  > 0 it is skewed toward the right, if
 < 0, it is skewed toward the left. The parameters  and  determine the shape of the
distribution. The parameter  is a scale parameter, it can be any positive number. The
parameter  is a location parameter, it shifts the distribution right if  > 0, and left if
 < 0.
A confusing issue with stable parameters is that there are multiple definitions of what
the parameters mean. There are at least 10 different definitions of stable parameters, see
Nolan (2003). The reader should be careful in reading the literature and verify what parameterization
is being used. We will describe two different parameterizations, which we
denote by S(,,,0;0) and S(,,,1;1). The first is what we will use in all our
applications, because it has better numerical behavior and intuitive meaning. The second
parameterization is more commonly used in the literature, so it is important to understand
it. The parameters ,  and  have the same meaning in the two parameterizations, only
108 J.P. Nolan
Fig. 1. Standardized stable densities for different  and  in the S(,,1,0;0) parameterization. The top
graph includes a Lévy(1,-1) = S(1/2,1,1,0;0) = S(1/2,1,1,-1;1) graph and the middle graph includes a
Cauchy(1,0) = S(1,0,1,0;0) = S(1,0,1,0;1) graph.
the location parameter is different. To distinguish between the two, we will sometimes
use a subscript to indicate which parameterization is being used: 0 for the location parameter
in the S(,,,0;0) parameterization and 1 for the location parameter in the
S(,,,1;1) parameterization.
Ch. 3: Modeling Financial Data 109
Definition 1. A random variable X is S(,,,0;0) if it has characteristic function
E exp(iuX) (2)
=



exp - 
|u|
1 + i tan

2
(signu) | u|1-
1 + i0u ,  = 1,
exp - |u| 1 + i
2

(signu)ln  |u| + i0u ,  = 1.
Definition 2. A random variable X is S(,,,1;1) if it has characteristic function
E exp(iuX) =



exp - 
|u|
1 - i tan

2
(signu) + i1u ,  = 1,
exp - |u| 1 + i
2

(signu)ln|u| + i1u ,  = 1.
(3)
The location parameters are related by
0 =



1 +  tan

2
,  = 1,
1 + 
2

 ln,  = 1,
1 =



0 -  tan

2
,  = 1,
0 - 
2

 ln,  = 1.
(4)
Note that if  = 0, the parameterizations coincide. When  = 0, the parameterizations differ
by a shift  tan 
2 , which gets infinitely large as   1. In particular, the mode of
a S(,,,1;1) density tends toward  (if sign( - 1) > 0) or - (otherwise) as
  1. When  is near 1, computing stable densities and cumulatives in this range is numerically
difficult and estimating parameters is unreliable. From the applied point of view,
it is preferred to use the S(,,,0;0) parameterization, which is jointly continuous in
all four parameters. The arguments for using the S(,,,1;1) parameterization are historical
and algebraic simplicity. It seems unavoidable that both parameterizations will be
used, so users of stable distributions should know both and state clearly which they are
using.
There are three cases where one can write down closed form expressions for the density
and verify directly that they are stable ­ normal, Cauchy and Lévy distributions.
Example 1 (Normal or Gaussian distributions). X  N(,2) if it has a density
f (x) =
1

2
exp (x
- )2
22
, - < x < .
Gaussian laws are stable with  = 2 and  = 0; more precisely N(,2) = S(2,0,/

2,
;0) = S(2,0,/

2,;1).
110 J.P. Nolan
Example 2 (Cauchy distributions). X  Cauchy(,) if it has density
f (x) =
1


 2 + (x - )2
, - < x < .
Cauchy laws are stable with  = 1 and  = 0; more precisely, Cauchy(,) =
S(1,0,,;0) = S(1,0,,;1).
Example 3 (Lévy distributions). X  Lévy(,) if it has density
f (x) =

2
1
(x - )3/2
exp -

2(x - )
,  < x < .
These are stable with  = 1/2,  = 1;
Lévy(,) = S
1
2
,1,, + ;0 = S
1
2
,1,,;1 .
The graphs in Figure 1 show several qualitative features of stable laws. First, stable
distributions have densities and are unimodal. These facts are not obvious: since there is
no general formula for stable densities, indirect arguments must be used and it is quite
involved to prove unimodality. Second, the - curve is a reflection of the  curve. Third,
when  is small, the skewness is significant, when  is large, the skewness parameter
matters less and less. The support of a stable density is either all of (-,) or a halfline.
The latter case occurs if and only if 0 <  < 1 and  = +1 or -1. More precisely, the
support of density f (x|,,,;k) for a S(,,, ;k) law is



 -  tan

2
, ,  < 1,  = +1, k = 0,
-, +  tan

2
,  < 1,  = -1, k = 0,
[,),  < 1,  = +1, k = 1,
(-,],  < 1,  = -1, k = 1,
(-,), otherwise.
In particular, to model a positive distribution, a S(,1,,0;1) distribution with  < 1 is
used.
When  = 2, the normal law has light tails and all moments exist. Except for the normal
law, all stable laws have heavy tails with an asymptotic power law (Pareto) decay. The term
stable Paretian distributions is used to distinguish the  < 2 cases from the normal case.
For X  S(,,1,0;0) with 0 <  < 2 and -1 <  1, then as x  ,
P(X > x)  c(1 + )x-
,
f (x|,;0)  c(1 + )x-(+1)
,
Ch. 3: Modeling Financial Data 111
where c = ()(sin 
2 )/. When  = -1, the right tail decays faster than any power.
The left tail behavior is similar by the symmetry property mentioned above.
One consequence of these heavy tails is that only certain moments exist. This is not
a property restricted to stable laws: any distribution with power law decay will not have
certain moments. When  < 2, it can be shown that the variance does not exist and that
when  1, the mean does not exist. If we use fractional moments, then the p-th absolute
moment E|X|p = |x|pf (x)dx exists if and only if p < . We stress that this is a population
moment, and by definition it is finite when the integral just above converges. If the
tails are too heavy, the integral will diverge. In contrast, the sample moments of all orders
will exist: one can always compute the variance of a sample. The problem is that it does
not tell you much about stable laws because the sample variance does not converge to a
well-defined population moment (unless  = 2).
If X, X1, X2 are i.i.d. stable, then for any a,b > 0,
aX1 + bX2
d
= cX + d,
for some c > 0, - < d < . This condition is equivalent to (1) and can be taken as a
definition of stability. More generally, linear combinations of independent stable laws with
the same  are stable: if Xj  S(,j ,j ,j ;k) for j = 1,...,n, then
a1X1 + a2X2 +  + anXn  S(,,,;k), (5)
where  = ( n
j=1 j (signaj )|aj j |)/ n
j=1 |aj j |,   = n
j=1 |aj j |, and
 =



j +  tan

2
, k = 0,  = 1,
j + 
2

 ln, k = 0,  = 1,
j , k = 1.
This is a generalization of (1): it allows different skewness, scales and locations in the
terms. It is essential that all the s are the same: adding two stable random variables with
different s does not give a stable law.
2. Appropriateness of stable models
Stable distributions have been proposed as a model for many types of physical and
economic systems. There are several reasons for using a stable distribution to describe
a system. The first is where there are solid theoretical reasons for expecting a non-Gaussian
stable model, e.g., reflection off a rotating mirror yielding a Cauchy distribution, hitting
times for a Brownian motion yielding a Lévy distribution, the gravitational field of stars
112 J.P. Nolan
yielding the Holtsmark distribution; see Feller (1975) and Uchaikin and Zolotarev (1999)
for these and other examples. The second reason is the Generalized Central Limit Theorem,
see below, which states that the only possible non-trivial limit of normalized sums of
independent identically distributed terms is stable. It is argued that some observed quantities
are the sum of many small terms, e.g., the price of a stock, and hence a stable model
should be used to describe such systems. The third argument for modeling with stable distributions
is empirical: many large data sets exhibit heavy tails and skewness. The strong
empirical evidence for these features combined with the Generalized Central Limit Theorem
is used to justify the use of stable models. Examples in finance and economics are
given in Mandelbrot (1963), Fama (1965), Embrechts, Klüppelberg and Mikosch (1997),
and Rachev and Mittnik (2000). Such data sets are poorly described by a Gaussian model,
some can be well described by a stable distribution.
The classical Central Limit Theorem says that the normalized sum of independent, identical
terms with a finite variance converges to a normal distribution. The Generalized Central
Limit Theorem shows that if the finite variance assumption is dropped, the only possible
resulting limits are stable. Let X1,X2,X3,... be a sequence of independent, identically
distributed random variables. There exists constants an > 0, bn and a non-degenerate random
variable Z with
an(X1 +  + Xn) - bn
d
-Z (6)
if and only if Z is stable. A random variable X is in the domain of attraction of Z if
there exists constants an > 0, bn such that (6) holds when X1,X2,X3,... are independent
identically distributed copies of X.
The Generalized Central Limit Theorem says that the only possible distributions with
a domain of attraction are stable. Characterizations of distributions in the domain of attraction
of a stable law are in terms of tail probabilities. The simplest is: if X is a random
variable with xP(|X| > x)  c > 0 for some 0 <  < 2 as x  , then X is in the
domain of attraction of an -stable law.
Even if we accept that large data sets have heavy tails, is it ever reasonable to use a
stable model? One of the arguments against using stable models is that they have infinite
variance, which is inappropriate for real data that have bounded range. However, bounded
data are routinely modeled by normal distributions which have infinite support. The only
justification for this is that the normal distribution gives a usable description of the shape
of the distribution, even though it is clearly inappropriate on the tails for any problem
with naturally bounded data. The same justification can be used for stable models: does a
stable fit gives an accurate description of the shape of the distribution? The variance is one
measure of spread; the scale  in a stable model is another. Perhaps practioners are so used
to using the variance as the measure of spread, that they automatically retreat from models
without a variance. The parameters  and  can play the role of the scale and location
usually played by the mean and variance. For the normal distribution, the first and second
moment completely specify the distribution; for most distributions they do not.
Ch. 3: Modeling Financial Data 113
We propose that the practitioner approach this dispute as an agnostic. The fact is that
until recently we have not really been able to compare data sets to a proposed stable model.
The next Section shows that estimation of all four stable parameters is feasible and that
there are methods to assess whether a stable model accurately describes the data. In some
cases there are solid theoretical reasons for believing that a stable model is appropriate; in
other cases we will be pragmatic: if a stable distribution describes the data accurately and
parsimoniously with four parameters, then we accept it as a model for the observed data.
3. Computation, simulation, estimation and diagnostics
Until recently, it was difficult to use stable laws in practical problems because of computational
difficulties. Most of these difficulties have been resolved by the program STABLE,1
which can compute stable densities, cumulative distribution functions and quantiles. The
basic method used in the program are described in Nolan (1997). Later improvements to
the program include incorporating the Chambers, Mallows and Stuck (1976) method of
simulating stable random variables, improved accuracy in the calculations, and estimation
of stable parameters from data sets. Except for  close to 0, it is now possible to quickly
and accurately work with stable distributions. We will not discuss details of these programs
here, but will focus on the practical problems of estimation and assessing goodness of fit.
The basic estimation problem for stable laws is to estimate the four parameters
(,,,) from an i.i.d. sample X1,X2,...,Xn. Because of numerical problems with the
1-parameterization, we will always use the 0-parameterization in estimation. If desired, the
parameter 1 can be estimated by using (4). There are several methods available for this
basic estimation problem: a quantile method of McCulloch (1986), a fractional moment
method of Ma and Nikias (1995), sample characteristic function (SCF) method of Kogon
and Williams (1998) based on ideas of Koutrouvelis, and maximum likelihood (ML) estimation
of DuMouchel (1971) and Nolan (2001). These methods have been compared in a
large simulation study, Ojeda (2001), who found that the ML estimates are almost always
more accurate, with the SCF estimates next best, followed by the quantile method, and
finally the moment method. The ML method has the added advantage that one can give
large sample confidence intervals for the parameters, based on numerical computations of
the Fisher information matrix.
Perhaps just as important as methods of estimation, are diagnostics for assessing the
fit. While a Kolmogorov­Smirnov goodness-of-fit test statistic can be computed, giving a
correct significance level to such a test when comparing a data set to a fitted distribution is
an involved problem. However, one can adapt standard exploratory data analysis graphical
techniques to informally evaluate the closeness of a stable fit. We have found that comparing
smoothed data density plots to a proposed fit gives a good sense of how good the
fit is near the center of the data. P­P plots allow a comparison over the range of the data.
1 The program STABLE is available at www.mathstat.american.edu and following the "Faculty" link to the
author's homepage.
114 J.P. Nolan
For technical reasons we recommend the "variance stabilized" P­P plot of Michael (1983).
We found Q­Q plots not as satisfactory for comparing heavy tailed data to proposed fit.
One reason for this is visual ­ by definition a heavy tailed data set will have many more
extreme values than a typical sample from finite variance population. This forces a Q­Q
plot to be visually compressed, with a few extreme values dominating the plot. Also, the
heavy tails imply that the extreme order statistics will have a lot of variability, and hence
deviations from an ideal straight line Q­Q plot are hard to assess. The next section shows
some examples of these techniques on financial data, more examples can be found in Nolan
(1999, 2001).
There are methods for more complicated estimation problems involving stable laws.
For example, regression models with stable residuals have been described by McCulloch
(1998) for the symmetric stable case and Ojeda (2001) for the general case. The problem
analyzing time series with stable noise is discussed in Section II of Adler, Feldman
and Taqqu (1998), in Nikias and Shao (1995), and in Rachev and Mittnik (2000). McCulloch
(1996) and Rachev and Mittnik (2000) give methods of pricing options under stable
models.
4. Applications to financial data
The first example we consider is the British Pound vs. German Mark exchange rate. The
data set has daily exchange rates for the 16 year period from 2 January 1980 to 21 May
1996. The log of the successive exchange rates was computed as yt = ln(xt+1/xt), yielding
4,274 yt values. The ML parameter estimates with 95% confidence intervals are 1.495 
0.047 for , -0.1820.085 for , 0.002440.00008 for  and 0.000190.00013for 0.
Figure 2 shows a P­P plot and density for the data vs. the stable fit. The third curve in the
density plot is the normal/Gaussian fit to the data.
The next example is another exchange rate one, this time from a developing country.
This data set consists of monthly exchange rates between the US Dollar and the Tanzanian
Shilling, from January 1975 to September 1997. The log of the successive exchange rates
were computed as above for this monthly data, giving a data set with n = 213 points. The
ML parameter estimates with 95% confidence intervals are 1.088  0.185 for , 0.112 
0.251 for , 0.0300  0.0055 for  and 0.00501  0.00621 for 0. The more extreme
fluctuations of the Tanzanian Shilling exchange rate show up in the smaller estimate of 
and in the larger estimate of  . Figure 3 shows the diagnostics, with the third curve again
showing a normal/Gaussian fit.
The third example is from the stock market. McCulloch (1997) analyzed 40 years of
monthly stock price data from the Center for Research in Security Prices (CRSP). The
data set is 480 values of the CRSP value-weighted stock index, including dividends and
adjusted for inflation. The ML estimates with 95% confidence intervals are 1.855  0.110
for , -0.5580.615 for , 2.7110.213 for  , and 0.8710.424 for 0. Figure 4 shows
the goodness of fit.
Stable distributions may be a useful tool in Value at Risk (VaR) calculations. The goal of
VaR calculations is to assess the risk in an asset by estimating population quantiles. Stable
Ch. 3: Modeling Financial Data 115
Fig. 2. P­P plot and density plot for Pound vs. Mark exchange rate data. On the density plot, the dotted curve is
the smoothed data, the solid curve is the stable fit, the dashed curve is a normal fit.
Fig. 3. P­P plot and density plot for the Tanzanian Shilling/US Dollar exchange rate.
distributions have two advantages over normal distributions: they can explicitly model both
the heavier tails and asymmetry that are frequently found in financial data. Sometimes
the normal distribution can give reasonable VaR estimates, because the sample variance
is inflated by the extreme values in the sample. If one is lucky, the poor fitting normal
distribution may approximate certain quantiles well, at the cost of poorly approximating
other quantiles. Additionally, some practioners compensate for the heavy tail behavior by
"adjusting" a normal quantile estimate by some empirical factor. If a stable distribution
gives a more accurate fit to the sample, then it is more likely to accurately predict the VaR
values. In order to compare different fits, a plot like Figure 5 can be useful. It uses the
Deutsch Mark exchange rate data (log ratios of successive values) described above.
116 J.P. Nolan
Fig. 4. P­P plot and densities for the CRSP stock price data.
Fig. 5. VaR comparison of quantiles for the Deutsch Mark exchange rate data (circles), quantiles predicted by
the stable fit (solid line), and quantiles predicted by the normal distribution (dotted line).
5. Multivariate stable distributions
This section is about d-dimensional stable laws. Such random vectors will be denoted by
X = (X1,...,Xd). The definition of stability is the same as in (1): for i.i.d. X,X1,X2,...,
X1 + X2 +  + Xn
d
= anX + bn, (7)
Ch. 3: Modeling Financial Data 117
for some an > 0, and some vector bn  Rd . As in one dimension, an equivalent definition
is that aX1 + bX2
d
= cX + d for all a,b > 0.
If X is a stable random vector, then every one-dimensional projection u  X = uiXi is
a one-dimensional stable random variable with the same index  for every u. The phrase
"jointly stable" is sometimes used to stress the fact that the definition forces all the components
Xj to be univariate -stable with one . Conversely, suppose X is a random vector
with the property that every one-dimensional projection u  X is one-dimensional stable,
e.g., u  X  S(,(u),(u), (u),(u);1). Then there is one  that is the index of all projections,
i.e., (u) =  is constant. If  1, then X is stable. If  < 1 and the location
parameter function (u) and the vector of location parameters  = (1,2,...,d) of the
components X1,X2,...,Xd (all in the 1 parameterization) are related by
(u) = u  , (8)
then X is stable. The point here is that we have a way of determining joint stability in terms
of univariate stability and, when  < 1, Equation (8).
We note that (8) holds automatically when  > 1, so the condition is only required
when  < 1. Furthermore, (8) is necessary when  = 1, so it cannot be dropped. There
are examples, e.g., Section 2.2 of Samorodnitsky and Taqqu (1994), where  < 1 and all
one-dimensional projections are stable, but (8) fails and X is not jointly stable.
One way of parameterizing multivariate stable distributions is to use the above results
about one dimensional projections. For any vector u  Rd,
u  X  S ,(u), (u),(u);k , k = 0,1.
Thus we know the (univariate) characteristic function of u  X for every u, and hence the
joint characteristic function of X. Therefore  and the functions (),  () and () completely
characterize the joint distribution. In fact, knowing these functions on the sphere
Sd = {u  Rd: |u| = 1} characterizes the distribution.
The functions (),  () and () must satisfy certain regularity conditions. The standard
way of describing multivariate stable distributions is in terms of a finite measure  on Sd ,
called the spectral measure. Let X = (X1,...,Xd) be jointly stable, say
u  X  S ,(u), (u),(u);k , k = 0,1.
Then there exists a finite measure  on Sd and a location vector   Rd with
 (u) =
Sd
|u  s|
(ds)
1/
,
(u) = Sd |u  s| sign(u  s)(ds)
Sd |u  s|(ds)
, (9)
118 J.P. Nolan
(u) =



  u, k = 1,  = 1,
  u -
2
 Sd
(u  s)ln|u  s|(ds), k = 1,  = 1,
  u + tan

2
(u) (u), k = 0,  = 1,
  u -
2
 Sd
(u  s)ln(u  s)(ds)
+
2

(u) (u)ln (u), k = 0,  = 1.
Thus another way to parameterize is X  S(,,;k), k = 0,1. If one knows , then
the above equations specify the parameter functions (),  () and (). Going the other
direction is more difficult. If one recognizes a certain form for the parameter functions,
then one can specify the spectral measure. In the general case, one can numerically invert
the map   ((), (),()) to get a discrete approximation to .
It is possible for X to be non-degenerate, but singular. For example, X = (X1,0) is
formally a two-dimensional stable distribution if X1 is univariate stable, but it is supported
on a one-dimensional subspace. In what follows, we will always assume that X is nonsingular
that is, it has a density on Rd. It can be shown that the following are equivalent:
(i) X is non-singular,
(ii)  (u) > 0 for all non-zero u  Rd , and
(iii) span support() = Rd .
For  1, the support of non-singular stable X is all of Rd. When  < 1, it can be
all of Rd or a cone, depending on the spectral measure. For A is a subset of Rd, define
CCH(A) = closed convex hull of A = closure of
x = a1b1 +  + anbn  Rd
: a1,...,an  A, b1,...,bn 0 .
Note that we only take positive linear combinations of elements of A, so this is not generally
the closed span of A. The translate of a cone is denoted by CCH(A) +  = {x + : x 
CCH(A)}. Then the support of X  S(,,;1) is
supportX =
CCH(support()) + ,  < 1,
Rd,  1.
For example, in the two-dimensional case, if the spectral measure is supported in the first
quadrant,  < 1, and  = 0, then the support of the corresponding stable distribution is
contained in the first quadrant, i.e., both components are positive.
The tail behavior of X is easiest to describe in terms of the spectral measure. It is best
stated in polar form: let A  Sd , then
lim
r
P(X  CCH(A),|X| > r)
P(|X| > r)
=
(A)
(Sd)
.
Ch. 3: Modeling Financial Data 119
The tail behavior of the densities is more intricate. In the radially symmetric case,
f (x)  c|x|-(d+) as |x|  . In other cases, the tail behavior can have very different
behavior in different directions. For example, in the bivariate independent case, the
joint density factors f (x1,x2) = f1(x1)f2(x2). The one-dimensional results above show
f (x,0)  c1x-(1+) along the x-axis, but f (x,x)  c2x-2(1+) along the diagonal. The
Fig. 6. Density surface and level curves for "triangle" example of a bivariate stable law.
Fig. 7. Contour plots for bivariate stable densities with independent S(,,1,0;1) components. The plots show
 = 0.6,  = 0 in upper left,  = 0.6,  = 1 in upper right,  = 1.6,  = 0 in lower left, and  = 1.6,  = 1 in
lower right.
120 J.P. Nolan
general case is complicated, depending on the nature (discrete, continuous) and spread of
the spectral measure.
We now give some examples of bivariate stable densities, see the next section for information
on their computation. In all cases, the shift vector  = 0.
Example 4. The first example uses  = 1.2 and a discrete spectral measure with three
unit point masses, distributed on the unit circle at angles /3,  and -/3. A plot of the
density surface and level curves are given in Figure 6. The triangular spread of the spectral
measure shows up in the triangular shape of the level curves. The contour plot reveals more
about the shape of the surface, so the following examples will show only the contour plots.
Example 5. Figure 7 shows the contour plots of the independent components cases when
 = 0.6, 1.6 and  = 0,1. Note that the upper right graph has  < 1 and is supported in
the first quadrant.
Example 6. Figure 8 shows a mix of different contours, mostly to show the range of
possibilities. The upper left plot shows an elliptically contoured stable distribution with
 = 1.5 and "covariation matrix"
R =
1.0 0.7
0.7 1.0
.
Fig. 8. Contours of miscellaneous bivariate stable distributions.
Ch. 3: Modeling Financial Data 121
The upper right plot shows a  = 0.8 stable distribution with discrete spectral measure
having point masses at angles -/9, /6, /3, /2 and uniform weight j = 0.3. The
lower left plot uses  = 0.7 with a discrete spectral measure with point masses at angles
/9, 4/9, 10/9, 13/9 of weight 0.75, 1, 0.25, 1. The lower right plot uses the same
discrete spectral measure as the lower left, but with  = 1.5.
There are some general statements that can be made about the qualitative behavior of
multivariate stable densities. For fixed , central behavior is determined by overall spread
of the spectral measure: if the spectral mass is highly concentrated the density is close
to singular, with large values near the center; if the spectral mass is more evenly spread
around the sphere, the density is less peaked. On the tails, behavior is determined by the
exact distribution of the spectral measure, with the contour lines bulging out in directions
where the spectral measure is concentrated. This tail effect is more pronounced for small
values of , where distributions can be highly skewed, and becomes less pronounced as 
approaches 2, where contours are all rounded into ellipses.
6. Multivariate computation, simulation, estimation and diagnostics
The computational problems are challenging, and not solved for general multivariate stable
distributions. The problems are caused by the both the usual difficulties of working
in d dimensions and by the complexity of the possible distributions: spectral measures
are an uncountable set of "parameters". The graphs above were computed by the program
MVSTABLE (available at the same web-site noted above), which only works in 2 dimensions
and has limited accuracy. Density calculations are based on either numerically inverting
the characteristic function as described in Nolan and Rajput (1995) or by numerically
implementing the symmetric formulas in Abdul-Hamid and Nolan (1998).
One class of accessible models is when the spectral measure is discrete with a finite
number of point masses:
() =
n
j=1
j 1{}(sj ). (10)
This class is dense in the space of all stable distributions: given an arbitrary spectral measure
1, there is a concrete formula for n and a discrete spectral measure 2 such that the
densities of the corresponding stable densities are uniformly close on Rd.
In the case of a discrete spectral measure, the parameter functions (),  () and ()
are computed as finite sums, rather than (d - 1)-dimensional integrals, which makes
all computations easier. It also makes simulation simple in an arbitrary dimension: X 
S(,,;k) where  is given by (10) can be simulated by the vector sum
X
d
=
n
j=1

1/
j Zj sj + ,
122 J.P. Nolan
where Z1,...,Zn are i.i.d. univariate S(,1,1,0;k) random variables.
Another example where computations are more accessible is the elliptically contoured,
or sub-Gaussian, stable distributions described in Section 8. Such densities are easier to
compute and simulation is straightforward. Certain sub-stable distributions are also easy to
simulate: if  < 1, X is strictly 1-stable and A is positive (/1)-stable, then A1/1X is
-stable. Since sums and shifts of multivariate stables are also multivariate stable, one can
combine these different classes to simulate a large class of multivariate stable laws.
There are several methods of estimating for multivariate stable distributions. If you know
the distribution is isotropic (radially symmetric), then Problem 4, p. 44 of Nikias and Shao
(1995) gives a way to estimate  and then the constant scale function/uniform spectral
measure from fractional moments. In general one should let the data speak for itself, and
see if the spectral measure  is constant. The general techniques involve some estimate of
 and some estimate of the spectral measure ^ = m
k=1 k1{}(sk), sk  Sd. Rachev and
Xin (1993) and Cheng and Rachev (1995) use the fact that the directional tail behavior
of multivariate stable distributions is Pareto, and base an estimate of  on this. Nolan,
Panorska and McCulloch (2001) define two other estimates of , one based on the joint
empirical/sample ch. f. and one based on the one-dimensional projections of the data.
Using the fact that one-dimensional projections are univariate stable gives a way of
assessing whether a multivariate data set is stable by looking at just one-dimensional projections
of the data. Fit projections in multiple directions using the univariate techniques
described above, and see if they are well described by a univariate stable fit. If so, and if
the 's are the same for every direction (and if  < 1, the location parameters satisfy (8)),
then a multivariate stable model is appropriate. We will illustrate this in examples below.
For the purposes of comparing two multivariate stable distributions, the parameter functions
(,(u), (u),(u)) are more useful than  itself. This is because the distribution
of X depends more on how  distributes mass around the sphere than exactly on the measure.
Two spectral measures can be far away in the traditional total variation norm (e.g.,
one can be discrete and the other continuous), but their corresponding parameter functions
and densities can be very close.
The diagnostics suggested for assessing stability of a multivariate data set are:
* Project the data in a variety of directions u and use the univariate diagnostics described
in Section 3 on each of those distributions. Bad fits in any direction indicate that the data
is not stable.
* For each direction u, estimate the parameter functions (u), (u),  (u), (u) by ML
estimation. The plot of (u) should be a constant, significant departures from this indicate
that the data has different decay rates in different directions. (Note that  (t) will be
a constant iff the distribution is isotropic.)
* Assess the goodness-of-fit by computing a discrete ^ by one of the methods above.
Substitute the discrete ^ in (9) to compute parameter functions. If it differs from the
one obtained above by projection, then either the data is not jointly stable, or not enough
points were chosen in the discrete spectral measure approximation.
These techniques are illustrated in the next section.
Ch. 3: Modeling Financial Data 123
Fig. 9. Projection diagnostics for the German Mark and Japanese Yen exchange rates.
124 J.P. Nolan
7. Multivariate application
Here we will examine the joint distribution of the German Mark and the Japanese Yen.
The data set is the one described above in the univariate example. We are interested in both
assessing whether the joint distribution is bivariate stable and in estimating the fit.
Figure 9 shows a sequence of smoothed density, q­q plot and variance stabilized p­p
plot for projections in 8 different directions: /2, /3, /4, /6, 0, -/6, -/4, -/3.
(We restrict to the right half plane because projections in the left half plane are reflections
of those in the right half plane.) These projections are similar to Figure 2, in fact the fifth
Fig. 10. Estimation results for the German Mark and Japanese Yen exchange rates.
Ch. 3: Modeling Financial Data 125
row of Figure 9 is exactly the same as Figure 2. Except on the extreme tails, the stable fit
does a good job of describing the data.
The projection functions (t), (t),  (t), and (t) were estimated and used to compute
an estimate of the spectral measure using the projection method. The results are shown in
Figure 10. It shows a discrete estimate of the spectral measure (with m = 100 evenly spaced
point masses) in polar form, a cumulative plot of the spectral measure in rectangular form,
and then four plots for the parameter estimates ((t),(t), (t),(t)). Also on the (t)
plot is a horizontal line showing the average value of all the estimated indices which is
taken as the estimate of the common  that should come from a jointly stable distribution.
The plots of (t) and  (t) also show the skewness and scale functions computed from the
estimated spectral measure substituted into (9). These curves, which are based on a joint
estimate of the spectral measure, are indistinguishable from the direct, separate estimates
of the directional parameters.
The fitted spectral measure was used to plot the fitted bivariate density shown in Figure
11. The spread of the spectral measure is spiky, and masks a pattern that is more obvious
in the density surface: the approximate elliptical contours of the fitted density. This
suggests modeling the data by a sub-Gaussian stable distribution, a topic discussed in the
next section.
Some comments on these plots. The polar plots of the spectral measure show a unit circle
and lines connecting the points (j ,rj ), where j = 2(j -1)/m and rj = 1 +(j /max),
where max = maxj . The polar plots are spiky, because we are estimating a discrete
object. What should be looked at is the overall spread of mass, not specific spikes in the
plot. In cases where the spectral measure is really smooth, it may be appropriate to smooth
these plots out to better show iťs true nature. In cases where the measure is discrete, i.e.,
the independent case, then one wants to emphasize the spikes. So there is no satisfactory
general solution and we just plot the raw data.
Finally, most graphing programs will set vertical scale so that the graph fills the graph.
This emphasizes minor fluctuations in the data that are not of practical significance. In the
graphs below, the vertical scales for the parameter functions (t), (t),  (t) are respectively
[0,2], [-1,1], and [0,1.2 × max (t)]. These bounds show how the functions vary
Fig. 11. Estimated density surface and level curves for a bivariate stable fit to the German Mark and Japanese
Yen exchange rates.
126 J.P. Nolan
over their possible range. For (t), we used the bounds [-1.2×max|(t)|,1.2×max|(t)|],
which visually exaggerates the changes in (t). A scale that depends on max (t) may be
more appropriate.
8. Classes of multivariate stable distributions
There may be cases where we believe that a multivariate sample has certain structure. If so,
we can fit a stable model that takes this into account. This may give a more parsimonious
fit to the model, especially if the data set is high dimensional. Below we fill focus on
elliptically contoured distributions and see that it is computationally accessible. The idea
here is to estimate an  and a matrix R so that the scale function is closely approximated by
 (u) = (uRu)/2. The principle can be generalized to other special classes of distributions.
Given some parametric model for the scale function  (), one can fit parameters, or use a
nonparametric model (smoothing or loess) for the scale. Or, one can assume a special form
of the spectral measure (), which determines the scale function  (). The methods of
estimation described above do this implicitly, by assuming  is discrete as in (10). This
can be adapted in many ways. If we assume the components of the data are independent,
then we can only allow point masses at "poles", i.e., where the coordinate axes intersect
the sphere. If we assume the spectral measure is concentrated on some smaller region, then
one can allow point masses only in that region.
If we assume the spectral measure is continuous, then one can use some particular model
for its density, say as a sum of terms like (ds) = n
k=1 k(s)ds, where the density terms
k() in the sum have some accessible form. If the goal is a computationally accessible
model, then an ad hoc approach may be useful. First compute a fit using a discrete spectral
measure. If there are clearly defined point masses that are isolated, then include them and
try to model the rest as an elliptical model, or using some spectral density.
Since the foreign exchange data seems to be approximately elliptically contoured, there
may be interest in categorizing such stable distributions. The main practical advantage to
this is that all d-dimensional elliptically contoured stable distributions are parameterized by
 and a symmetric, positive definite d × d matrix. Since the matrix is symmetric, there are
a total of 1 + d(d + 1)/2 parameters. This is quite different from the general stable case,
which involves an infinite dimensional spectral measure. Even a discrete approximating
measure involves a much larger number of terms: if a "polar grid" is used with each of the
angle directions divided up evenly with k subintervals, then there are kd-1 point masses to
be estimated.
For X an non-singular symmetric -stable random vector, the following are equivalent:
* X is elliptically contoured around the origin.
* X is sub-Gaussian, i.e., X
d
= A1/2G, where A  S(,1,,0;1) and G  N(0,R).
* The characteristic function is E exp(iu  X) = exp(-(uRuT)/2), for some symmetric,
positive definite matrix R.
There is a "random volatility" interpretation of sub-Gaussian distributions. Think of G
as an underlying multivariate normal model for the returns on d assets with random scale
Ch. 3: Modeling Financial Data 127
A1/2. In general, A can be any positive random variable, but the product will be -stable
only when A is itself a positive (/2)-stable random variable.
Computations with elliptically contoured stable distributions is much simpler than the
general stable case. All calculations are essentially reduced to one-dimensional problems:
the linear transformation Y = R-1/2X gives a radially symmetric distribution. With a radially
symmetric density, one only needs to compute it along some one-dimensional ray. In
symbols, f (x) = det(R)-1/2f (|R-1/2xT|,0,0,...,0) = c(R)g(|R-1/2x |). The univariate
function g can be computed for arbitrary dimension d by numerically evaluating the
univariate integral
g(x) = (2)-d/2

0
e-x2/(2t)
f t

2
,1,2
cos
4
2/
,0;1 dt.
We next describe ways of assessing a d-dimensional data set to see if it is approximately
sub-Gaussian and then estimating the parameters of a sub-Gaussian vector.
First perform a one-dimensional stable fit to each coordinate of the data using one of the
methods described above, to get estimates ^i = (^i, ^i, ^i, ^i). If the i's are significantly
different, then the data is not jointly -stable, so it cannot be sub-Gaussian. Likewise, if
the i's are not all close to 0, then the distribution is not symmetric and it cannot be sub-
Gaussian.
If the i 's are all close, form a pooled estimate of  = ( d
i=1 i)/d = average of the
indices of each component. Then shift the data by ^ = (^1, ^2,..., ^d) so the distribution
is centered at the origin.
Next, test for sub-Gaussian behavior. This can be accomplished by examining twodimensional
projections because of the following result. If X is a d-dimensional subGaussian
-stable random vector, then every two-dimensional projection
Y = (Y1,Y2) = (a1  X,a2  X), (11)
(a1,a2  Rd) is a two-dimensional sub-Gaussian -stable random vector. Conversely,
suppose X is a d-dimensional -stable random vector with the property that every twodimensional
projection of form (11) is non-singular sub-Gaussian. Then d-dimensional X
is non-singular sub-Gaussian -stable.
Estimating the d(d + 1)/2 parameters (upper triangular part) of R can be done in at
least two ways. For the first method, set rii =  2
i , i.e., the square of the scale parameter
of the i-th coordinate. Then estimate rij by analyzing the pair (Xi,Xj ) and take rij =
( 2(1,1)-rii -rjj )/2, where  (1,1) is the scale parameter of (1,1)(Xi,Xj ) = Xi +Xj .
This involves estimating d + d(d - 1)/2 = d(d + 1)/2 one-dimensional scale parameters.
For the second method, note that if X is -stable sub-Gaussian, then E exp(iu  X) =
exp(-(uRuT)/2), so
-lnE exp(iu  X)
2/
= uRuT
=
i
u2
i rii + 2
i<j
uiuj rij .
128 J.P. Nolan
This is a linear function of the rij 's, so they can be estimated by regression. This method
may be more accurate because it uses multiple directions, whereas the first method uses
only three directions: (1,0), (0,1) and (1,1). If a two-dimensional fit has already been done,
then one has already estimated  (u) on a grid. Note that uRuT =  2(u) is the square of
the scale parameter in the direction u. Sample estimates of  2(u) on a grid of u points can
be used for the middle term above. In both methods, checks should be made to test that the
resulting matrix R is positive definite.
The first method was used to estimate the matrix R for the Deutsche Mark­Japanese
Yen data set considered above. The estimated matrix R was
R = 10-6 5.9552 4.0783
4.0783 13.9861
.
The plot of  (t) shown in the lower left corner of Figure 10 also shows tRtT as a dashed
line. It is virtually indistinguishable from the curve of  (t), supporting the idea that a subGaussian
stable fit does a good job of fitting the bivariate data.
9. Operator stable distributions
A brief discussion of operator stable laws is given next. The class of operator stable distributions
allows different components of X to be stable with different indices j . It is defined
by replacing the real scale term an in (7) with a matrix scale term An, see Jurek and Mason
(1993) or Meerschaert and Schefler (2001). This may be of use in analyzing a portfolio,
where different assets have different characteristics, e.g., some have Gaussian behavior and
some have heavy tailed behavior, possibly with different tail behavior.
One subclass of the operator stable distributions is obtained by building up from independent
groups of -stable laws: suppose (X1,...,Xd1 ) has a d1-dimensional 1-stable
distribution, (Xd1+1,...,Xd1+d2 ) has a d2-dimensional 2-stable distribution, ...,
(Xd1+d2++dk-1+1,...,Xd1++dk ) has a dk-dimensional k-stable distribution. If all
these groups of distributions are independent, then the vector X = (X1,...,Xd),
d = d1 +  + dk, has a d-dimensional operator stable law. Also, for any d × d matrix
A, the vector Y = AX is an operator stable law. (One usually requires A to be invertible,
otherwise the resulting Y will not be d-dimensional.)
10. Discussion
We have shown that estimation of general stable parameters is now feasible. The diagnostics
show that some financial sets with heavy tails are well described by stable distributions.
While they do not give a perfect fit, stable models can give a much better fit than Gaussian
models.
Ch. 3: Modeling Financial Data 129
In practice, the decision to use a stable model should be based on the purpose of the
model. In cases where a large data set shows close agreement with a stable fit, confident
statements can be made about the population. In other cases where there is a poor fit, one
should not use a stable model. These models are not a panacea ­ not all heavy tailed data
sets can be well described by stable distributions. In intermediate cases, one could tentatively
use a stable model as a descriptive method of summarizing the general shape of the
distribution, but not try to make statements about tail probabilities. In such problems, it
may actually be better to use the quantile parameter estimates rather than ML estimates,
because the former tries to match the shape of the empirical distribution and ignores the
top and bottom 5% of the data.
In multivariate problems where the dimension is large, it will be very difficult to model
with a stable distribution unless there is some special structure. If some components are
independent, then they should be separated out and analyzed alone. If the dependent components
are elliptically contoured or have some other special structure, then Section 8
discusses a way to analyze them. In the general stable case, one may try to group the components
into smaller dependent groups, estimate within groups, and then try to characterize
dependence between groups. We are not aware of work on this topic.
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DuMouchel, W.H., 1973a. On the asymptotic normality of the maximum-likelihood estimate when sampling from
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Jurek, Z.J., Mason, J.D., 1993. Operator-Limit Distributions is Probability Theory. Wiley, New York, NY.
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Chapter 4
STATISTICAL ISSUES IN MODELING MULTIVARIATE STABLE
PORTFOLIOS
TOMASZ J. KOZUBOWSKI
University of Nevada, Reno
ANNA K. PANORSKA
University of Nevada, Reno
SVETLOZAR T. RACHEV
Department of Statistics and Applied Probability, University of California, Santa Barbara, USA
Institute of Statistics and Mathematical Economics, University of Karlsruhe, Germany
e-mail: rachev@lsoe-4.wiwi.uni-karlsruhe.de
Contents
Abstract 132
1. Introduction 133
2. Multivariate stable laws 134
2.1. Domains of attraction 135
2.2. Strictly stable and symmetric stable vectors 136
2.3. One-dimensional case 136
2.4. Discrete spectral measure 137
2.5. Linear combinations and risk of a financial portfolio 138
2.6. Densities 139
2.7. An alternative parameterization 139
2.8. Association 140
3. Estimation of the index of stability 141
3.1. Estimation of univariate stable parameters 142
3.2. Estimation of the tail index  143
3.2.1. The Hill estimator 143
3.2.2. A shifted Hilĺs estimator 144
3.2.3. The Pickands estimator and its modifications 144
3.2.4. Least-squares estimators 145
3.2.5. The M­S method 146
4. Estimation of the stable spectral measure 147
4.1. Tail estimators 147
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
132 T.J. Kozubowski et al.
4.2. The empirical characteristic function method 148
4.3. The projection method 150
5. Estimation of the scale parameter 151
6. Extensions to other stable models 152
6.1. -stable laws 152
6.1.1. Geometric stable laws 153
6.1.2. Statistical issues 154
6.2. Operator stable laws 155
6.2.1. Statistical issues 156
7. Applications 157
Acknowledgment 162
References 162
Abstract
Paretian stable distributions have had a relatively successful career in modeling of financial
data. We discuss statistical issues common in modeling multivariate portfolios with focus
on the estimation of the spectral measure that is important for estimation of the risk and
dependence structure of a portfolio. We also briefly discuss alternative multivariate stable
models for financial portfolios and estimation of their parameters.
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 133
1. Introduction
Statistical analysis of multivariate Paretian stable portfolios presents numerous practical
and theoretical challenges. Perhaps the most common practical issues in the modeling of
stable portfolios include diagnostics of the stable hypothesis and estimation of the index of
stability (or tail index)  and the stable spectral measure  . The index of stability  determines
the overall properties of a multivariate stable distribution and the spectral measure
 governs the dependence structure between the components of a stable portfolio. In this
work, although we will briefly discuss diagnostics for a stable model and estimation of the
index of stability, we will focus on the estimation of the spectral measure.
The fundamental work in the sixties of Mandelbrot [see, e.g., Mandelbrot (1963a, 1963b,
1967)] and Fama (1965a) led to development of a large field of research in the theory
and applications of Paretian stable models in finance and economics. For example, the
problem of derivative pricing for stable Paretian returns was considered in Dostoglou and
Rachev (1999), Janicki et al. (1997), Hurst, Platen and Rachev (1999), Karandikar and
Rachev (1995), Rachev and Rüschendorf (1994), Rachev and Samorodnitsky (1993), riskmanagement
issues were treated in Bassi, Embrechts and Kafetzaki (1988), Gamrowski
and Rachev (1996), Mittnik, Rachev and Paolella (1998), while the problem of computing
optimal portfolios when the returns have Paretian stable distributions was presented
in Bawa, Elton and Gruber (1979), Belkacem, Véhel and Walter (2000), Chamberlain,
Cheung and Kwan (1990), Gamba (1999), Fama (1965b), Press (1982), Rachev and Han
(2000), Ziemba (1974). For an extensive exposition of this subject and further references
we invite the reader to peruse a recent volume of Rachev and Mittnik (2000) containing
over 1000 references.
The properties of stable distributions that make them attractive for modeling include
domains of attraction and stability. Domains of attraction add robustness to the stable
model. We can not expect that the observed data follows exactly the distribution specified
by the modeler. In fact, any model is an approximation of the underlying distribution of the
process generating the data. As an approximation any model has a domain of applicability
where its fit to the observations is reasonable and justified. Domain of attraction for a
stable model contains many distributions with properties close to the specified stable law.
Therefore, a stable distribution provides good approximation for a wide range of observed
data. More importantly, decisions will essentially not be affected by using a stable approximation
as the model instead of the true distribution. Additionally, it is possible to check
whether or not a distribution is in the domain of attraction of a stable model by examining
only its tails because the tails completely determine domain of attraction (see Sections 2.1).
Stability implies existence of an overall parameter, the index of stability , that remains
unchanged across all scales (sampling intervals). This is beneficial because in univariate
modeling we can focus on the estimation of only one parameter that controls the main
properties of the underlying distribution. In the multivariate case, the index of stability is
still crucial, but it is not enough to describe all of the important properties of the stable
model. To describe a multivariate portfolio, it is necessary to estimate its spectral measure
which carries the information about the dependence structure and risk. The dependence
134 T.J. Kozubowski et al.
structure and tests for association between the returns on different assets require an estimate
of the spectral measure [see Lee, Rachev and Samorodnitcky (1990a, b)]. The risk
of a stable portfolio is usually measured by its scale parameter which is defined as a functional
of the spectral measure [see Section 2.5 and Rachev and Mittnik (2000)] and . Here
again, estimates of the spectral measure prove necessary for both modeling and decision
making processes.
Our chapter is organized as follows. In Section 2 we define stable distributions and review
their properties, particularly those relevant to financial modeling and estimation of
parameters. In Sections 3, 4, and 5 we discuss estimation of the index of stability (the tail
index), spectral measure, and scale parameter, respectively. Then, in Section 6, we present
some other heavy tailed multivariate laws related to Paretian stable distributions that have
found applications in mathematical finance. We review existing estimation procedures for
these laws and introduce new estimators. Finally, in Section 7, we apply some of the procedures
discussed in this work to financial data sets and further discuss some practical
statistical issues related to stable modeling.
2. Multivariate stable laws
A random vector X = (X1,...,Xd) in Rd is stable (Paretian stable, -stable) if it obeys the
stability property, that is for any n 2 there is some   (0,2] and a vector Dn such that
X1 +  + Xn
d
= n1/
X + Dn, (1)
where the Xi's are i.i.d. copies of X [see, e.g., Samorodnitsky and Taqqu (1994)]. Parameter
 is called index of stability. Stable vectors do not admit densities or distribution functions
in a closed form (with a few exceptions) and are usually described in terms of their
characteristic functions (Fourier transforms), which are of the form [see, e.g., Samorodnitsky
and Taqqu (1994)]:
(t) = Eei t,X
= e-I(t)+i t,m
, (2)
where m  Rd is the shift parameter (the mean for  > 1) while
I(t) =
Sd
,1 t,s  (ds). (3)
Here, Sd is the unit sphere in Rd,  is a finite measure on Sd, called the spectral measure,
the quantity t,s = j tj sj is the inner product in Rd, and
,(u) =
|u| 1 - i sign(u)tan 
2 for  = 1,
|u| 1 + i 2
 sign(u)log|u| for  = 1.
(4)
We denote the distribution of a stable r.v. X with the ch.f. (2) by S(m, ).
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 135
The index of stability   (0,2] determines the tail of the stable law and can be thought
of as a shape parameter. When  = 2 we obtain the special case of multivariate normal distribution,
while when  < 2, the probability P(Xj > x) associated with each component
Xj of an -stable r.v. X decreases like the power function x- as x increases to infinity.
Spectral measure  controls the dependence among the components of X. The latter are
independent if and only if  is discrete and concentrated on the intersection of Sd with the
coordinate axes.
2.1. Domains of attraction
Stable laws are the only possible limiting distributions of scalar-normalized sums of i.i.d.
random vectors. A random vector X is said to be in the domain of attraction of a multivariate
stable r.v. Y if for some an > 0 and bn  Rd the following convergence in distribution
holds
an(X1 +  + Xn) + bn
d
 Y as n  , (5)
where the Xi's are i.i.d. copies of X. By the stability property (1) it is clear that any stable
r.v. belongs to its own domain of attraction. The domain of attraction of a stable law with
index  = 2 (the normal law) includes all distributions with finite second moments for
which the convergence in (5) coincides with the classical Central Limit Theorem. The
domain of attraction of a nonnormal stable law admits the following characterization due
to Rvaˇceva1 (1962), and plays a crucial role in estimating the spectral measure  .
Proposition 2.1. A random vector X on Rd belongs to the domain of attraction of some
full2 stable S(m, ) law with  < 2 if and only if V (r) = P(||X|| > r) is regularly varying
at infinity with index - and
P
X
X
 D given X > r =
P(X/ X  D, X > r)
V (r)

 (D)
 (Sd)
(6)
as r   for all Borel subsets D of the sphere Sd with  (D) = 0.
In other words, the tail behavior of X in the direction of D is determined by the spectral
measure of the set D.
1 The original proof in Rvaˇceva (1962) seems to contain an error; for a corrected proof and a more modern
treatment (in terms of regular variation), see Meerschaert and Scheffler (2001).
2 The probability distribution of a random vector X on Rd is full if t,X is nondegenerate for every t = 0.
136 T.J. Kozubowski et al.
2.2. Strictly stable and symmetric stable vectors
A r.v. X is strictly stable if the relation (1) is valid with Dn = 0. This holds if the shift
vector m is zero for  = 1 and if
Sd
s (ds) = 0 (7)
if  = 1 [see, e.g., Samorodnitsky and Taqqu (1994)]. A r.v. X is said to be symmetric
stable if it is stable and the probabilities P(X  A) and P(-X  A) are the same for all
Borel sets A of Rd. Then, the spectral measure  of X is symmetric and the ch.f. (2)
reduces to
(t) = e
- Sd
| t,s | (ds)
. (8)
2.3. One-dimensional case
In one dimension, the unit sphere is the set {-1,1} and the ch.f. (2) reduces to
(t) = EeitX
= eit-, (t)
, (9)
where the parameter  is the index of stability as before,   [-1,1] is the skewness
parameter, parameters   R and  > 0 control location and scale, respectively, and ,
is given by (4). We shall use the notation S(,,) to denote the stable distribution given
by the ch.f. (9). Strictly stable laws in one dimension correspond to  = 0 for  = 1 and
 = 0 for  = 1. Symmetric univariate stable laws are strictly stable with  =  = 0. Stable
distributions are supported on the entire real line, except when  < 1 and || = 1, when
we obtain totally skewed distributions concentrated on (,) for  = 1 and (-,) for
 = -1.
The following moment formula from Samorodnitsky and Taqqu (1994), is useful in estimating
parameters of multivariate stable laws [cf. Nikias and Shao (1995)].
Proposition 2.2. Let X  S(,,0) with   (0,2) and  = 0 for  = 1. Then for any
p  (0,) we have
E|X|p
= p
C, (10)
where
C = C(,,p) =
2p-1 (1 - p/)
p

0 u-p-1 sin2
udu
1 + 2
tan2 
2
p/(2)
× cos
p

arctan  tan

2
. (11)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 137
2.4. Discrete spectral measure
An important special class of stable laws are those with a discrete spectral measure. A measure
 is discrete if
 (A) =
k
j=1
sj (A)j , (12)
where the sj 's are k points on the unit sphere, s denotes a point mass at s,
s(A) =
1 if s  A,
0 otherwise,
(13)
and j > 0 for j = 1,2,...,k. If the spectral measure  has form (12), then the corresponding
ch.f. is straightforward to compute, because in this case I in (3) takes the form:
I(t) =
k
j=1
,1 t,sj j . (14)
Because of the simple form of their ch.f.'s, stable laws with discrete spectral measures are
much easier to handle in practice than the general ones. In particular, their computer simulation
is straightforward, whereas exact algorithms for simulation of general stable vectors
are not available. The simulation of stable variates with discrete spectral measure is based
on the following representation from Modarres and Nolan (1994) [see also Samorodnitsky
and Taqqu (1994), Example 2.3.6].
Proposition 2.3. Let X  S(m, ) with  of the form (12). Then
X
d
=



m +
k
j=1

1/
j Vj sj if  = 1,
m +
k
j=1
j Vj +
2

logj sj if  = 1,
(15)
where the Vj 's are i.i.d. totally skewed, one-dimensional standard stable variables
S(1,1,0).
Since there exist exact algorithms for simulating one-dimensional stable variates [see,
e.g., Weron (1996)], representation (15) can be used to generate d-dimensional stable vectors
with discrete spectral measure.
138 T.J. Kozubowski et al.
Another important aspect of stable laws with discrete spectral measure is their role
in approximating general stable distributions. As shown in Byczkowski, Nolan and Rajput
(1993) every stable distribution can be approximated by one with a discrete spectral
measure.
Proposition 2.4. Given a stable vector X  S(m, ) in Rd with density p, for every
 > 0 there exists a positive integer k = k(,d,, ), points s1,...,sk on the unit sphere
Sd , and positive constants 1,...,k such that
sup
xRd
p(x) - p
(x) < , (16)
where p is the density of the stable distribution on Rd with a discrete  given by (12).
The value of k is given explicitly in Byczkowski, Nolan and Rajput (1993). Because of
the above approximation, in practice one usually restricts attention to laws with discrete
spectral measure, see Nolan (1998) for further discussion.
2.5. Linear combinations and risk of a financial portfolio
Return on a d-asset portfolio can be modeled as a linear combination
b,X = b1X1 +  + bdXd (17)
of the stable vector of returns on individual assets X and the vector of weights b indicating
the portion with which each asset enters the portfolio. The properties of a portfolio can
then be studied via properties of linear combinations of stable random variables.
It is well known that all linear transformations (17), which include marginal distributions
of stable vectors, are again stable. In particular, linear combinations of a stable r.v. X =
(X1,...,Xd)  S(m, ) are univariate stable S(b,b,b), where
b =
Sd
b,s

 (ds)
1/
, (18)
b =
Sd
| b,s | sign( b,s ) (ds)
Sd
| b,s | (ds)
, (19)
b =



b,m for  = 1,
b,m -
2
 Sd
b,s log b,s  (ds) for  = 1.
(20)
Parameter 
b is often called the risk of a stable portfolio. We would like to note here, that
it is necessary to have information about the spectral measure  in order to estimate that
risk. For the motivation and more discussion of the definition of risk please see Rachev and
Mittnik (2000).
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 139
2.6. Densities
All fully d-dimensional stable laws are absolutely continuous and admit bounded unimodal
densities. In general there are no closed form expressions for stable densities. For numerical
stable density computations one can use the following integral representation of stable
densities due to Abdul-Hamid and Nolan (1998).
Proposition 2.5. Let X  S(m, ) be a nondegenerate stable random vector in Rd with
d 1, and let s, s and s be given by (18)­(20). Then the density of X admits the
following form:
(i) For  = 1,
p(x) =
Sd
g,d
x - m,s
s
,s -d
s ds, (21)
where
g,d(v,) =
1
(2)d

0
cos vu - u
tan

2
ud-1
e-u
du. (22)
(ii) For  = 1,
p(x) =
Sd
g1,d
x - m,s - s + (2/)ss logs
s
,s -d
s ds, (23)
where
g1,d(v,) =
1
(2)d

0
cos vu -
2

ulogu ud-1
e-u
du. (24)
As remarked by Nolan (1998), this representation is more suitable for approximating
multivariate stable densities than the numerical inversion of the stable ch.f. [see Nolan and
Rajput (1995)], since g,d is a function of two variables regardless of the dimension d and
it is the same for any stable random vector.
2.7. An alternative parameterization
Note that the spectral measure is not necessarily a probability measure on Sd . An alternative
parameterization introduces a scale parameter
 =  (Sd)
1/
(25)
and the normalized measure
 (ds) = -
 (ds), (26)
140 T.J. Kozubowski et al.
so that  (Sd) = 1 [see, e.g., Davydov and Paulauskas (1999)]. With this new normalized
spectral measure, the ch.f. (2) takes the form
(t) = e-I(t)+i t,m
, (27)
where I is as before (with  in place of  ). We now have four parameters: the stability index
  (0,2], the scale parameter  > 0, the shift parameter m  Rd, and the normalized
spectral measure  . We shall use the notation S
(,m, ) for the distribution corresponding
to the ch.f. (27).
2.8. Association
A strong form of positive dependence of the components of a d-dimensional r.v. X =
(X1,...,Xd) is the association, introduced in Esary, Proschan and Walkup (1967). The
components of X are said to be associated if for any functions f,g :Rd  R, nondecreasing
in each coordinate, we have
Cov f (X),g(Y) 0 (28)
whenever covariance exists. Normal variables are associated if and only if they are nonnegatively
correlated (Pitt, 1982). Association of stable variables has been characterized in
terms of the spectral measure in Lee, Rachev and Samorodnitsky (1990a).
Proposition 2.6. Let X = (X1,...,Xd)  S(m, ), where 0 <  < 2. Then X1,...,Xd
are associated if and only if
 Sd
= 0, (29)
where
Sd
= s = (s1,...,sd)  Sd: for some i,j  {1,...,d}, si > 0 and sj < 0 . (30)
Thus, bivariate stable vectors are associated if and only if their corresponding spectral
measure is concentrated on the first and third quadrants.
Remark. Other notions of positive dependence include positive upper orthant dependence
(PUOD) and positive lower orthant dependence (PLOD). The variables X1,...,Xd are
PUOD if
P(X1 > x1,...,Xd > xd) P(X1 > x1)P(Xd > xd) (31)
for any x1,...,xd, and they are PLOD if
P(X1 x1,...,Xd xd) P(X1 x1)P(Xd xd), (32)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 141
so that the variables are likely to take on larger or smaller values together. It is well known
that association implies both PUOD and PLOD, but one cannot in general reverse these
implications. However, as shown in Lee, Rachev and Samorodnitsky (1990a), for stable
random vectors association is equivalent to PUOD and also to PLOD, so that all of the
above notions of positive dependence are equivalent.
The components of X = (X1,...,Xd) are said to be negatively associated if for any
1 k < d and any functions f :Rk  R, g :Rd-k  R, nondecreasing in each coordinate,
we have
Cov f (Y),g(Z) 0 (33)
whenever the covariance exists, where Y and Z are any k and (d - k)-dimensional subvectors
of X [see Alam and Saxena (1982)]. The negative association of stable random
vectors was characterized in Lee, Rachev and Samorodnitsky (1990a).
Proposition 2.7. Let X = (X1,...,Xd)  S(m, ), where 0 <  < 2. Then X1,...,Xd
are negatively associated if and only if
 S+
d = 0, (34)
where
S+
d = s = (s1,...,sd)  Sd: sisj > 0 for some i = j . (35)
Thus, a bivariate stable vector has negatively associated components if and only if the
corresponding spectral measure is concentrated on the second and forth quadrants.
3. Estimation of the index of stability
In this section we address the issue of estimating the tail index . We start with the case
when the sample comes from a univariate -stable distribution, and then consider a more
general situation where the observations are not necessarily stable, but asymptotically have
a stable-Pareto tail with index , that is
P(X1 > x) = 1 - F(x)  xL(x),
(36)
where L is some slowly varying function. Given a multivariate heavy tailed data set
X1,...,Xn, one can apply the methods of this section to one-dimensional samples corresponding
to the norms ||Xj || or the projections Xj ,b for some b  Rd.
142 T.J. Kozubowski et al.
3.1. Estimation of univariate stable parameters
Estimating the parameters of stable distributions is a challenging problem due to the fact
that the densities and distributions functions of these laws are not available in closed form.
Various estimation methods have been developed over the last 30 years, most of them
requiring numerical approximations.
Since the stable characteristic function can be written in a closed form, several estimation
techniques are based on fitting the sample characteristic function to its theoretical
counterpart. The substantial collection of papers in this area started with Press (1972b),
and include Arad (1980), Feuerverger and McDunnough (1977, 1981a, b), Kogon and
Williams (1998), Koutrouvelis (1980, 1981), Paulson and Delehanty (1984, 1985), Paulson,
Holcomb and Leitch (1975). As noted by McCulloch (1996), these estimation procedures
were reported by practitioners to have high efficiency relative to the maximum
likelihood approach. However, some of these methods are quite complex and require the
practitioner to choose certain arbitrary parameters. A discussion and comparative study of
these approaches can be found in Kogon and Williams (1998).
The maximum likelihood (ML) method for the stable case was first proposed by DuMouchel
(1971, 1973), who also discussed the asymptotic properties of the estimators. To
approximate the loglikelihood function DuMouchel (1971) employed fast Fourier transform
(FFT) for the central part of the data and series expansions for the tails. See also DuMouchel
(1975, 1983) for numerical approximation of the Fisher information matrix and
further comments on this approach. Since this early work, various numerical procedures for
approximating stable densities have been developed, which now permit an efficient computation
of the likelihood function without the grouping procedure of DuMouchel (1971).
For the ML in the symmetric case, see Brorsen and Yang (1990), McCulloch (1979, 1998).
Asymmetric stable ML was treated in Brorsen and Preckel (1993), Liu and Brorsen (1995),
Mittnik et al. (1999), Nolan (2001), Stuck (1976). As noted in Mittnik et al. (1999), one advantage
of the ML approach over most other methods is its ability to handle generalizations
to dependent or not identically distributed data arising in financial modeling (for example,
regression or various time series models with stable disturbances). An implementation of
the ML method for such generalizations can be found in Liu and Brorsen (1995) (stable
GARCH), Mittnik, Rachev and Paolella (1998) (ARMA models driven by asymmetric stable
distributions), and Brorsen and Preckel (1993), McCulloch (1998) (linear regression).
In the last section of our chapter, we utilize the maximum likelihood numerical procedures
of Nolan (1998), applicable for the most general i.i.d. stable case (available on the author's
web site).
Numerous other methods of estimating stable parameters have been suggested. Perhaps
the most commonly used estimators in empirical work are quantile procedures of
Fama and Roll (1971) for the symmetric case and their modifications to the general case
obtained by McCulloch (1986). Buckle (1995) proposed sampling based Bayesian inference
for stable laws, see also Qiou and Ravishanker (1995), Ravishanker and Qiou
(1998) for further extensions and discussion of the Bayesian approach. Nikias and
Shao (1995) derived moment estimators based on sample fractional moments. Computationally
simple estimators based on the modified method of scoring were proposed in
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 143
Klebanov, Melamed and Rachev (1994). For further references on estimating stable parameters,
see, e.g., McCulloch (1996), Rachev and Mittnik (2000). Comparative studies of
various estimators for stable parameters include Akgiray and Lamoureux (1989) and more
recent Höpfner and Rüschendorf (1999), Kogon and Williams (1998).
3.2. Estimation of the tail index 
Assume that we have a one-dimensional random sample X1,...,Xn satisfying (36) and
belonging to the domain of attraction of an -stable distribution. There is a large body
of literature concerning estimation of the tail index . Many common estimators of  are
based on a subset of the sample order statistics,
X(1)  X(n). (37)
Below we sketch few standard and some recent methods for estimating  and give references
for many others.
3.2.1. The Hill estimator
The Hill estimator [see Hill (1975)] along with its various modifications is perhaps the most
common way of estimating the tail thickness  of a financial data set [see, e.g., Jansen and
de Vries (1991), Koedijk, Schafgans and de Vries (1990), Loretan and Phillips (1994),
Phillips (1993)]. The estimator uses the k largest order statistics,
^Hill =
1
k
k
j=1
logX(n+1-j) - logX(n-k)
-1
, (38)
and arises as the conditional maximum likelihood estimator for the Pareto distribution
P(X > x) = Cx-. With the proper choice of the sequence k = k(n), the estimator is consistent
and asymptotically normal, see, e.g., Beirlant and Teugels (1989), Csörg˝o and Mason
(1985), de Haan and Resnick (1998), Deheuvels, Haeusler and Mason (1988), Goldie
and Smith (1987), Haeusler and Teugels (1985), Hall (1982), Hall and Welsh (1984, 1985),
Mason (1982). For further discussion and extensions, see, e.g., Csörg˝o, Deheuvels and Mason
(1985), Csörg˝o and Viharos (1995), Dekkers and de Haan (1993), Dekkers, Einmahl
and de Haan (1989).
An obvious problem with the Hill estimator and its generalizations discussed below is
the practical choice of k. Generally, we must have
k   and
k
n
 0 as n   (39)
to achieve strong consistency and asymptotic normality. In practice, one usually plots values
of the estimator against the values of k (obtaining the so-called Hill plot) and looks for
144 T.J. Kozubowski et al.
a stabilization (flat spot) in the graph. An alternative, more informative method of doing a
Hill plot, is described in Drees, de Haan and Resnick (2000), Resnick and St˘aric˘a (1997).
We refer the readers to Danielsson, Jansen and de Vries (1996), Embrechts, Klüppelberg
and Mikosh (1997), Kratz and Resnick (1996), Mittnik and Paolella (1999), Rachev and
Mittnik (2000), Resnick (1998), Resnick and St˘aric˘a (1997) and references therein for
more details on this and related tail estimators.
3.2.2. A shifted Hilĺs estimator
Noting that the Hill estimator is scale invariant but not shift invariant, Aban and Meerschaert
(2001) proposed the modification that is shift invariant. Their method consists of
conditional maximum likelihood estimation for the shifted Pareto distribution P(X > x) =
C(x - s)-, and yields the estimators:
^ =
1
k
k
j=1
log(X
(j) - ^s) - log(X
(k+1) - ^s)
-1
, (40)
^c =
k
n
(X
(k+1) - ^s)^
, (41)
where ^s is obtained by solving the equation
^(X
(k+1) - ^s)-1
= (^ + 1)k-1
k
j=1
(X
(j) - ^s)-1
(42)
over the set ^s < X
(k+1). Here the starred variables indicate the order statistics taken in the
decreasing order:
X
(1)  X
(n). (43)
Numerical procedures are required to compute the estimators.
3.2.3. The Pickands estimator and its modifications
Pickands (1975) introduced a tail estimator of the form
^Pick =
log2
log(X(n-k+1) - X(n-2k+1)) - log(X(n-2k+1) - X(n-4k+1))
, 4k < n,
(44)
see also Drees (1996), Rosen and Weissman (1996). Noting its poor performance on samples
from stable distributions, Mittnik and Rachev (1996) introduced a modification of (44)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 145
based on Bergström expansion of stable distribution function [see Bergström (1952), and
also Janicki and Weron (1994)]. Their unconditional Pickands estimator is of the form
^UP =
log2
logX(n-k+1) - logX(n-2k+1)
. (45)
We refer the readers to Rachev and Mittnikl (2000) for further discussion on the practical
performance and other modifications of the Pickands estimator.
3.2.4. Least-squares estimators
Taking the logarithm of both sides in relation (36) we observe that for large values of x
the points with abscissa logx and ordinate log(1 - F(x)) should approximately fall on a
straight line with slope -. Using the k largest order statistics Xn+1-j , j = 1,...,k, we
can examine the plot of logXn+1-j versus
log
j
n
 log 1 - F(Xn+1-j ) (46)
and visually estimate the slope of the resulting line. This graphical approach was suggested
by Mandelbrot (1963b).
Using these upper order statistics one can estimate the slope by the classical leastsquares
method [see Kratz and Resnick (1996), Schultze and Steinebach (1996)]. Below
we briefly describe the estimators obtained in Schultze and Steinebach (1996). Assuming
that in (36) we have L(x) = ec (which is the case for stable distributions), Schultze and
Steinebach (1996) applied the method of least squares to estimate the intercept c/ and
the slope 1/ of a straight line fit to
logXn+1-j 
c

+
1

log
n
j
, j = 1,...,k. (47)
This resulted in the following estimator of :
^(1)
LS =
1
k
k
j=1
log
n
j
logXn+1-j -
1
k2
k
j=1
log
n
j
k
j=1
logXn+1-j
-1
×
1
k
k
j=1
log2 n
j
-
1
k
k
j=1
log
n
j
2
. (48)
146 T.J. Kozubowski et al.
Another estimator was obtained in Schultze and Steinebach (1996) by the least squares
method under the assumption of zero intercept in (47):
^(2)
LS =
k
j=1
log
n
j
logXn+1-j
k
j=1
log2 n
j
. (49)
Finally, Schultze and Steinebach (1996) proposed yet another estimator of  resulting from
expressing (47) in the form
 logXn+1-j  c + log
n
j
, j = 1,...,k, (50)
and minimizing the sum of squares
k
j=1
 logXn+1-j - c - log
n
j
2
.
This produced:
^
(3)
LS =
1
k
k
j=1
log
n
j
logXn+1-j -
1
k2
k
j=1
log
n
j
k
j=1
logXn+1-j
×
1
k
k
j=1
log2
Xn+1-j -
1
k
k
j=1
logXn+1-j
2 -1
. (51)
Consistency and asymptotic normality of the above estimators are established in Schultze
and Steinebach (1996) and Csörg˝o and Viharos (1997), respectively [see also Kratz and
Resnick (1996) for similar results on their QQ estimator].
3.2.5. The M­S method
Meerschaert and Scheffler (1998) introduced a simple robust estimator for the tail index 
that is based on the asymptotics of the sum and utilizes the entire sample not just the largest
order statistics. The estimator is based on the idea that if Xi 's are i.i.d. and belong to the
domain of attraction of an -stable law with 0 <  < 2 (and their distribution function
satisfies (36)), then their sample variance,
^2
=
1
n
n
j=1
Xj - X
2
, (52)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 147
converges to an /2-stable (totally skewed) r.v. Y:
n1-2/
^2 d
 Y. (53)
Taking the logarithm on both sides of (53) we obtain the convergence
2 logn
1
^
-
1

d
 logY, (54)
where
1
^
=
logn + log ^2
2 logn
(55)
is the Meerschaert­Scheffler (M­S) estimator of 1/. The estimator is consistent and its
asymptotic distribution is that of logY for some totally skewed /2 positive stable r.v. Y.
Moreover, the estimator applies to certain dependent data. Comparing its performance with
that of Hilĺs estimator, Meerschaert and Scheffler (1998) concluded that it works as well
as the latter in most cases, and substantially better when applied to stable data, see Meerschaert
and Scheffler (1998) for further details.
4. Estimation of the stable spectral measure
4.1. Tail estimators
A method of estimating the spectral measure of a stable r.v. Y based on a random sample
X1,...,Xn (56)
from the domain of attraction of Y was proposed by Rachev and Xin (1993) and Cheng
and Rachev (1995). The method, referred to as the Rachev­Xin­Cheng (RXC) method
by Nolan and Panorska (1997), is based on the limiting relation in Proposition 2.1. To
estimate  (D), where  is the (normalized) spectral measure of Y [cf. parameterization
(27)], choose a large value of r and calculate the proportion of the Xi's with the norm
exceeding r that belong to the set D when normalized, that is
 (D) =
{Xi/ Xi  D and Xi > r}
{ Xi > r}
. (57)
Equivalently, we can choose an integer k = k(n) n/2 and consider the set
Xi1 ,..., Xik (58)
148 T.J. Kozubowski et al.
of the k largest order statistics connected with the corresponding sample of the norms:
X1 ,..., Xn . (59)
Then, the RXC estimator of  is the discrete measure on Sd that assigns the mass of 1/k
to each of the unit vectors
Xi1
Xi1
,...,
Xik
Xik
. (60)
The authors suggest taking about 20% of the largest order statistics. Under appropriate
technical conditions the estimator is strongly consistent and asymptotically normal.
A similar method was recently proposed by Davydov et al. (2000) and discussed further
in Davydov and Paulauskas (1999). We refer to this approach as the Davydov­PaulauskasRackauskas
(DPR) method. Assuming that the sample (56) is actually from an -stable
distribution with a zero shift vector m and a symmetric (normalized) spectral measure  ,
and the sample size n is a perfect square n = k2 for some integer k, the method consists
of splitting the data into k groups of k variables each, choosing a vector with the largest
norm within each group, leading to a set of k vectors Xi1 ,...,Xik , and again estimating 
by the empirical measure based on the unit vectors (60). The consistency and asymptotic
normality of the resulting estimators,
 (D) =
1
k
k
j=1
ID
Xij
Xij
, (61)
is established in Davydov and Paulauskas (1999).
Both RXC and DPR methods do not assume any prior knowledge of  and are well
suited for the S
(m,, ) parameterization, as they provide estimators for the normalized
spectral measure. Once the spectral measure and the index  are estimated, the scale
parameter  can be estimated by methods described in Section 5.
4.2. The empirical characteristic function method
The method described below, proposed in Nolan, Panorska and McCulloch (2001) and
investigated in Nolan and Panorska (1997), assumes that the sample comes from an stable
distribution with shift vector m equal to zero. First, estimate the index of stability
and center the data by the sample mean (if  > 1) or sample median (if  < 1). In Nolan,
Panorska and McCulloch (2001) the value of  was estimated by the average 1
d
d
j=1 ^j ,
where ^j is an estimate of the index obtained from a univariate sample X1j ,...,Xnj (the
quantile method of McCulloch (1986) was used to obtain these). Then, the method uses
the sample to estimate the exponent I of the stable ch.f. (2) (with m = 0):
^I(t) = -logn(t), (62)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 149
where the quantity n is the sample characteristic function,
n(t) =
1
n
n
j=1
ei t,Xj . (63)
For some grid t1,...,tk  Sd, the quantity
^IECF = ^In(t1),..., ^In(tk) (64)
is the empirical ch.f. (ECF) estimate of I. If  is a discrete measure of the form (12),
then the exponent I is given by (14), and we can estimate  = (1,...,k) by solving
the following system of linear equations:
I = A, (65)
where I = ^IECF is an estimate of I given by (64) and A is a k × k (complex) matrix
[aij ]i,j=1,...,k with
aij = ,1 ti,sj . (66)
If the grid is chosen so that the inverse of A exists, then the solution of the system (65)
is  = A-1I.
For a general spectral measure, divide the unit sphere into k non-overlapping patches
Aj with some central points sj , where j = 1,...,k, and consider an approximation of 
of the form (12), where i =  (Aj ) (which is always possible in view of Proposition 2.4).
When d = 2, it is convenient to take the arcs
Aj =
2(j - 3/2)
k
,
2(j - 1/2)
k
, j = 1,...,k, (67)
centered at
sj = cos
2(j - 1)
k
,sin
2(j - 1)
k
 Sd, j = 1,...,k. (68)
We would again estimate I by (64) and solve the system (65) to obtain the estimates of
the weights j .
As reported in Nolan and Panorska (1997), in practice there are some problems with
the direct implementation of the above method; the matrix A may be ill-conditioned and
the solution of the system (65) may include negative or complex numbers (although the
values of j must be real and positive). Thus, in practice one should restate the problem as
a constrained quadratic programming problem,
minimize I - A = (I - A ) (I - A ) subject to  0, (69)
150 T.J. Kozubowski et al.
which guarantees a nonnegative solution  . We refer the readers to Nolan (1998), Nolan
and Panorska (1997), Nolan, Panorska and McCulloch (2001) for examples and further
discussion of these issues.
4.3. The projection method
The projection (PROJ) method was introduced in McCulloch (1994) and studied in Nolan
and Panorska (1997), Nolan, Panorska and McCulloch (2001). As before, assume that the
data have been shifted so that the parameter m is zero. The method is similar to the ECF
method, since we estimate the weights j at sj of a discrete spectral measure  of the
form (12) by solving the linear system of Equations (65). However, the PROJ method uses
a different value of I, the estimate of I, obtained from estimators of univariate stable
parameters applied to a one-dimensional sample
X1,tj ,..., Xn,tj , j = 1,...,k, (70)
where t1,...,tk  Sd is a suitably chosen grid on the unit sphere. More precisely, for each
t  Rd the r.v. X1,t is one-dimensional stable with parameters given by (18)­(20) and
ch.f.
(u) = Eeiu t,X
= Eei ut,X
= (ut) = e-I(ut)
, (71)
where I is the characteristic exponent of the Xj 's. Now, we can estimate the scale ^(tj )
and skewness ^(tj ) (and also the shift ^(tj ) if  = 1) of the univariate stable law corresponding
to the sample (70), and use them to estimate the ch.f. (9) of this univariate law.
Then, we can equate the above estimate with the right-hand side of (71) with u = 1 to
estimate the quantity I on the grid t1,...,tk:
^In(tj ) =



^
(tj ) 1 - i ^(tj )tan

2
for  = 1,
^(tj ) 1 - i ^(tj ) for  = 1.
(72)
For the index  McCulloch (1994) recommend using the pooled estimate obtained by
averaging the univariate estimates obtained for each of the univariate samples (70). Thus,
the PROJ estimate of I on the grid t1,...,tk is the quantity
^IPROJ = ^In(t1),..., ^In(tk) . (73)
Now, the weights j of the spectral measure are obtained as before by solving the system
(65). For examples and further discussion, please see McCulloch (1994), Nolan and
Panorska (1997), Nolan, Panorska and McCulloch (2001).
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 151
5. Estimation of the scale parameter
Let us consider the problem of estimating the scale parameter  of the stable S
(,m, )
distribution given by the ch.f. (27) (where  is the normalized spectral measure). As before,
we shall assume that the distribution is strictly stable with  > 1, so that m = 0. We
extend the moment estimators of Davydov and Paulauskas (1999) who considered the case
of symmetric spectral measure.
Note that if X  S
(,0, ) then Y = -1X  S
(1,0, ), so that for any 0 < p < 
we have
E X p
= p
C(, ,p), (74)
where
C(, ,p) = E Y p
(75)
is independent of  and can be computed for a given values of  and  . Then, approximating
E X p by the corresponding sample moment we obtain
^n =
1
nC(, ,p)
n
j=1
Xj
p
1/p
. (76)
Alternatively, we might use moment estimator for univariate stable variables on the i.i.d.
observations
X1,t ,..., Xn,t (77)
for some t  Rd. Then, by (18), (19), the above variables are univariate S(t,t,0), where
t = 
Sd
t,s

 (ds)
1/
(78)
and
t =
Sd
| t,s | sign( t,s ) (ds)
Sd
| t,s | (ds)
. (79)
Then, for any 0 < p < , we have
E X,t
p
= p
C(,t,p)C1(, ,p), (80)
152 T.J. Kozubowski et al.
where C(,t,p) is given by (11) and
C1(, ,p) =
Sd
t,s

 (ds)
p/
. (81)
Now, approximating (80) with the sample p-moment we obtain the following estimator
of :
~n =
1
nC(,t,p)C1(, ,p)
n
j=1
Xj ,t
p
1/p
. (82)
6. Extensions to other stable models
In this section we briefly discuss two generalizations of multivariate stable laws that often
compete with them in modeling financial data: the -stable laws that arise as limiting distributions
in the random summation scheme and operator stable laws arising as limits in
ordinary summation (5) but normalized by linear operators an.
6.1. -stable laws
Let X1,X2,... be a sequence of i.i.d. random vectors in Rd and let p, p  (0,1), be
a family of integer-valued random variables independent of the Xi's. Assuming that p
converges to infinity (in probability) as p  0, we can study the limiting distributions of
the random sums
ap
p
j=1
(Xj + bp), (83)
where ap > 0 and bp  Rd. It follows from transfer theorems [see, e.g., Rosínski (1976)]
that if the variables pp converge in distribution to a positive r.v. Z with the Laplace
transform (s) = Eexp(-sZ) and the Xj 's are in the domain of attraction of some -stable
distribution with ch.f. , then the random sums (83) will converge to a random variable
with the ch.f. of the form
 (t) =  -log(t) . (84)
The variables with the ch.f. (84), referred to as the -stable laws ­ see, e.g., Klebanov
and Rachev (1996), Kozubowski and Panorska (1998, 1999b), can be described by the
same parameters as the corresponding stable laws: the tail index , location vector m, and
spectral measure  . Strictly -stable laws are given by (84) with a strictly stable ch.f. .
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 153
We use the notation (m, ) for the distribution corresponding to the ch.f. (84) with 
given by (2).
The -stable laws are essentially location-scale mixtures of stable laws [see, e.g.,
Kozubowski and Panorska (1998)] and for a light-tailed r.v. Z have the same tail behavior
as the corresponding stable laws. More precisely, the tail behavior of each coordinate
of a -stable r.v. X is of the form P(Xk > x) = O(x-) as x   under the following
conditions [see Kozubowski and Panorska (1996, 1998)]:
* EZ <  if X is strictly -stable,
* EZ1 <  and  = 1 or E|Z logZ| <  and  = 1, if X is not strictly -stable.
Under the above conditions, the same tail behavior applies to every linear combinations
X,b of X, the order statistics of the vector X (as well as their absolute values), and the
norm X , see Kozubowski and Panorska (1998) for details. Note that these conditions are
satisfied, for example, by the geometric stable laws discussed below.
Remark. Although the tails of -stable laws are essentially of the same type as those of
stable distributions, -stable densities may behave very differently near the mode than their
stable counterparts (may be more peaked, or even infinite) which may lead to an improved
fit when modeling financial data.
Kozubowski and Panorska (1999b) showed that if the spectral measure is discrete, then
truly d-dimensional -stable random vectors admit a representation similar to that of stable
laws given in Proposition 2.3:
Proposition 6.1. Let Y  (m, ) with  of the form (12) and 0 <  < 2. Then
Y
d
=



Zm + Z1/
k
j=1

1/
j Vj sj if  = 1,
Zm + Z
k
j=1
Vj +
2

log(j Z) j sj if  = 1,
(85)
where the Vj 's are i.i.d. totally skewed, one-dimensional standard stable variables
S(1,1,0), independent of Z.
Thus, -stable random variates are straightforward to simulate if  is discrete. Distributions
with general  can be approximated by those with discrete spectral measure [see
Kozubowski and Panorska (1999b)] as in the stable case, so that in practice we can restrict
attention to the case with discrete  .
6.1.1. Geometric stable laws
An important special case are the limiting distributions of (83) when the variables p are
geometric with mean 1/p in which case the variables pp converge to a standard expo-
154 T.J. Kozubowski et al.
nential variable with the Laplace transform (s) = (1 + s)-1. We then obtain the class of
geometric stable law (GS) laws GS(m, ) with the ch.f.
 (t) = 1 + I(t) - i t,m
-1
, (86)
where m  Rd and I is given by (3). In financial applications, where these laws have
been successfully applied [see, e.g., Kozubowski and Panorska (1999a), Kozubowski and
Rachev (1994), Mittnik and Rachev (1991, 1993a)] the r.v. p represents the moment when
the probabilistic structure governing the returns changes, so that the random sum
p
j=1
Xj (87)
represents the total return up to this random time. In case  = 2, we obtain the multivariate
Laplace distribution [see, e.g., Kozubowski and Podgórski (2000)], which may be particularly
well suited for financial applications due to its simplicity and flexibility [see, e.g.,
Kozubowski and Podgórski (2001)], although the tails of these laws, being heavier than
Gaussian tails, are not as heavy as those of stable and geometric stable laws. More information
on theory and applications of GS laws can be found in Kozubowski and Rachev
(1999).
6.1.2. Statistical issues
Most estimation procedures for stable laws can be extended to the corresponding -stable
distributions. For simplicity we consider the problem of estimating  and  of a strictly
geometric stable distribution given by the ch.f. (86) with m = 0 and  = 2, based on a
random sample
Y1,...,Yn. (88)
For estimating , the tail estimators of Section 3.2 can be applied to one-dimensional
samples corresponding to (88) by taking the norms of the Yi's or their projections Yi,b
for some b  Rd. These apply regardless of whether the sample is actually geometric stable
or only belongs to a geometric stable domain of attraction. Alternatively, assuming that the
Yi's are geometric stable, one can use estimators for univariate geometric stable parameters
[see, e.g., Kozubowski (1983, 2001), Rachev and Mittnik (2000)] applied to the projections
Yi,b .
To estimate the spectral measure  , one can use the RXC tail estimator discussed in
Section 4.1 since geometric stable distributions have the same domains of attraction as the
corresponding stable laws (that have the same  and  ), see, e.g., Klebanov and Rachev
(1996). Alternatively, the empirical characteristic function method discussed in Section 4.2
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 155
can be modified to accommodate the geometric stable case. Assuming that the sample (88)
is from a GS distribution, we estimate the exponent I of the GS ch.f. (86) as follows:
^I(t) =
1
 n(t)
- 1, (89)
where  n is the sample characteristic function (63) based on the Yi's. The rest is the same
as in the stable case. For some grid t1,...,tk  Sd , the quantity (64) is the empirical ch.f.
(ECF) estimate of I. If  is a discrete measure of the form (12), then I is given by (14),
and we can estimate  = (1,...,k) by solving the system of linear equations of the
form (65), where I = ^IECF is an estimate of I given by (64) and A is a k × k (complex)
matrix with the entries specified in (66). If the inverse of A exists, then the solution of the
system (65) is  = A-1I. To avoid the same numerical problems as in the stable case, in
practice one should restate the problem as a constrained quadratic programming problem
(69). The projection method of Section 4.3 can be extended similarly.
6.2. Operator stable laws
If we have a heavy-tail multivariate data with different tail indexes in different directions,
then the multivariate stable (as well as the -stable) laws are no longer appropriate to
model such data. Instead, we can consider the class of multivariate laws with stable marginal
distributions, introduced in Resnick and Greenwood (1979), that arise as limiting
distributions in the summation scheme (5) where the scaling factors are diagonal matrices,
diag(an1,...,and), for some positive ani 's. The resulting limiting marginally stable
random vectors X possess a stability property similar to (1),
X1 +  + Xn
d
= nE
X + Dn, (90)
where the Xi's are i.i.d. copies of X, E is a diagonal matrix
E = diag
1
1
,...,
1
d
, 0 < i 2, i = 1,...,d, (91)
called the characteristic exponent of X, and nE denotes the diagonal matrix
nE
= diag n1/1 ,...,n1/d . (92)
Remark. More general operator stable (OS) laws arise as the limits in (5) when the sums
are normalized by some linear operators an [see Sharpe (1969)]. For a comprehensive
review of the theory of OS laws see Jurek and Mason (1993).
156 T.J. Kozubowski et al.
Marginally stable OS laws satisfying (90) with the characteristic exponent E of the form
(91) can be described in terms of their characteristic function. If all 's are strictly between
one and two, we have
(t) = exp C
Sd

0
ei t,rEs
- 1 - i t,rE
s
dr
r2
 (ds) + i t,m , (93)
where m  Rd is the shift parameter, C > 0 controls the scale, and  is a probability
measure on the unit sphere Sd (the normalized spectral measure, also called the mixing
measure). If all the 's of the characteristic exponent E in (91) are equal, then (93) reduces
to the stable ch.f. with the same spectral measure. We use the notation OS(m,C,E, ) to
denote the distributions with the ch.f. (93) with E given by (91). Similarly to the stable
case, the measure  determines the dependence among the components of a marginally
stable vector. For example, if X  OS(m,C,E, ) is positively or negatively associated,
then the spectral measure  satisfies the condition (29) or (34), respectively [see Mittnik,
Rachev and Rüschendorf (1999)].
6.2.1. Statistical issues
Estimating the parameters of an OS(m,C,E, ) distribution is similar to the stable case.
Since all marginal distributions are univariate stable, one can obtain estimates of the i 's
by using the methods for univariate stable laws (see Section 3.1) for each of the d samples
X1j ,...,Xnj , j = 1,...,d. (94)
For samples from a domain of attraction of an OS law we can again consider univariate
samples (94) and apply the methods of Section 3.2, or use the moment estimator of E
based on the sample covariance matrix [see Meerschaert and Scheffler (1999)].
To estimate C and  , one can use a generalization of the tail estimator of the spectral
measure for stable laws described in Section 4.1 [see Mittnik, Rachev and Rüschendorf
(1999), Scheffler (1999)]. First, write each of the data points (different than zero) in the
unique form
Xi = (Xi)E
si, (95)
where (Xi) > 0 is the "radius" of Xi and si is a point on the unit sphere Sd [these are the
so-called Jurek coordinates, see Jurek (1984)]. Next, for some integer k = k(n) consider
the k largest of the (Xi)'s, that is the k largest order statistics
(Xi1 ),...,(Xik ) (96)
corresponding to the random sample
(X1),...,(Xn). (97)
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 157
Then, the estimator of  is the discrete measure on Sd that assigns the mass of 1/k to each
of the unit vectors
si1 ,...,sik (98)
corresponding to these order statistics via (95). Thus, the probability assigned by the estimated
spectral measure  to a set A  Sd is the fraction of the points (98) falling in the set
A.
The corresponding estimator of C is
C =
k
n
 Y
, (99)
where Y is the k-th largest of the values (97). More details regarding the estimation of 
(including the asymptotic properties of estimators) can be found in Mittnik, Rachev and
Rüschendorf (1999), Scheffler (1999).
7. Applications
In this section we present an example of fitting bivariate financial data sets with stable models.
We fit a bivariate stable and a bivariate operator stable models to two data sets. Our
data consists of 1162 daily DAX30 Index (DAX), FTSE100 Index (UKX), and S&P500
Index (SPX) prices for the period from 1/1/95 to 11/3/99. The raw indexes are first transformed
into log-returns by taking natural logarithms of the quotients of their consecutive
values. We analyze log-returns (1161 observations) Xt = ln(Yt /Yt-1), where the Yt 's are
the raw daily index prices. The goal is to fit reasonable models to the bivariate vectors
(DAX, UKX) and (UKX, SPX). This section is modeled after Nolan and Panorska (1997).
We start with Exploratory Data Analysis (EDA) which focuses on general properties
of the data with particular attention to the amount of variability in each data set. We first
plot the log-returns of individual indexes as time series (see Figure 1). The plots show
relatively large number of high spikes in the returns which points out to high volatility and
the possibility that the log-returns' innovations are non-Gaussian.
The next step is to plot density histograms (total area under a density histogram equals
one) of the log-returns and check for indications of long tails which again suggest more
variability than allowed by a Gaussian distribution. It helps at this time to fit a Gaussian
distribution to the data and overlay the histogram with the fitted Gaussian density curve.
Fitting a Gaussian model amounts to estimating its mean and standard deviation from the
data using the sample mean and sample standard deviation. We also check for unimodality
and symmetry of the data. The density histograms of the univariate log-returns overlayed
with the Gaussian (and stable) models' densities are presented in Figure 2.
We note that the histograms are much more peaky in the center and have heavier tails
than the Gaussian models. Since the histograms are fairly symmetric and unimodal, the
158 T.J. Kozubowski et al.
Fig. 1. Time series plots of the daily log-returns for the three indexes (1/1/95­11/3/99). Top panel: DAX
log-returns. Middle panel: UKX log-returns. Bottom panel: SPX log-returns.
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 159
two distributional problems (sharp center peaks and long tails) with the Gaussian model
could be alleviated by the stable approach. The next step in model fitting is deciding if we
should use stable or operator stable models. To answer that question we have to estimate
the tail indexes  for all three financial indexes' log-returns. If the 's for a pair of indexes
are the same, we fit a stable law to their bivariate distribution, otherwise we work with
an operator stable model. To fit univariate stable models to the indexes' log-returns we
estimated their parameters using maximum likelihood procedure of Nolan (2001) and its
numerical implementation (STABLE 2.16) due to Nolan and available on his web page. We
report the parameters according to the parametrization used by Samorodnitsky and Taqqu
(see (9)). Estimation results are summarized in Table 1.
We used STABLE 2.16 to compute densities of the stable models with the estimated parameters.
To evaluate and compare stable and Gaussian fit we overlayed density histograms
of the data with stable and Gaussian densities of the models. The results appear in Figure 2.
We note much better fit of the stable models. From now on we will work under the
assumption that the individual stock indexes have univariate stable distributions. Since the
tail parameters for DAX and UKX and for DAX and SPX are different, we model bivariate
distribution of DAX and UKX using an operator stable distribution. The tail parameters for
UKX and SPX appear to be the same and thus we will fit a bivariate stable model to UKX
and SPX data. To fit these bivariate models we need estimates of the spectral measures
for both DAX­UKX and UKX­SPX portfolios. To fit an operator stable distribution, we
estimated the spectral measure using the method described in Section 6.2. The estimated
cumulative normalized (total mass equal to one) spectral measure in radian coordinates
is presented in Figure 3. Since the spectral measure seems to be concentrated on the first
and third quadrants, we conclude that these variables are positively associated (see our
comments in Section 6.2). Conversion to Jurek coordinates was performed using a Fortran
program due to Meerschaert (personal communication), all other numerical and graphical
work was done by the authors in Splus2000 Professional.
To estimate the bivariate stable spectral measure for the UKX­SPX portfolio we used
all four methods described in Section 4: the tail estimators (RXC and DPR), the projection
method (PROJ) and the empirical characteristic function method (ECF). The numerical
implementation of the RXC, PROJ and ECF estimation procedures was done with the
program MVSTABLE (Version 2.0) of Nolan available on J.P. Nolan's web site3 with 40
weights, that is using a 40 points estimation grid on the unit circle. The DPR estimator
Table 1
Index alpha beta gamma delta
UKX 1.28 0 0.0055 0.0004
DAX 1.57 0.31 0.0076 0.0041
SPX 1.28 0 0.0058 0.001
3 See http://academic2.american.edu/jpnolan/ for Stable 2.16 and MVSTABLE programs.
160 T.J. Kozubowski et al.
Fig. 2. Density histograms with Gaussian (solid line) and stable (dashed line) fitted densities. Top panel: DAX
log-returns. Middle panel: UKX log-returns. Bottom panel: SPX log-returns.
Ch. 4: Statistical Issues in Modeling Multivariate Stable Portfolios 161
Fig. 3. Estimated cumulative (normalized) operator spectral measure for (DAX, UKX) data.
Fig. 4. Estimates of the stable spectral measure for (UKX, SPX) vector: Solid line ­ RXC estimator, dotted line
­ ECF estimator, long-dashed line ­ PROJ estimator, and short-dashed line ­ DPR estimator.
was computed for the first 1,156 (= 342) observed vectors of UKX­SPX log-returns (from
1/1/95 to 10/26/99). Numerical work for the DPR estimator was done by the authors. The
graph of the estimated cumulative normalized spectral measure in radian coordinates for
DAX-SPX is given in Figure 4.
As the spectral measure appears to be concentrated in the first and third quadrants we
believe that UKX and SPX are positively associated (see Proposition 2.6).
To check the goodness of fit of our model we suggest methods described in Nolan and
Panorska (1997). These include plotting parameters (e.g., scale) of one-dimensional projections
of the sample (in several directions) computed first directly from the projected
sample and then using the estimate of the spectral measure. A good fit will be indicated by
162 T.J. Kozubowski et al.
general agreement between the parameters of these projections computed using two different
methods. For a more detailed discussion of the choice of gridsize and its relationship
with the goodness of fit, we refer the reader to Nolan and Panorska (1997).
To summarize, we performed EDA and fit two data sets (DAX, UKX) and (UKX, SPX)
with bivariate operator stable and stable models. The indexes seem to be positively associated,
which is an important information in constructing a portfolio. The estimates of
the stable spectral measures can be used to estimate risk of a portfolio using the methods
described in Section 2.5.
Acknowledgment
We thank Dr. Mark Meerschaert, Department of Mathematics, University of Nevada for
helpful comments and for his Fortran routines. The research of the first two authors was
partially supported by NSF grant DMS-0139927.
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Chapter 5
JUMP-DIFFUSION MODELS
WOLFGANG J. RUNGGALDIER
Dipartimento di Matematica Pura ed Applicata, Universit di Padova, 7 Via Belzoni, I-35131, Padova, Italy
web: http://www.math.unipd.it/runggaldier/index.html
Contents
Abstract 170
Keywords 170
1. Introduction 171
2. Preliminaries 173
2.1. Univariate point processes (Poisson jump processes) 173
2.2. Multivariate and marked point processes 175
2.3. Martingale representation 177
2.4. Exponential formula; generalized Ito formula 178
2.5. Absolutely continuous transformation of measures 180
3. Market models with jump-diffusions 182
3.1. Asset-price and term structure models with additive jumps 182
3.1.1. Asset price models with jumps 182
3.1.2. Term structure models with jumps 183
3.2. Jump-diffusion models driven by hidden jump processes 184
3.3. Asset prices as diffusions sampled at the jump times of a jump process 185
4. Martingale measures: Existence and uniqueness (Market price of risk and market
completion) 186
4.1. The case of jump-diffusion asset price models 187
4.2. The case of jump-diffusion term structure models 190
5. Hedging in jump-diffusion market models 193
5.1. Hedging when the market is completed 194
5.1.1. Asset-price models 194
5.1.2. Term structure models 196
5.2. Hedging when the market is not complete 198
6. Pricing in jump-diffusion models 201
6.1. General aspects 201
6.2. Computational aspects 203
References 207
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
170 W.J. Runggaldier
Abstract
We discuss jump-diffusion type models for financial market as well as methods for pricing
and hedging of contingent claims in such markets. We consider both, asset price and term
structure models, and deal also with situations when there is a stochastic volatility correlated
with the jumps and when one has very small time scales, i.e., high frequency data.
To make the presentation possibly self-contained, in a preliminary section we recall some
basic notions from stochastic analysis for jump-diffusions.
Keywords
jump-diffusions, Poisson point processes, marked point processes, Cox processes, hidden
processes, martingale measures, market price of risk, pricing and hedging in incomplete
markets, market completion, risk minimization, stochastic volatility, high frequency data,
computing expectations of functionals of jump-diffusions
Ch. 5: Jump-Diffusion Models 171
1. Introduction
Most of the standard literature in Finance, in particular for pricing and hedging of contingent
claims, is based on the assumption that the prices of the underlying assets follow a
diffusion-type process, in particular a geometric Brownian motion (GBM).
Documentation from various empirical studies shows that such models are inadequate,
both in relation to their descriptive power, as well as for the mispricing that they might
induce. The contributions to the present volume deal with various generalizations of the
basic GBM; here we concentrate on the fact that returns of various asset prices and interest
rates may exhibit a jumping behaviour. We thus study possible superpositions of jump and
diffusion processes, namely what is called jump-diffusion processes. Jump-diffusions form
a particular class of Levy processes. Our purpose here is not to study the general case of
a Levy driving process, but rather to concentrate on the specific aspects of the subclass
of jump-diffusions. Jump-diffusion models have also some intuitive appeal in that they
let prices and interest rates change continuously most of the time, but they also take into
account the fact that from time to time larger jumps may occur that cannot be adequately
modeled by pure diffusion-type processes.
Among the earlier empirical studies, documenting a jumping behaviour in prices and
interest rates, one may quote Ball and Torous (1985), Jorion (1988). There are also studies,
such as Babbs and Webber (1997), putting forward specific sources of jumps in interest
rates like moves by central banks. On the other hand, a first approach developing further
the basic Black and Scholes (BS) model with the inclusion of jumps appears to be that of
Merton (1976). Since the introduction of jumps in the BS model implies that derivative
prices are no longer determined by the principle of absence of arbitrage alone, Merton
solved the pricing problem by assuming that the jump risk was not systematic. This was
later criticized by showing that such an assumption is equivalent to the existence of a
market portfolio, that contains the underlying asset and that does not present a jumping
behaviour [for a discussion on this point see, e.g., Björk and Näslund (1998)]. Further
studies then appeared showing that jumps in stock returns are indeed systematic. Another
early approach is that in Cox and Ross (1976), where the market remains however complete
since the authors consider just a simple jump-type process with fixed jump amplitude and
thus with a single source of randomness. One of the major purposes of this chapter is now
to try to give an overview of the state of the art of jump-diffusion modeling in stock and
bond markets as well as of the corresponding approaches for pricing and hedging.
It was further documented in empirical studies [see, e.g., Bakshi, Cao and Chen (1997)]
that a combination of jumps and stochastic volatility leads to even better fits and allows to
avoid implied volatility skews. Stochastic volatility models are treated elsewhere and so in
this chapter we limit ourselves to stochastic volatility in conjunction with jump-diffusion
modeling, also because empirical documentation gives evidence for a jump-type behaviour
in the volatility and of a correlation between jumps in volatility and jumps in prices. In fact,
as mentioned, e.g., in Naik (1993), it is natural to expect that, if the volatility jumps, also
the price should jump. A further purpose of the present chapter is then to discuss issues
related to such more general jump-diffusion-stochastic-volatility modeling.
172 W.J. Runggaldier
On very small time scales actual prices do not really change continuously over time,
but rather at discrete random points in time in reaction to trade and/or significant new
information. We shall show that also such situations can be captured by models featuring a
combination of diffusion and jump processes, that is however different from the canonical
jump-diffusion processes.
The outline of the chapter is as follows. In Section 2 we recall some preliminary notions
from stochastic analysis for jump-diffusion processes, such as a martingale representation
result and generalized versions of the Ito formula as well as of the Girsanov measure transformation.
We limit ourselves to those notions that will be used in the sequel. In Section 3
we then describe various market models based on jump-diffusion representations. More
precisely, in line with the introductory remarks above, we shall consider first canonical
jump-diffusion models for stock and bond markets, then consider jump-diffusions correlated
with stochastic volatility and, finally, combinations of diffusions and jumps to describe
high frequency data. In Section 4 we discuss existence and uniqueness of martingale
measures in a jump-diffusion setting, exhibiting also the market price of (jump-diffusion)
risk. Some emphasis is given to the notion of completion of the market as a tool to obtain
a unique martingale measure. In this context it is also pointed out that uniqueness of a
martingale measure does not necessarily always imply completeness of the market in the
sense of hedging, namely that every claim can be duplicated by a self financing portfolio.
In Section 5 we then concentrate on hedging in jump-diffusion market models having two
goals in mind: first, investigating whether and when a market, that has been completed to
yield a unique martingale measure, is also complete in the sense of hedging. Second, to
study the hedging problem when the market cannot be completed or market completion
is inappropriate. In such cases there is always some residual risk and so one may want to
choose a strategy such as to minimize a criterion related to this risk. Finally, Section 6 is
devoted to the problem of pricing in jump-diffusion market models. With jumps and/or
stochastic volatility, the market is incomplete. The principle of absence of arbitrage alone
is then insufficient to define uniquely a price and so the preference structure of investors
has to come into play to determine a pricing measure. From the point of view of pure pricing,
the problem reduces formally to that of determining a specific martingale measure. In
Section 6.1 we mention various approaches to this effect, related to the literature, in particular
approaches based on market completion and on the relationship between the choice
of a hedging criterion and that of a martingale measure. Once a martingale measure has
been chosen, there remains the problem of the actual computation of the expectation of the
discounted claim and this is dealt with in Section 6.2.
Unavoidably, this overview of the state of the art may turn out to be incomplete and
reflects the specific interests and competences of the author. Among the topics that are
not discussed here, we just mention the American-type options in a jump-diffusion setting
[for this see, e.g., Mulinacci (1996), Pham (1997)] and Portfolio Optimization [see, e.g.,
Framstad, Oeksendal and Sulem (2001) and references therein]. The same has to be said
about the references to the literature: while we have tried to take into account a good deal
of recent papers on the subject, we have only quoted a small selection of previous papers
in order to keep the list within a reasonable size. Still, we hope to have succeeded in giving
Ch. 5: Jump-Diffusion Models 173
a sufficiently comprehensive account on models and methods related to jump-diffusions in
financial markets.
2. Preliminaries
In this section we recall basic definitions and results needed for the study of jump-diffusion
models, limiting ourselves to multivariate (univariate) and marked point processes and
assuming that the reader is familiar with the corresponding notions concerning diffusion
processes. In addition to the basic definitions we recall here a martingale representation
result and discuss the Ito formula and Girsanov's measure transformation, generalized to
jump-diffusion processes. The main reference for this section is Brémaud (1981), from
which most of the contents of the section are taken.
2.1. Univariate point processes (Poisson jump processes)
A point process is intended to describe events that occur randomly over time. It can be
represented as a sequence of nonnegative random variables
0 = T0 < T1 < T2 < ,
where the generic Tn is the n-th instant of occurrence of an event. One makes the usual
assumption of nonexplosion, according to which
T = lim  Tn = +.
The process may equivalently be represented via its associated counting process Nt where
Nt = n if t  [Tn,Tn+1), n 0, or, equivalently, Nt =
n 1
1{Tn t}. (1)
It counts the number of events up to and including time t. The nonexplosion condition
becomes Nt <  for t 0. Both, Tn and Nt , are defined on some probability space
(,F,P) with a filtration Ft to which Nt is adapted.
A point process Nt is called a Poisson point process if
(i) N0 = 0;
(ii) Nt is a process with independent increments;
(iii) Nt - Ns is a Poisson random variable with a given parameter s,t .
Usually one assumes s,t =
t
s u du for a deterministic function t ; the latter is called
the intensity of the Poisson point process Nt . If Ft is the filtration FN
t , generated by Nt ,
and t  1, then Nt is called a standard Poisson process. It is also easily seen that, if Nt
is a Poisson process with intensity t  , then Tn+1 - Tn are i.i.d., exponential random
174 W.J. Runggaldier
variables with parameter . A natural interpretation of the intensity and of this latter property
comes from relating the above setup with the usual Poisson model, that is based on the
following assumptions:
(a) the probability of one change/jump in an interval of length  is    + o();
(b) the probability of two or more changes/jumps in an interval of length  is o();
(c) the number of changes/jumps in nonoverlapping intervals are stochastically indepen-
dent.
In this setup one can in fact consider the two "dually" related random variables:
(1) A discrete random variable X describing the number of changes/jumps in a time interval
of given length T and having as distribution the usual Poisson distribution, i.e.,
P{X = k} =
(T )k
k!
e-T
, k  N.
(2) A continuous random variable T describing the time that is needed to obtain k successive
changes/jumps and for which the distribution is of the Gamma-type with density
fT (t) =
k
(k)
tk-1
e-t
, T > 0.
The parameter  is the same in both cases and corresponds to the  in assumption (a)
above.
It will be convenient to consider also the case when the intensity of a Poisson process is
itself an adapted process being driven by some background process. This can be explained
by a two-step randomization procedure: first one draws at random a trajectory of the background
process, say Zt ; then one generates a Poisson process with intensity t = (t,Zt),
where the dependence also on t allows to incorporate seasonality effects. We now have a
Poisson process Nt conditionally on Zt and it is called a doubly stochastic Poisson process,
or a Cox process [see Cox (1955)]. Formally, we require that the random intensity t is F0measurable,
i.e., FZ
  F0. For additional details of the intensity of a Poisson process we
refer to Brémaud (1981).
Notice that the above characterizations (i)­(iii) of a Poisson process parallel those of
a Wiener process: both are processes with independent increments; the increments of a
Wiener process are normally distributed, while those of a Poisson process are Poisson distributed.
The Wiener process is the basic building block for processes with continuous
trajectories, the Poisson process is a basic building block for processes with jumping trajectories.
On the other hand, while the Wiener process is itself a martingale, a Poisson
process as such is not. It becomes a martingale if one subtracts from Nt the process given
by its mean. Indeed,
Mt := Nt -
t
0
s ds (2)
Ch. 5: Jump-Diffusion Models 175
is an Ft -martingale by the F0-measurability of t , assuming in addition that E{
t
0 u du}
< . By (iii) one then has in fact
E{Nt - Ns|Fs} = E
t
s
u du|Fs (3)
which implies that E{Nt} <  and that Mt in (2) is an Ft -martingale. Equality (3) admits
a generalization in the form
E

0
Cs dNs = E

0
Css ds (4)
that has to be valid for all nonnegative, Ft -predictable processes Ct and as such characterizes
a doubly stochastic Poisson process with intensity t [see Brémaud (1981)].
2.2. Multivariate and marked point processes
Let Tn be a (univariate) point process and Yn, n 1, a sequence of random variables with
values in {1,2,...,K}, all defined on the same (,F,P). For each k = 1,...,K we may
then consider the counting process
Nt(k) :=
n 1
1{Tn t}1{Yn=k}.
Each Nt(k) is a univariate point process and the various Nt (k)'s have no common jumps,
i.e., Nt(k) Nt(h) = 0, t 0 and all k = h. Analogously to the case of univariate point
processes, here too we have now two equivalent representations, either as the double sequence
(Tn,Yn)n 1 or as the K-vector process Nt = (Nt (1),...,Nt (K)) and this process
is called a multivariate, more precisely a K-variate point process. As in the univariate case,
here too we shall mainly use the representation as the K-vector process Nt and we have
formula (2) with Mt a K-vector and t the K-vector intensity process whose components
are the individual intensities of the components Nt (k) of Nt .
Considering the representation (Tn,Yn), we may interpret Tn as the n-th occurrence of
some phenomenon and Yn as an attribute or mark of this phenomenon. We may then speak
of (Tn,Yn) as a marked point process, or space-time point process and extend its definition
to allow Yn to take values in a general measurable mark space (E,E). We synthesize the
foregoing in the following
Definition 2.1. An E-marked point process is a double sequence (Tn,Yn)n 1 where
(i) Tn is a (univariate) point process;
(ii) Yn is a sequence of E-valued random variables.
Obviously, the univariate and multivariate point processes are special cases of a marked
point process.
176 W.J. Runggaldier
Generalizing the representation of a multivariate point process in the form of the
K-vector process Nt , we associate to each A  E the counting process
Nt (A) :=
n 1
1{Tn t}1{YnA}
and let simply Nt := Nt (E). Considering the filtration
FN
t :=  Ns(A); s t, A  E
define the associated (random) counting measure
p (0,t],A = Nt (A), t 0, A  E, (5)
which is -finite under the assumption of nonexplosion of Tn. This measure allows to
obtain more concise expressions via integrals of the form
t
0 E
H(s,y)p(ds,dy) =
n 1
H(Tn,Yn)1{Tn t} =
Nt
n=1
H(Tn,Yn). (6)
Again, we may represent an E-marked point process equivalently as the double sequence
(Tn,Yn) or as the counting measure p(ds,dy).
To introduce now the intensity process in this more general setup, assume that for each
A  E, the point process Nt (A) admits the intensity t (A). This then leads to a measurevalued
intensity t (dy) so that, generalizing (4), one has
E

0 E
H(s,y)p(ds,dy) = E

0 E
H(s,y)s(dy)ds (7)
that has to be valid for all nonnegative Ft -predictable E-marked processes H (given a
filtration Ft on , Ft -predictability here means measurability with respect to P(Ft)  E
where P(Ft) is the predictable -field on (0,) × ). We have also the generalization of
(2) in the form
q(ds,dy) = p(ds,dy) - s(dy)ds, (8)
where q(ds,dy) is a (signed) measure-valued martingale in the sense that
t
0 E
H(s,y)q(ds,dy)
is a (P,Ft )-martingale (local martingale) for each Ft -predictable E-marked process H ,
satisfying appropriate integrability conditions. The most common form of intensity is
t (dy) = t mt(dy), (9)
Ch. 5: Jump-Diffusion Models 177
where t is nonnegative Ft -predictable and represents the intensity of the Poisson process
Nt(E), while mt(dy) is a probability measure on E (typically, the Yn will be i.i.d., independent
of Nt (E)). The pair (t ,mt(dy)) is called the (P,Ft )-local characteristics of
p(ds,dy).
Notice finally that, as in the univariate case, we may let t(dy) depend on some driving
F0-measurable random process Zt, leading to a doubly stochastic marked point process.
If, in the representation (9), t is a deterministic time function, the marked point process is
called a marked Poisson process.
2.3. Martingale representation
Martingale representation results are widely used in Finance, especially when it comes to
solving hedging problems. For pure "Wiener-martingales" we have in fact the well-known
result that every square integrable martingale with respect to the filtration generated by a
Wiener process is, up to an additive constant, a stochastic integral of the Ito type. We shall
now recall a corresponding result for point-process martingales that we formulate in the
most general case of a marked point process. We have in fact the following theorem [see
Theorem VIII, T8 in Brémaud (1981)]
Theorem 2.2. Let (,F,Ft,P) be a probability space satisfying the "usual assumptions"
where Ft = F0  F
p
t with F
p
t the filtration generated by a marked point process,
represented by the counting measure p(dt,dy). Then any (P,Ft )- martingale Mt admits
the representation
Mt = M0 +
t
0 E
H(s,y)q(ds,dy) (10)
with q() as in (8) and H an integrable (with respect to t (dy)) Ft -predictable E-marked
process. This representation is essentially unique.
In the case of a multivariate (and therefore also univariate) point process, the representation
(10) becomes
Mt = M0 +
K
k=1
t
0
Hs(k) dNs(k) - s(k)ds , (11)
where [Ht(1),...,Ht(K)] is Ft -predictable with Ht(k) integrable with respect to t (k).
This representation result can be generalized according to Jacod and Shiryaev (1987) to
include martingales that are simultaneously "Wiener" and point-process martingales and
that will have some relevance later on.
Theorem 2.3. Given a Wiener process wt and a marked point process p(ds,dy), let
Ft :=  ws,p (0,s],A ,B; 0 s t, A  E, B  N
178 W.J. Runggaldier
with N the collection of P -null sets from F. Then any (P,Ft )-local martingale Mt has
the representation
Mt = M0 +
t
0
s dws +
t
0 E
H(s,y) p(ds,dy) - s(dy)ds , (12)
where t is predictable and square integrable and H is an Ft-predictable E-marked
process, integrable with respect to t (dy).
2.4. Exponential formula; generalized Ito formula
With the definition of a marked point process and of integrals in the form of (6), we may
now consider processes of the general type
Xt = X0 +
t
0
s ds +
t
0
s dws +
t
0 E
 (s,y)p(ds,dy) (13)
that are called jump-diffusion processes and where the coefficients satisfy the implicit integrability
conditions, t is adapted and  (t,y) is predictable in the sense as defined previously.
As usual, we may rewrite (13) in differential form and consider, more specifically,
differential equations of the type
dXt = Xt- t dt + t dwt +
E
 (t,y)p(dt,dy) , (14)
where we write Xt- with t- because of the predictability requirement in the last coefficient
and where  (t,y) > -1. Notice that [see (6)] the last term in (14) can also be written as
E
 (t,y)p(dt,dy) =  (t,Yt )dNt, (15)
where Nt = Nt (E) = p((0,t],E) is the total number of jumps and Yt denotes the piecewise
constant, left-continuous time interpolation of the sequence Yn. Notice also that, in
the case of a multivariate (in particular univariate) point process, this last term in (14) takes
the form
E
 (t,y)p(ds,dy) =
K
k=1
t(k)dNt(k). (16)
We shall not discuss here in detail equations of the form (14), in particular the uniqueness of
their solutions, but limit ourselves to show that a solution to (14) is given by the following
Ch. 5: Jump-Diffusion Models 179
Exponential formula
Xt = X0 exp
t
0
s -
1
2
2
s ds +
t
0
s dws +
t
0
log 1 +  (s,Ys) dNs
= X0 exp
t
0
s -
1
2
2
s ds +
t
0
s dws
Nt
n=1
1 +  (Tn,Yn) . (17)
While the diffusion part in this expression follows from the usual Ito formula, the jump
part follows from the so-called exponential formula of Stieltjes­Lebesgue Calculus [see
Theorem T4 of Appendix A4 in Brémaud (1981)], but it can also be obtained from the
generalized Ito formula as we shall show next. For this purpose let a process Xt satisfy the
general Equation (13). Given a C1,2-function F(t,X), we have the generalized Ito formula
dF(t,Xt) = Ft ()dt + FX()t dt +
1
2
FXX()2
t dt + FX()t dwt
+ F t,Xt- +  (t,Yt) - F(t,Xt-) dNt (18)
that, in the specific case of (14), becomes
dF(t,Xt) = Ft ()dt + FX()Xt t dt +
1
2
FXX()X2
t 2
t dt + FX()Xt t dwt
+ F t,Xt- 1 +  (t,Yt) - F(t,Xt-) dNt (19)
and where, again, Nt = Nt (E) = p((0,t],E) and () stands for (t,Xt); the pedices in F
denote partial derivatives. Notice that, if (19) is written in integral form, for the last term
on the right we have the two equivalent representations
t
0
F s,Xs- 1 +  (s,Ys) - F(s,Xs-) dNs =
Nt
n=1
F(Tn,XTn ) - F(Tn,XT -
n
) ,
where the right-hand side remains the same also in the more general case of (18).
We shall now use the generalized Ito formula (19) to obtain the solution (17) of Equation
(14). Choosing F(t,X) = logX, from (19) and (14) we have
dF = t dt -
1
2
2
t dt + t dwt + log 1 +  (t,Yt ) dNt
from which
logXt = logX0 +
t
0
s -
1
2
2
s ds +
t
0
s dws +
t
0
log 1 +  (s,Ys-) dNs,
(20)
i.e., we obtain (17) by taking the exponential on both sides in (20).
180 W.J. Runggaldier
2.5. Absolutely continuous transformation of measures
We recall that in the classical case of Wiener driven diffusion processes, the Girsanov-type
measure transformation concerns a translation of the Wiener process that in turn induces a
change in the drift of the diffusion equation. In view of its generalization below, we recall
here the basic result of Girsanov's transformation, conveniently reformulated for a finite
time horizon t  [0,T ].
Theorem 2.4 (Girsanov's measure transformation). Given a filtered probability space
(,F,Ft,P) with F = t Ft , let t  [0,T ] with T given and t be a square integrable
predictable process. Define L = (Lt ) by
dLt = Ltt dwt, L0 = 1, (21)
and suppose that, for all t, EP {Lt } = 1. Then there exists a probability measure Q on F,
equivalent to P , with dQ = LT dP such that
dwt = t dt + dw
Q
t , (22)
where wQ
t is a Q-Wiener process.
Conversely, if Ft = Fw
t , then every probability measure Q, equivalent to P , has the
above structure.
Notice that the second statement relies on martingale representation and requires thus
the filtration Ft to be the one generated by the Wiener process.
As mentioned, Girsanov's measure transformation allows to change the drift in a diffusion
equation. In fact, suppose that under P we have
dXt = atXt dt + tXt dwt
and that we would like to change to a measure Q  P ( meaning equivalent to), under
which the same Xt satisfies
dXt = rtXt dt + t Xt dwQ
t .
In this case just take t = -1
t (rt - at).
If, besides a Wiener wt , we now have also a marked point process represented by a
counting measure p(dt,dy), a Girsanov-type measure transformation allows, in addition
to the translation of the Wiener, to perform also a change in the intensity process of the
point process part. We have [see Theorem VIII, T10 in Brémaud (1981), see also Björk,
Kabanov and Runggaldier (1997)]
Theorem 2.5. On the finite time interval [0,T ] let p(dt,dy) be an E-marked point process
with (P,Ft)-local characteristics (t ,mt(dy)). Let t 0 be Ft - predictable and ht (y)
0 an Ft -predictable E-indexed process such that, P -a.s. and for all t  [0,T ],
t
0
ss ds < ;
E
ht(y)mt(dy) = 1.
Ch. 5: Jump-Diffusion Models 181
Define Lt = L(1)
t  L(2)
t where L(1)
t satisfies (21) and L(2)
t satisfies
dL
(2)
t =
E
tht (y) - 1 L
(2)
t- q(dt,dy) (23)
with q(dt,dy) = p(dt,dy) - t mt(dy)dt ­ the martingale measure associated with
p(dt,dy). If EP {L
(2)
t } = 1 for all t, then all the statements of Theorem 2.4 hold true in addition
to the fact that p(dt,dy) has the (Q,Ft)-local characteristics (t t ,ht(y)mt(dy)).
Notice that, using (21) and (23), we have for the Radon­Nikodym derivative Lt
dLt = d L(1)
t  L(2)
t = L(1)
t- dL(2)
t + L(2)
t dL(1)
t
= Ltt dwt + Lt-
E
t ht(y) - 1 q(dt,dy), L0 = 1. (24)
Using the exponential formula (17), we have that a solution of (24) is given by
Lt = exp -
1
2
t
0
2
s ds +
t
0
s dws
× exp
t
0 E
1 - shs(y) sms(dy)ds
Nt
n=1
Tn hTn (Yn) . (25)
In the case of a multivariate (in particular univariate) point process (Nt(1),...,Nt (K))
with (P,Ft )-intensities (t (1),...,t(K)), consider an Ft -predictable process (t (1),...,
t (K)) such that, P -a.s. and for t  [0,T ], K
k=1
t
0 s(k)s(k)ds < . Define then L
(2)
t
by
dL(2)
t =
K
k=1
t (k) - 1 L(2)
t- dNt(k) - t(k)dt (26)
instead of by (23) making also corresponding changes in (24) and (25) for the RadonNikodym
derivative Lt , namely
Lt = exp -
1
2
t
0
2
s ds +
t
0
s dws
K
k=1
exp
t
0
1 - s(k) s(k)ds
Nt (k)
n=1
Tn (k) .
(27)
Then, under Q, the intensities become (t(1)t (1),...,t (K)t (K)).
Notice, finally, that a condition to have EP {L
(2)
t } = 1 can be found in Theorem VIII,
T11 of Brémaud (1981).
182 W.J. Runggaldier
3. Market models with jump-diffusions
In this section we introduce various jump-diffusion type models that were studied in the
literature and that we shall be dealing with in the sequel. In the first two subsections we discuss,
for asset price and term structure models respectively, the canonical jump-diffusion
models in which there are two additive terms: a diffusion term and a jump term. In the last
two subsections we then discuss diffusion/jump-diffusion models with stochastic volatility,
where the latter is also described in terms of a jumping process. In addition, in the last
subsection we model asset price behaviour on very small time scales where actual prices
do not change continuously in time but rather at discrete random time points in reaction
to trades and significant information. This then leads to a rather peculiar combination of
diffusion and jump processes.
3.1. Asset-price and term structure models with additive jumps
As mentioned in the Introduction, the asset price evolution can perhaps be adequately
described by a GBM for most of the time, but from time to time a large jump may occur and
this cannot be adequately captured by a GBM. It appears thus natural to introduce models,
where a jump process can be superimposed on a GBM, e.g., by adding to the diffusion
term also a jump term. In a first subsection we discuss this modeling issue in the context
of asset prices, while in the second subsection we concentrate on interest rate modeling.
3.1.1. Asset price models with jumps
In this section we adapt the outline of Section 7.2 in Lamberton and Lapeyre (1997). Let
the price St of a risky asset jump at the random times T1,...,Tn,... and suppose that the
relative/proportional change in its value at a jump time is given by Y1,...,Yn,... respectively.
We may then assume that, between two jump times, the price St follows a Black and
Scholes model for a Wiener process wt , that Tn are the jump times of a Poisson process
Nt with intensity t and that Yn is a sequence of random variables with values in (-1,).
This description can be formalized by letting, on the intervals [Tn,Tn+1),
dSt = St (t dt + t dwt) (28)
while, at t = Tn, the jump is given by Sn = STn - ST -
n
= ST -
n
Yn so that
STn = ST -
n
(1 + Yn) (29)
which, by the assumption that Yn > -1, leads always to positive values of the prices. Using
the standard Ito formula to obtain the solution to (28) as well as a recursive argument based
on (29), it is easily seen that, at the generic time t, St can be given the following equivalent
representations
St = S0 exp
t
0
s -
2
s
2
ds +
t
0
s dws
Nt
n=1
(1 + Yn)
Ch. 5: Jump-Diffusion Models 183
= S0 exp
t
0
s -
2
s
2
ds +
t
0
s dws +
Nt
n=1
log(1 + Yn)
= S0 exp
t
0
s -
2
s
2
ds +
t
0
s dws +
t
0
log(1 + Ys)dNs , (30)
where, as before, Yt is obtained from Yn by a piecewise constant and left continuous time
interpolation. By the generalized Ito formula (19), the process St in (30) is easily seen to
be a solution of
dSt = St-[t dt + t dwt + Yt dNt ]. (31)
This equation corresponds to (28) with the addition of a jump term and is a particular case
of the general jump-diffusion model (14) ((15)) when  (t,y) = y. In what follows we shall
thus consider the more general version of (31) given by
dSt = St- t dt + t dwt +  (t,Yt )dNt (32)
that corresponds to (14) in the version of (15) and can thus equivalently be represented as
dSt = St- t dt + t dwt +
E
 (t,y)p(dt,dy) . (33)
If the marked point process is in particular a multivariate (or univariate) point process
(Nt(1),...,Nt (K)), then (32) ((33)) takes the form (see also (16))
dSt = St- t dt + t dwt +
K
k=1
t(k)dNt(k) . (34)
We finally point out that the marked point process in (32) ((33)) may be doubly stochastic
in the sense specified in Sections 2.1 and 2.2 and this allows for further flexibility when it
comes to modeling.
Remark 3.1. Occasionally, in the financial literature one finds model (32) ((33)) written
in the form
dSt = St-[t dt + t dwt + dJt],
where, in the specific case when (32) reduces to (31), Jt := Nt
n=1 Yn, while in the general
case Jt := Nt
n=1  (Tn,Yn). Furthermore, in models of the form (31) one may find the last
term Yt dNt written as (Yt - 1)dNt; in this latter case, instead of (29), we would then have
STn = ST -
n
Yn = ST -
n
YTn .
3.1.2. Term structure models with jumps
Among the basic objects in term structure models we have the zero-coupon bonds with
prices p(t,T ) (the price, at t, of a bond maturing at T ), forward rates f (t,T ) (the
rate, contracted at t, for instantaneous borrowing at T ), and the short rate r(t). There
184 W.J. Runggaldier
exist some well-known relationships among these quantities, in particular f (t,T ) =
- logp(t,T )/T ; r(t) = f (t,t). Since interest rates, and therefore also bond prices may
indeed jump, one may consider the following jump-diffusion models for the above three
quantities
dr(t) = at dt + bt dwt +
E
c(t,y)p(dt,dy), (35)
df (t,T ) = (t,T )dt + (t,T )dwt +
E
(t,T ;y)p(dt,dy), (36)
dp(t,T ) = p(t-,T ) m(t,T )dt + v(t,T )dwt +
E
n(t,T ;y)p(dt,dy) , (37)
where the differential is with respect to the time argument t, not the maturity T . Notice
that only (37) has the factor p(t-,T ) also in the right-hand side. This guarantees (see the
exponential formula (17)) positivity of p(t,T ) as it should be since p(t,T ) is the price of
an asset; the interest rates r(t), f (t,T ) need not necessarily be positive. Given the wellknown
relationships between the three quantities in (35)­(37), there obviously has to exist
a relationship also between the coefficients in these models. This relationship can be found
in Proposition 2.2. of Björk, Kabanov and Runggaldier (1997).
So far we have mentioned only continuously compounded interest rates. In financial
markets also discretely compounded or simple rates such as LIBOR rates play an important
role. Given a fixed accrual period , denote by L(t,T ) the forward rate, contracted at
t < T , for the interval from T to T + . Jump-diffusion models for L(t,T ) are studied in
Glasserman and Kou (1999) under the form
dL(t,T ) = L(t-,T ) (t,T )dt + (t,T )dwt + dJ(t,T ) , (38)
where (see Remark 3.1) J(t,T ) = Nt
n=1  (Tn,Yn) for a given marked point process represented
by the double sequence (Tn,Yn) [for a more general setup beyond jump-diffusions
see Jamshidian (1999)]. Notice that the relationship
L(t,T ) =
1

exp
T +
T
f (t,s)ds - 1 (39)
between discretely and continuously compounded forward rates induces a relationship between
the coefficients of the corresponding dynamic equations (36) and (39).
3.2. Jump-diffusion models driven by hidden jump processes
As mentioned in the introduction, empirical studies have led to consider also combinations
of jumps and stochastic volatility, where the volatility presents a jump-type behaviour and
is possibly also correlated with the jumps in the prices. As pointed out in Naik (1993), it
is in fact natural to expect that, when the volatility jumps, also the price should jump. One
can capture these aspects by a jump-diffusion model, where the coefficients depend on a
Ch. 5: Jump-Diffusion Models 185
hidden/latent jump process Zt that affects also the intensity of the marked point process in
the jump term (doubly stochastic marked point process). Formally, and limiting ourselves
to asset price models of the form of (33) (that are equivalent to (32) and include (34)), we
then have
dSt = St- t(Zt)dt + t(Zt)dwt +
E
 (t,y;Zt-)p(dt,dy) , (40)
where Zt is any jump process with non-predictable jumps (could also be a Markov jump
process) and p(dt,dy) is the counting measure of a doubly stochastic marked point process
with intensity t (Zt-,dy). Notice that Zt affects the jump part both through the intensity
as well as through the proportional jump sizes and it affects them in a predictable way.
3.3. Asset prices as diffusions sampled at the jump times of a jump process
As was mentioned in the Introduction, on very small time scales the real asset prices do not
change continuously over time, but rather only at discrete random points in time in reaction
to trades and/or significant new information. This makes jump processes attractive also for
modeling high frequency data and here we give a description of such a modeling approach
according to Frey and Runggaldier (2001, 1999). Marked point processes as models for
high frequency data were also studied independently by various authors in the recent literature
[see, e.g., Geman, Madan and Yor (1999), Rogers and Zane (1998), Rydberg and
Shephard (1999)]. The models in Frey and Runggaldier (2001, 1999) are more in the spirit
of jump-diffusions in that they consider a combination, although not an additive one, of a
diffusion and a jump process as follows: given is a background price process of the diffusion
type and this process is then sampled according to the random jump times of a jump
process. This setup allows also to incorporate a possible correlation between (stochastic)
volatility and price jumps in the way mentioned in the previous section, by letting again
Zt be a hidden process that drives the volatility of the background diffusion process and at
the same time also the intensity of the (doubly stochastic) jump process that determines the
random sampling times. In more formal terms, the logarithm t of the background price
process is supposed to satisfy
dt = vt (Zt)dwt (41)
with wt a Wiener process independent of Zt . The process Zt is the hidden or latent state
variable process that can be interpreted as modeling the rate at which new information is
absorbed by the market. It may be given as a diffusion or as a finite state Markov process.
Next consider a univariate doubly stochastic Poisson process (a Cox process) Nt with
intensity t = t (Zt-). The time dependence of this  as well as of v in (41) is introduced
to incorporate systematic patterns in trading activity. The actual price process is now such
that its logarithm Lt satisfies
Lt = Tn-1 for t  [Tn-1,Tn) (42)
186 W.J. Runggaldier
with Tn the jump times of Nt . The given model can thus be interpreted as a stochastic
volatility model, evaluated at random times Tn. It is easily seen that the process Lt in (42)
satisfies
dLt = (t - TNt - )dNt, (43)
where TNt- is the time of the last jump strictly prior to t and it is thus a marked point
process with local characteristics (t (Zt),N(0,
t
TNt vs
ds)) where N(m,2) denotes a
Gaussian r.v. with mean m and variance 2.
Notice that we may choose an intensity of the form
t (Zt) = (1)
t + (2)
t Zt (44)
so that Nt can be seen as the sum Nt = N(1)
t + N(2)
t of two independent jump processes:
N(1)
t with deterministic intensity (1)
t corresponding to noise trading and N(2)
t corresponding
to informed trading.
One interesting aspect of the above model is that it makes it clear how sample path properties
matter when it comes to volatility estimation: the volatility in a diffusion model, i.e.,
its quadratic variation, can be approximated arbitrarily well by the sum of the observed
squared increments. For the given piecewise constant processes the empirical quadratic
variation is useless for volatility estimation, even if computed over very small time inter-
vals.
We finally point out that the definition, that was given in Section 2 concerning a doubly
stochastic Poisson process, in particular that t is F0-measurable, has as consequence the
fact that Nt and Zt cannot have common jumps and that the actual trading activity, namely
the realization of the point process Nt , does not affect the law of Zt . In economic terms
this means that, in the given model, trading is caused purely by exogenous factors such as
fundamental information, and not by the observed past trading activity.
4. Martingale measures: Existence and uniqueness
(Market price of risk and market completion)
In each of the models discussed in Section 3, individual asset prices are driven by at least
two independent sources of randomness so that the corresponding market models are incomplete.
Based on the extended Girsanov-type measure transformation recalled in Section
2.5, in this section we shall discuss existence and, where applicable, uniqueness of
martingale measures, thereby exhibiting also the market price of (jump-diffusion) risk.
Uniqueness of the martingale measure will be mainly related to completion of the market.
We want to point out that, as will be shown in more detail in the next Section 5 on hedging,
it is not necessarily true that, if a market is completed to yield a unique martingale
measure, then it is also genuinely complete in the sense that every contingent claim can be
duplicated by a self financing portfolio. In fact, for marked point process with an infinite
mark space, i.e., with an infinite number of sources of randomness, it will be shown in
Ch. 5: Jump-Diffusion Models 187
Section 5.1.2 that uniqueness of the martingale measure implies only some form of approximate
completeness. In this Section 4 we shall limit ourselves to the jump-diffusion
asset price and term structure models of Section 3.1. In Section 4.1 below we treat the case
of jump-diffusion models for asset prices and show that the market can relatively easily be
completed to yield a unique martingale measure if the jump part corresponds to a marked
point process with a finite number of marks (multivariate point processes). For an infinite
number of marks the situation is studied in more detail in Section 4.2 below in the context
of term structure models.
4.1. The case of jump-diffusion asset price models
We start with a jump-diffusion model, where the jump part corresponds to a univariate
Poisson point process with P -intensity t , namely (see (34) for K = 1)
dSt = St-[t dt + t dwt + t dNt]
= St- (t + tt )dt + t dwt + t dMt (45)
with (see (2)) Mt = Nt -
t
0 s ds the P -martingale corresponding to Nt . The RadonNikodym
derivative for an absolutely continuous change of measure from P to Q, that
implies a translation of the Wiener by t and a change of the Poisson intensity from t to
t t , is (see (27) for K = 1)
Lt = exp
t
0
(1 - s)s -
1
2
2
s ds +
t
0
s dws +
t
0
logs dNs . (46)
Defining the Wiener and Poisson martingales wQ
t and MQ
t by (see (22) and (2))
dwQ
t = dwt - t dt,
dMQ
t = dNt - t t dt
(47)
the dynamics of St under Q become
dSt = St- (t + t t + tt t )dt + t dw
Q
t + t dM
Q
t . (48)
Taking as numeraire the usual money market account Bt , where dBt = rtBt dt, we immediately
see that Q is a martingale measure, i.e., a measure under which St := B-1
t St is a
martingale, if t and t 0 are chosen such that t + t t + tt t = rt . From here we
see that, for each pair (t,t ) with t 0 arbitrary and
t = -1
t (rt - t - tt t ), (49)
we obtain a martingale measure, i.e., we can obtain infinitely many martingale measures,
one for each choice of t .
Concerning the market price of risk t , from (45) and (49) we have
t := t + tt - rt = tt - t t - tt t = -t t - tt(t - 1) (50)
188 W.J. Runggaldier
from where we see that (-t) can be interpreted as risk premium per unit of diffusion
volatility, whereas -t (t -1) can be interpreted as risk premium per unit of jump volatility.
On an arbitrage-free market all assets have, at a given time t, the same diffusion- and
jump-risk premia and they determine, via the Girsanov transformation, i.e., via (46), the
equivalent martingale measure Q.
We obtained infinitely many martingale measures because, for a single risky asset, we
had two independent sources of randomness. One may thus expect that, by adding a further
asset, one can complete the market to obtain a unique martingale measure. Consider then,
in addition to St in (45), an asset with price St satisfying
dSt = St-[ t dt + t dwt + t dNt]. (51)
Notice that St could correspond to the price of a derivative asset with underlying St . In
fact, if one is given the explicit expression of this derivative price in terms of St , i.e.,
St = F(t,St ), then (51) is straightforwardly obtained from (45) by use of the generalized
Ito formula (19). Since the two risk premia t and t (t - 1) have to be the same for all
assets, we may impose (49) on both assets with prices St and St respectively, namely
t = -1
t (rt - t - tt t ) = -1
t (rt - t - tt t ) (52)
from where one immediately gets
t t =
rt (t - t ) + (t t - t t)
t t - t t
. (53)
Inserting this expression in (49) it follows that
t =
t( t - rt) - t(t - rt)
t t - t t
. (54)
We have thus obtained unique risk premia and, consequently, a unique martingale measure
provided the coefficients in (45) and (51) are such that t t - t t = 0 and that t t in
(53) is positive.
With the unique martingale measure we may expect to have also obtained a complete
market in the sense that, by investing in a self financing way in the two assets with prices
St and St , one can duplicate any claim. In Section 5.1.1 we shall show that, for the given
market model, this is indeed the case.
It is easily seen that, if the jump part in the jump-diffusion model corresponds to a
multivariate Poisson process, i.e., if instead of (45) we have (see (34))
dSt = St- t dt + t dwt +
K
k=1
t(k)dNt(k)
= St- t +
K
k=1
t(k)t(k) dt +
K
k=1
t(k)dMt(k) (55)
Ch. 5: Jump-Diffusion Models 189
with Mt(k) = Nt (k) -
t
0 s(k)ds, then the previous results admit a straightforward extension.
In particular, (49) becomes
t = -1
t rt - t -
K
k=1
t (k)t(k)t (k) (56)
and the market price of risk is
t = -tt -
K
k=1
t(k)t (k) t (k) - 1 . (57)
This time the generic k-th term t(k)t (k)(t(k) - 1) on the right can be interpreted as
risk premium per unit of jump volatility of type k.
Again we obtain infinitely many martingale measures by choosing freely t (k) 0,
(k = 1,...,K), and t according to (56). Having now K + 1 independent sources of randomness,
we may expect that one can complete the market by adding K further assets to
obtain a unique equivalent martingale measure. This can be done along the lines of (52)(54)
although this time the calculations are more complicated and the conditions on the
coefficients more cumbersome.
Finally, we consider the more general model (33) (or, equivalently, (32)) with a possibly
infinite number of marks. Using the P -martingale measure q() in (8), by analogy to (45)
and (55) we may rewrite (33) as
dSt = St- t dt + t dwt +
E
 (t,y)p(dt,dy)
= St- t +
E
 (t,y)t(dy) dt + t dwt +
E
 (t,y)q(dt,dy) . (58)
Using the particular form of the intensity given in (9), we also have
E
 (t,y)t(dy) = tt with t =
E
 (t,y)mt(dy) (59)
and so (58) becomes, quite analogously to (45),
dSt = St- (t + tt )dt + t dwt +
E
 (t,y)q(dt,dy) . (60)
Consider then, instead of (46), the more general Radon­Nikodym derivative (25) that we
rewrite here in the form analogous to (46) as
Lt = exp
t
0
(1 - s hs)s -
1
2
2
s ds +
t
0
s dws +
t
0
log shs(Ys) dNs ,
(61)
190 W.J. Runggaldier
where hs = E hs(y)ms(dy). Define next the Wiener and jump martingales wQ
t and
qQ(dt,dy) by (see (22) and (8), (9) as well as (47))
dwQ
t = dwt - t dt,
qQ(dt,dy) = p(dt,dy) - t t ht(y)mt(dy)dt.
(62)
The dynamics of St under Q then become
dSt = St- (t + t t + tt t )dt + t dw
Q
t +
E
 (t,y)qQ
(dt,dy) , (63)
where t = E  (t,y)ht(y)mt(dy). The measure Q is now a martingale measure if t and
t 0 as well as ht (y) 0 are chosen so that t + t t + tt t = rt , which leads to the
following relation corresponding to (49)
t = -1
t (rt - t - tt t ). (64)
Again, this leads to infinitely many martingale measures but, unless the mark space is finite,
to complete the market in order to obtain a unique equivalent martingale measure one needs
infinitely many assets. We shall discuss this situation in more detail in the context of bond
markets in the next subsection.
To complete the analogy with the previous cases, notice that this time the market price
of risk becomes (by (60) and (64))
t := t + tt - rt = t t - t t - tt t = -tt - t (tt - t)
= -t t - t
E
 (t,y) t ht(y) - 1 mt(dy). (65)
This time one may interpret [tht (y)-1]mt(dy) as risk premium per unit of jump volatility
of type y.
In this latter context of a more general model of type (45) we want to point out that
a methodology to obtain all equivalent martingale measures has also been worked out in
Prigent (2001).
We close this subsection by mentioning that, depending on the purpose, one can single
out some specific martingale measures among the various possible ones in a jump-diffusion
model, where the market has not been completed. As an example, the construction of the
so-called minimal martingale measure in a univariate Poisson jump diffusion model can
be found in Runggaldier and Schweizer (1995). From a more practical point of view, an
obvious possibility is always that of calibrating the model to market data.
4.2. The case of jump-diffusion term structure models
Consider first a term structure model where, under a given measure P , the (continuously
compounded) forward rates f (t,T ) and the (zero coupon) bond prices p(t,T ) satisfy (36)
and (37) respectively, namely
df (t,T ) = (t,T )dt + (t,T )dwt +
E
(t,T ;y)p(dt,dy), (66)
Ch. 5: Jump-Diffusion Models 191
dp(t,T ) = p(t-,T ) m(t,T )dt + v(t,T )dwt +
E
n(t,T ;y)p(dt,dy) . (67)
We shall also make the ad hoc assumptions that all objects are specified in a way to guarantee
the validity of the various operations that will have to be performed, such as differentiation
under the integral sign and interchange of the order of integration.
For later use we recall from Björk, Kabanov and Runggaldier (1997) the relationship
between the coefficients in (66) and (67): if f (t,T ) satisfies (66), then p(t,T ) satisfies
(67) with



m(t,T ) = r(t) + A(t,T ) +
1
2
S(t,T )
2
,
v(t,T ) = S(t,T ),
n(t,T ;y) = eD(t,T ;y) - 1,
(68)
where r(t) = f (t,t) is the short rate and



A(t,T ) = -
T
t
(t,s)ds,
S(t,T ) = -
T
t
(t,s)ds,
D(t,T ;y) = -
T
t
(t,s;y)ds.
(69)
In the given bond market there are, at least theoretically, infinitely many assets, namely the
bonds for all possible maturities T > t. A martingale measure Q is now a measure under
which all these bond prices, discounted with respect to the money market account, are
(local) martingales. We are therefore not even sure whether in such a given market model
there exists a martingale measure and so our first purpose is to investigate the existence of
such a measure.
Following essentially Björk, Kabanov and Runggaldier (1997) and considering general
marked point processes, we also take the general form of the Radon­Nikodym derivative
Lt , namely (see (24) where, for simplicity, we put t  1)
dLt = Lt t dwt + Lt-
E
hs(y) - 1 q(ds,dy), (70)
where (see (8) and (9))
q(ds,dy) = p(ds,dy) - sms(dy), (71)
i.e., we assume that, under P , the local characteristics of the marked point process
p(ds,dy) are (t ,mt(dy)). By Theorem 2.5 we know that, under the measure Q that corresponds
to Lt in (70), the local characteristics become (t ,ht (y)mt(dy)) so that, defining
(see also (62))
dw
Q
t = dwt - t dt,
qQ(dt,dy) = p(dt,dy) - t ht(y)mt(dy)ds
(72)
192 W.J. Runggaldier
the bond prices p(t,T ) satisfy, under Q, the dynamics
dp(t,T ) = p(t-,T ) m(t,T ) + v(t,T )t + t
E
n(t,T ;y)ht(y)mt(dy) dt
+ v(t,T )dw
Q
t +
E
n(t,T ;y)qQ
(dt,dy) . (73)
A necessary condition for the existence of martingale measure Q is then that there exist
a predictable process t and a predictable E-indexed process ht(y) 0 such that the
conditions of Theorem 2.5 hold and
m(t,T ) + v(t,T )t + t
E
n(t,T ;y)ht(y)mt(dy) = r(t). (74)
Notice that this implies for the market price of risk a relation analogous to (65), namely
t := m(t,T ) + t
E
n(t,T ;y)mt(dy) - r(t)
= -v(t,T )t - t
E
n(t,T ;y) ht (y) - 1 mt(dy). (75)
We shall now translate condition (74), involving the coefficients of (67), into a condition
involving the coefficients of (66), namely of the forward rates. Using (68), condition (74)
becomes
A(t,T ) +
1
2
S(t,T )
2
+ S(t,T )t +
E
hs(y)(t,T ;dy) = 0 (76)
with (t,T ;dy) := (eD(t,T ;y) - 1)tmt(dy) and with A, S and D as in (69).
When building a term structure model it is often convenient to specify all objects directly
under a martingale measure Q and this obviously imposes some restrictions on the
coefficients in the models. Concentrating on forward rates, assume that we want model
(66) to be valid under a martingale measure Q, i.e., we are postulating that P = Q and so
we have to choose t  0, ht(y)  1. Notice now that (76) has to hold for all maturities so
that, inserting the above choices of t and ht(y) and differentiating with respect to T , we
obtain (using also (69)) the following necessary condition
(t,T ) = (t,T )
T
t
(t,s)ds -
E
(t,T ;y)eD(t,T;y)
tht (y)mt(dy) (77)
which is a clear extension of the classical Heath­Jarrow­Morton drift condition for the
pure diffusion case.
Having investigated the existence of a martingale measure, we may next look for conditions
implying its uniqueness. Concentrating again on forward rates, a necessary condition
for the existence of a martingale measure has been seen to be the existence of a predictable
t and a predictable E-indexed ht(y) 0 such that relation (76) holds. Quite obviously
then, if (76) admits a unique solution in t and ht(y), the martingale measure is unique.
Ch. 5: Jump-Diffusion Models 193
To formalize this fact, consider the following linear operator [for technical details, that for
simplicity we neglect here, we refer to Björk, Kabanov and Runggaldier (1997)]
Kt : ,h(y)  S(t,) +
E
h(y) eD(t,;y)
- 1 t mt(dy). (78)
The operator Kt is an integral operator of the first kind and we refer to it as martingale
operator. The martingale measure is then unique if and only if, dP, dt-a.e., we have
KerKt = 0. (79)
We may now wonder whether, in the present context of infinitely many sources of randomness,
the uniqueness of the martingale measure implies completeness in the sense that
every contingent claim can be replicated by a self financing portfolio. The answer is no;
in fact, as we shall mention in Section 5.1.2 below, we obtain only a form of approximate
completeness.
We finally remark that the relationship (39) between discretely and continuously compounded
forward rates has allowed Glasserman and Kou (1999) to carry over the just mentioned
results for continuously compounded forward rates also to the case when one has
simple forwards instead. In fact, a model of the term structure of simple forwards L(t,T )
(see (38)) is defined in Glasserman and Kou (1999) to be arbitrage-free, if it can be embedded
in an arbitrage-free model of instantaneous forwards f (t,T ) via (39).
5. Hedging in jump-diffusion market models
In the previous section we have seen that, as a consequence of its incompleteness, in a
jump-diffusion market model we have in general infinitely many martingale measures.
We have then investigated the method of market completion as a tool to obtain a unique
martingale measure. On the other hand, from the second fundamental theorem of asset
pricing one has that, in general, if a market admits a unique equivalent martingale measure,
then it is also complete in the sense that every contingent claim can be hedged by a self
financing portfolio.
We shall investigate the hedging problem in a jump-diffusion market model having in
mind two goals: for the first goal, in the context of asset price models, we shall show in
Section 5.1.1 that completed market models with a unique martingale measure are complete
also in the sense of hedging if there are only a finite number of marks for the jumping
component (there is a finite number of sources of randomness). If however there are an infinite
number of marks (an infinite number of sources of randomness) then, in the context
of bond markets, in Section 5.1.2 we shall show that the completed market models with a
unique martingale measure are only approximately complete in the sense of hedging.
In the context of the first goal we also want to add here that Jensen (1999) approximates
a given jump-diffusion market model, having an infinite number of marks, by a sequence
of jump-diffusion models with a finite number of marks that are therefore complete also in
the sense of hedging.
194 W.J. Runggaldier
For the second goal, in Section 5.2 we shall consider the case when one cannot have
a complete market or when it is not appropriate to complete it. In such a case one has
to determine the hedging strategy according to some specific hedging criterion. We shall
consider the (local) risk minimization and the related minimum variance criteria and show
that they lead to hedging strategies that are quite natural extensions of those in complete
markets. While so far only the models of Section 3.1 have been further investigated, the
discussion in Section 5.2 will center mainly around the model of Section 3.3.
In part, this section can also be seen as preliminary to the next Section 6 on pricing.
In fact, if a market is complete in the sense of hedging, then by the criterion of absence
of arbitrage the initial value of the self financing and hedging strategy has to correspond
to the arbitrage-free price of the contingent claim. If the market cannot be completed, the
criterion of absence of arbitrage alone is not sufficient to define a price and the preference
structure of the investors has to come into play. Since, typically, the initial value of a hedging
portfolio satisfying a specific hedging criterion can be expressed as expectation of the
discounted claim under a specific martingale measure, the choice of a hedging criterion implies
also the choice of a martingale measure and thus of a pricing kernel. We shall discuss
these issues in more detail in Section 6.1 below.
5.1. Hedging when the market is completed
5.1.1. Asset-price models
In this subsection we consider the univariate jump-diffusion model of Section 4.1. We
had seen that, considering in addition to the asset with price St satisfying (45), also the
asset with price St satisfying (51) with coefficients such that t t in (53) is positive and
t t - t t = 0, then there exists a unique martingale measure Q corresponding to the
choice of t and t according to (53) and (54). Basing ourselves on Jeanblanc-Piqué and
Pontier (1990), we show now that in this situation any claim can be hedged with a self
financing portfolio.
Given a maturity T , consider as claim a (square-integrable) random variable HT , measurable
with respect to FT , where Ft := {S0,S0,ws,Ns, s t}, completed with the null
sets. In addition to the two risky assets with prices St and St , we suppose given also a
nonrisky asset, whose price we take for simplicity identically equal to 1 (equivalent to assuming
all prices discounted with respect to the nonrisky asset). An investment strategy is
then a triple t = [t, t ,t], where t denotes the number of units of the nonrisky asset
held in the portfolio at time t and t, t are the number of shares of the two risky assets
respectively. Let t , t be predictable and t be adapted. The value, at time t, of a portfolio
corresponding to the strategy  is then
V(t) = t St + tSt + t. (80)
We want  to be such that the corresponding portfolio is self financing and duplicates the
claim, i.e., that it satisfies
dV(t) = t dSt + t dSt,
V(T ) = HT .
(81)
Ch. 5: Jump-Diffusion Models 195
It follows from Section 4.1 that, under the unique martingale measure Q, the discounted
prices of the two risky assets, that for simplicity we continue denoting by St and St , are the
martingales satisfying
dSt = St- t dw
Q
t + t dM
Q
t ,
dSt = St- t dw
Q
t + t dM
Q
t ,
(82)
where w
Q
t and M
Q
t are as in (47) with t t and t according to (53) and (54). Replacing
dSt and dSt from (82) in (81), it follows that also V(t) is a (Q,Ft)-martingale satisfying
V(t) = V(0) +
t
0
[sSss + sSs s]dwQ
s +
t
0
[sSs-s + sSs- s]dMQ
t .
(83)
Consider next the (Q,Ft)-martingale
M(t) := EQ
{HT |Ft}. (84)
By the martingale representation theorem (see Theorem 2.3 applied here to the particular
case of a univariate Poisson point process) there exist two Ft -predictable processes 
(1)
t
and (2)
t such that
M(t) = M(0) +
t
0
(1)
s dwQ
s +
t
0
(2)
s dMQ
s . (85)
Comparing (83) and (85), one sees immediately that, by putting
V(0) = M(0) = EQ
{HT |F0} (86)
and choosing t, t such that (integrating with respect to a Wiener process one may change
St into St-)
t St-t + tSt- t = 
(1)
t ,
t St-t + tSt- t = 
(2)
t
(87)
we have V(t) = M(t). Since M(T ) = HT by definition, with the choices (86) and (87) we
obtain a self financing and hedging strategy (the value of t follows from (80)). Notice that,
in order to obtain a unique solution of (87), we have to require that t t - t t = 0, which
is exactly one of the conditions required after (53) and (54) to obtain a unique equivalent
martingale measure.
What we have just shown is an existence result leading to the completeness (in the
sense of hedging) of the given market when the martingale measure is unique. To actually
determine the hedging strategy, we need an explicit expression for the processes (1)
t and
(2)
t that, in the case of a simple claim of the form HT = H(ST ,ST ), can be obtained
by analogy to the pure diffusion case using the generalized Ito formula (19). Due to the
Markov property of (St ,St ), we may in fact put
M(t) = M(t;St ,St) = EQ
H(ST ,ST )|Ft . (88)
196 W.J. Runggaldier
Formula (19) then leads to
dM(t) = Mt() +
1
2
MSS()S2
t-2
t +
1
2
MS S()S 2
t- 2
t + MS SSt-St-t t
+ M t;St-(1 + t),St-(1 + t) - M(t;St-,St-)
- MS()t - MS() t t t dt + MS()St t + MS()St t dwQ
t
+ M t;St-(1 + t),St-(1 + t) - M(t;St-,St-) dM
Q
t . (89)
Since M(t) is a Q-martingale, the drift (finite variation) term in (89) has to vanish and so
it follows from (89) and (85) that

(1)
t = MS(t;St ,St )St t + MS(t;St ,St )St t,

(2)
t = M t;St-(1 + t),St-(1 + t) - M(t;St-,St-).
(90)
For a related result see also Shirakawa (1990). We conclude this subsection by pointing out
that, analogously to Section 4.1, the procedure that we have described here for the case of
a univariate point process can quite naturally be extended to the case of multivariate point
processes, provided the market is completed with the addition of an appropriate number of
further assets.
5.1.2. Term structure models
We consider the term structure model discussed in Section 4.2 assuming that the condition
for uniqueness of the martingale measure given by the injectivity (see (79)) of the integral
operator Kt in (78) is satisfied. This subsection is mainly based on Björk, Kabanov and
Runggaldier (1997) [see also Jarrow and Madan (1999) for a related approach].
In this market, where the basic assets are zero-coupon bonds with prices p(t,T ) for any
maturity T > t in addition to a nonrisky asset (money market account Bt ), we have first to
define a portfolio.
Definition 5.1. On the given bond market a portfolio is a pair (t,t (dT )) where
(i) t is predictable;
(ii) t, t () is a signed finite measure on [t,).
Intuitively, t is the number of units of the riskfree asset held in the portfolio at time t,
t (dT ) is the "number" of bonds, with maturities in [T,T + dT ), held at time t. Some
integrability assumptions are also required, but we leave them here as implicit. The value
process of the portfolio (,), discounted with respect to Bt , is
Vt(,) = t +

t
p(t,T )t (dT ) (91)
where, with some abuse of notation, we denote by p(t,T ) also the discounted value of a
T -bond.
Ch. 5: Jump-Diffusion Models 197
Definition 5.2. The portfolio (,) is self-financing if
dVt(,) =

t
t (dT )dp(t,T ). (92)
The integral in the right-hand side in (92) needs an appropriate definition. Justified by
the development in Björk et al. (1997), we shall simply replace here dp(t,T ) in (92) by
its expression under the (unique) martingale measure. To obtain this expression, recall
the condition (77) (or, equivalently, (76) with t = 0, ht(y) = 1) on the coefficients of
the forward rate dynamics in order that these dynamics hold under a martingale measure.
Translating, via (68), these conditions back to the bond price dynamics and taking also into
account the definition of qQ(dt,dy) in (72), one has
dp(t,T ) = p(t-,T ) S(t,T )dw
Q
t +
E
eD(t,T ;y)
- 1 qQ
(dt,dy) (93)
(recall that we take here for p(t,T ) the discounted values). Given a contingent claim
HT  FT , that we assume here to be bounded, the conditions for self financing and perfect
hedging can be expresses as (combining (92) with (93))



Vt(,) = V0(,) +
t
0

s
s(dT )p(s,T )S(s,T )dw
Q
s
+
t
0 E

s
s(dT )p(s-,T ) eD(s,T ;y) - 1 qQ(ds,dy),
VT (,) = HT ,
(94)
where the inner integral is with respect to T and the outer with respect to s.
Paralleling the development in the previous Section 5.1.1, consider next the (Q,Ft)-
martingale
M(t) := EQ
{HT |Ft} (95)
which, by the martingale representation Theorem 2.3, admits the representation (see (12)
under the measure Q)
M(t) = M(0) +
t
0
s dwQ
s +
t
0 E
H(s,y)qQ
(ds,dy) (96)
for predictable (and appropriately integrable)  and H . Comparing (94) with (96) one sees
that, by putting
V0(,) = M(0) = EQ
{HT |F0} (97)
and choosing t(dT ) such that




t
t (dT )p(t,T )S(t,T ) = t,

t
t (dT )p(t-,T ) eD(t,T ;y) - 1 = H(t,y)
(98)
198 W.J. Runggaldier
we have Vt(,) = M(t) and, in particular, VT (,) = HT , i.e., we have obtained a self
financing and hedging strategy (the value of t follows from (91)). Everything now hinges
upon the (unique) solvability of (98). To this effect consider the integral operator K
t implicit
in the left-hand side of (98), namely
K
t : 

t p(t,T )S(t,T )(dT )

t p(t-,T ) eD(t,T ;) - 1 (dT )
(99)
so that the conditions (98) become
K
t  =
t
H(t,)
. (100)
The integral operator K
t will be called hedging operator and the market is complete if
K
t is surjective. Combining this result with that of Section 4.2 on the uniqueness of the
martingale measure, namely (79), we may synthesize them into
Proposition 5.3. For the given term structure model (66), (67) we have that
(i) the martingale measure is unique, if the martingale operators Kt in (78) are injective;
(ii) the market is complete if the hedging operators K
t in (99) are surjective.
It turns out that the operators K
t are adjoint to Kt . If the spaces, on which they act,
are finite-dimensional, then the injectivity of Kt implies surjectivity of K
t and thus that
uniqueness of the martingale measure implies completeness. Unfortunately, our spaces
here are infinite-dimensional and so, due to the duality relationship (KerK) = cl(ImK)
between bounded linear operators, the injectivity of Kt implies denseness of K
t . In other
words, the uniqueness of the martingale measure implies only an approximate completeness.
For details we refer to Björk, Kabanov and Runggaldier (1997).
For the case when the mark space E is infinite, Björk, Kabanov and Runggaldier (1997)
also give a characterization of the hedgeable claims, based on a Laplace-transform technique
and under assumptions that hold, e.g., in the case of an affine term structure. When
the mark space E is finite, in Björk, Kabanov and Runggaldier (1997) it is furthermore
shown that, under appropriate assumptions, any claim can be hedged with a finite number
of bonds, whose maturities can be chosen in an essentially arbitrary way and such that they
remain fixed as the running time t varies.
5.2. Hedging when the market is not complete
If one cannot have a complete market or market completion is not appropriate, one has
to accept some residual risk, due either to non-self-financing or nonperfect hedging, and
choose an investment strategy that minimizes the unhedgeable risk. For this purpose various
criteria have been proposed and here we describe one such criterion for the case of a
slight variant of the market model described in Section 3.3.
We assume here that the actual price St of the risky asset satisfies a model of the form
of (41), namely
dSt = St vt (Zt)dwt, (101)
Ch. 5: Jump-Diffusion Models 199
where Zt is supposed to be a diffusion-type process of the form
dZt = t (Zt)dt + t(Zt)dwt (102)
for a Wiener wt , independent of wt . Given a univariate, doubly stochastic Poisson process
Nt with intensity t = t (Zt), suppose that the prices of the risky asset can only be observed
at the jump times Tn of Nt , i.e., the observation process Yt is given by (see (43))
dYt = (St - STNt)dNt
(103)
so that the information of the hedger can be modeled by the filtration
FY
t = {Ns,Ys; s t}  Ft = {S0,Z0,ws,ws,Ns; s t}.
Notice that the only difference with respect to the model described in Section 3.3 is that
here the actual price process St varies continuously in time according to (101), but is observed
only at the discrete time points Tn; there, the process according to (101) is only a
background process and the actual price process is given by the values of the background
process, sampled at the time points Tn according to (103). Notice also that, according to
(101), the process St is implicitly assumed to be a (P,Ft )-martingale. On one hand, this
will make our hedging procedure below applicable; on the other hand it can be justified by
assuming that [see, e.g., Becherer (2001)] St is discounted with respect to a P -numeraire
portfolio, which is a tradable numeraire such that the discounted assets become martingales
with respect to the original measure P .
Our hedging criterion will be that of (local) risk minimization according to Föllmer
and Sondermann (1986), Föllmer and Schweizer (1991), that keeps the requirement of
perfect hedging and relaxes the self financing requirement into mean self financing. More
precisely, considering as strategy a pair (t,t ) of FY
t -predictable processes with t and
t denoting the number of units of the numeraire and the given asset respectively, that are
held in the portfolio at time t, we give the following
Definition 5.4. Assuming prices are discounted with respect to the numeraire, define
Vt = Vt(,) := tSt + t as value process,
Ct = Ct(,) := Vt -
t
0 s dSs as cost process.
Notice that, if Ct(,) = const., the strategy (,) is self financing. We shall now relax
this assumption by allowing Ct (,) to be a (P,FY
t )-martingale and, given a (squareintegrable)
claim H(ST ) (already discounted with respect to the numerarire), determine a
hedging strategy (,) that, for all t = Tn (n = 1,2,...), minimizes
RY
t (,) := E CT (,) - Ct (,)
2
|FY
t (104)
with respect to the hedging strategies (,) for which Ct (,) is a (P,FY
t )-martingale.
The strategy (,) will be called an FY
t -risk minimizing strategy.
Notice that there is a close relationship between risk minimizing strategies in the just
specified sense and variance-minimizing strategies that are self financing and minimize the
variance of the residual hedging error.
200 W.J. Runggaldier
To compute an FY
t -risk minimizing strategy we shall proceed in two steps following
Frey and Runggaldier (1999) [see also Fischer, Platen and Runggaldier (1999) and Frey
(2000)]. In the first step we determine an Ft -risk minimizing strategy, namely a risk minimizing
strategy where the (hypothetical) information of the hedger corresponds to the full
filtration Ft , instead of the subfiltration FY
t . For this purpose define the P -martingale
g(t,St ,Zt) := E H(ST )|Ft , (105)
where the notation is justified by the Markov property of (St ,Zt). Assuming sufficient
regularity of g(), we proceed analogously to the last part of Section 5.1.1 applying Ito's
formula to g(t,St ,Zt) thereby obtaining
H(ST ) = g(0,S0,Z0) +
T
0
gt () + gZ()t () dt
+
T
0
1
2
gSS()vt ()S2
t +
1
2
gZZ()2
t () dt
+
T
0
gS()dSt +
T
0
gZ()t()dwt. (106)
Since g(t,St ,Zt) is a P -martingale, the finite variation terms in (106) vanish, leading to
H(ST ) = g(0,S0,Z0) +
T
0
gS(t,St ,Zt)dSt + MH
T (107)
which is of the form of a Kunita­Watanabe decomposition of H(ST ), namely a decomposition
of the form
H(ST ) = H0 +
T
0
H
t dSt + MH
T , (108)
where MH is a P -martingale that, due to the independence of wt and wt , is orthogonal to
the P -martingale S. It then follows from Föllmer and Sondermann (1986) and Föllmer and
Schweizer (1991) that the Ft -risk minimizing strategy is given by
F
t = H
t = gS(t,St ,Zt),
F
t = g(t,St ,Zt) - F
t St
(109)
so that Vt(F ,F ) = g(t,St ,Zt). This strategy appears as a very natural extension of the
classical Black Scholes strategy in the pure diffusion case. Notice that, to actually determine
(F
t ,F
t ) and its value, one needs to compute g(t,St ,Zt), which can be achieved
either by computing the expectation in (105) (numerical simulations may be used) or by
solving the PDE that results from (106) by setting equal to zero the finite variation terms.
Details can be found in Frey and Runggaldier (1999).
Coming to the second step, it follows from a general result in Schweizer (1994) [see also
Di Masi, Platen and Runggaldier (1995)] that the FY
t -risk minimizing strategy is obtained
by projecting the Ft -risk minimizing strategy onto the subfiltration FY
t . This projection
Ch. 5: Jump-Diffusion Models 201
property, which is due to the quadratic nature of the risk minimizing criterion, makes this
latter criterion very attractive every time one has to deal with partial information. More
precisely, the FY
t -risk minimizing strategy (,) is given by

t = E vt(Zt)S2
t F
t (St ,Zt)|FY
t- E vt(Zt)S2
t |FY
t- ,

t = E H(ST ) - 
t St |FY
t .
(110)
Notice that, according to the model, the hedger will compute the strategy (,) only at
the jump times Tn of Nt , when he receives new information [for details and a stochastic
filtering-type algorithm to compute the projection in (110) see again Frey and Runggaldier
(1999)].
We close the section mentioning that, for a standard jump-diffusion model of the type
of Section 3.1.1 with a marked point process, a self financing strategy that minimizes the
variance of the residual hedging error can be found in Chapter 7 of Lamberton and Lapeyre
(1997).
6. Pricing in jump-diffusion models
6.1. General aspects
With the introduction of jumps and/or stochastic volatility the market becomes incomplete.
Consequently, the principle of absence of arbitrage does not lead to a uniquely defined
price. One obtains actually an entire range of prices [see Eberlein and Jacod (1997), Bellamy
and Jeanblanc (2000)] and the preference structure of the investors has to come into
play to determine the pricing measure. From the point of view of pure pricing, the problem
then reduces to determining a specific martingale measure or, equivalently, the market
price of risk. To this effect there are various possibilities and in this section we mention
some of them, the last two of which will be discussed in more detail.
(i) Historically it appears that a first approach to pricing in markets that are incomplete
due to jumps in the prices and to a jumping volatility has been based on general
equilibrium with a representative agent [see, e.g., Ahn and Thompson (1988), Naik
and Lee (1990), Ahn (1992)].
(ii) A somewhat related and rather recent approach is that of pricing by utility maximization,
in which the density of the martingale measure (the pricing kernel) is related to
the marginal utility of terminal wealth [see, e.g., Frittelli (2000) and the references
therein; for a specific jump-diffusion setting see Miyahara (1998)].
(iii) An alternative possibility is given by more econometric-type approaches based on estimating/filtering
the market price of risk on the basis of market data. Related to such
an approach is the approach described in Herzel (1998) for a diffusion model with a
volatility that may jump at a random time and where the price of a European call turns
out to be a monotone function of a parameter  characterizing the martingale measures.
There exists then a unique  consistent with the option price thus allowing to
price all the other derivatives consistently with this option. This corresponds basically
to completing the market with the given option.
202 W.J. Runggaldier
(iv) Approaches based on market completion. In the previous Section 4.1 we have discussed
various ways to complete both stock as well as bond markets of the jumpdiffusion
type. As we have seen, this completion leads always to a unique martingale
measure, but it does not necessarily imply also completeness in the sense that every
claim can be hedged with a self financing portfolio. On the other hand, the uniqueness
alone of an equivalent martingale measure is already sufficient to obtain a unique
arbitrage-free price of a claim as the expectation of its discounted value under this
measure. In all cases where one achieves also completeness in the sense of hedging
(essentially all cases except when there are an infinite number of sources of randomness)
then, always by absence of arbitrage, the (unique) initial value of the self financing
and hedging portfolio has to coincide with the price computed as expectation
under the unique martingale measure. The approach based on market completion has
been widely used an implemented in various economic setups and here we mention
just Shirakawa (1990, 1991), Jeanblanc-Piqué and Pontier (1990), Naik (1993), Mercurio
and Runggaldier (1993), Jarrow and Madan (1995, 1999). It has the advantage to
lead to a unique price on the basis of the principle of absence of arbitrage alone, without
having to make assumptions on a non-priced jump risk and without the need to
introduce a general equilibrium model. On the other hand it requires that the stochastic
evolution of more than just the underlying asset has to be specified and, without
specific criteria, the completion may occasionally be rather arbitrary.
(v) In the previous Section 5, in the context of hedging it was mentioned that, if the
market cannot be completed, then one has to accept some residual risk and it becomes
natural to determine the hedging strategy on the basis of a risk minimization
criterion. On the other hand, in the previous point (iv) we recalled the fact that, in a
complete/completed market the initial value of a self financing and hedging portfolio
has to coincide with the arbitrage-free price of the claim. By analogy, it appears
then natural to define as price of a claim in a noncomplete market the initial value of
a portfolio minimizing a given hedging criterion. Quite typically, the initial value of
such a portfolio turns out to be the expectation of the discounted value of the given
claim under a specific martingale measure. In other words, there is a correspondence
between hedging criteria and martingale measures and the choice of a specific pricing
measure can be based on the choice of a specific hedging criterion. An approach
along these lines appears thus related to the pricing approach by utility maximization,
mentioned in point (ii) above. As an example, let us point out that the criterion of
risk minimization discussed in Section 5.2 leads to the so-called minimal martingale
measure that was already mentioned at the end of Section 4.1. It has been further
shown in Runggaldier and Schweizer (1995) that, if in a jump-diffusion model claims
are priced according to the minimal martingale measure, then convergence of asset
prices implies convergence of option prices. This stability result for prices computed
according to the minimal martingale measure makes the risk minimization criterion
discussed in Section 5.2 an attractive criterion for hedging. [For further extensions of
this stability property see Prigent (1999), Hubalek and Schachermayer (1998).]
Ch. 5: Jump-Diffusion Models 203
6.2. Computational aspects
Assume that for a jump-diffusion model we have selected a specific martingale measure
according to one of the approaches mentioned in the previous Section 6.1. We have then
to compute the expectation of the (discounted value of the) claim under this martingale
measure. In this section we shall mention some of the possible methods to accomplish this.
We consider first the univariate jump-diffusion model (45) under a generic martingale
measure Q with intensity of the Poisson process Nt given by t t . If Q corresponds to the
unique martingale measure obtained from a market completion as in Section 4.1, then t t
has to be taken according to (53). For simplicity we assume that all the prices are already
discounted and so we can put rt  0. The dynamics of St under Q are given by (see (48),
(49), (47))
dSt = St- -tt t dt + t dwQ
t + t dNt . (111)
We want to compute the value of a European call option, namely EQ{(ST -K)+}. For this
purpose we adapt an approach from Mercurio and Runggaldier (1993), assuming first that
in (111) we have t   , i.e., the jump coefficient is constant [for this case see also Aase
(1988)]. We have
EQ
(ST - K)+
= EQ
EQ
(ST - K)+
|NT . (112)
For a fixed k, i.e., when NT = k (k = 0,1,...), using the exponential formula (17) for the
specific case when (14) is given by (111), we have
S(k)
T = S0 ek log(1+ )
exp -
T
0
 ss +
1
2
2
s ds +
T
0
s dwQ
s (113)
namely
logS(k)
T  N ;mT ,2
T (114)
with



mT = logS0 + k log(1 +  ) -
T
0
 ss +
1
2
2
s ds,
2
T =
T
0
2
s ds,
(115)
i.e., S(k)
T is lognormal with mean and variance given by mT and T respectively. Next
compute (with () the cumulative standard Gaussian distribution function)
V (k)
0 := EQ
S0
S(k)
T - K
+
=
+
logk
ex
- K dN x;mT ,2
T dx
=
1
22
T
+
logk
ex
e
- 1
22
T
(x-mT )2
dx -
K
22
T
+
logk
e
- 1
22
T
(x-mT )2
dx
204 W.J. Runggaldier
= emT - 1
2 2
T 
mT + 2
T - logK
T
- K
mT - logK
T
:= (1 +  )k
G(k,S0) (116)
with
G(k,S0) = S0 exp -
T
0
 ss ds (x) -
K
(1 +  )k
(y), (117)
where



x =
log(S0(1 +  )k/K)
T
0 (- ss + 1
22
s )ds
T
0 2
s ds
,
y = x -
T
0 2
s ds.
(118)
Coming back to (112) we then have
EQ
(ST - K)+
= EQ
V (NT )
0 =

k=0
(1 +  )k
G(k,S0)
Hk
k!
e-H
(119)
with H =
T
0 ss ds. Notice that, for actual computations, the infinite sum in the right in
(119) has to be truncated at a sufficiently large positive integer.
The result for t   can be easily extended to the case when t is a piecewise constant
deterministic time function. To this effect, given a positive integer m and a subdivision
0 = tm
0 < tm
1 <  < tm
m = T , let

(m)
t = 01{0}(t) +
m
j=1
j 1(tm
j-1,tm
j ](t); j > -1. (120)
Furthermore, let Pj (j = 1,...,m), be independent Poisson random variables with parameters
Hj =
tm
j
tm
j-1
ss ds. The generalization of formula (119) is then
EQ
(ST - K)+
=

k1,...,km=0
exp
m
j=1
kj log(1 + j ) G(k1,...,km,S0)
m
j=1
(Hj )kj
(kj )!
e-Hj (121)
with
G(k1,...,km,S0) = S0 exp -
T
0
 (m)
s ss ds (x) -
K
m
j=1(1 + j )kj
(y),
(122)
Ch. 5: Jump-Diffusion Models 205
where



x =
log(S0
m
j=1(1 + j )kj /K)
T
0 (- (m)
t ss + 1
2 2
s )ds
T
0 2
s ds
,
y = x -
T
0 2
s ds.
(123)
Coming finally to the case of a more general deterministic time function t for the jump
coefficient, we assume that there exist piecewise constant deterministic time functions 
(m)
t
and 
(m)
t such that
 (m)
t  t, (m)
t  t as m  . (124)
Consider then a sequence of fictitious risky assets, whose (discounted) values S(m)
t are
martingales with respect to the same martingale measure Q as is St in (111), namely they
satisfy
dS
(m)
t = S
(m)
t- -
(m)
t t t dt + 
(m)
t dw
Q
t + 
(m)
t dNt . (125)
For each of the processes S
(m)
t we can compute
v(m)
0 = EQ
S(m)
T - K
+
(126)
according to (121)­(123). In Mercurio and Runggaldier (1993) it is now shown that
lim
m
v
(m)
0 = v0 = EQ
(ST - K)+
, (127)
i.e., if t is a generic time function, that can be approximated from below by a sequence
of piecewise constant time functions, then the corresponding option value can be approximated
arbitrarily closely by computable expressions. In Mercurio and Runggaldier (1993)
it is also shown that, for given m, v(m)
0 can be interpreted as initial value of a mean self
financing and risk minimizing portfolio in the sense of Section 5.2 when the asset price
evolves in discrete time according to the process S
(m)
t of (125), evaluated at the discrete
time points tj . In line with the last part of point (v) of the previous Section 6.1, we may thus
consider the approximating values v
(m)
0 as option values themselves, computed according
to the minimal martingale measure.
After having discussed the univariate jump-diffusion model (45), we turn now to the
general jump-diffusion model with a marked point process and which can equivalently be
represented either by (32) or (33). We opt here for the representation (32). i.e.,
dSt = St- t dt + t dwt +  (t,Yt )dNt . (128)
In what follows we shall make the further
Assumption 6.1.
(i)  (t,Yt )   (Yt ), i.e.,  is independent of the current time;
206 W.J. Runggaldier
(ii) considering the representation of the marked point process as double sequence
(Tn,Yn), assume that Tn is independent of Yn and the Yn form a sequence of independent
random variables, the generic one Yn having law m(dy).
The driving marked point process has thus local P -characteristics (t ,m(dy)).
Suppose that we have chosen a specific martingale measure Q and that we want to
compute v0 = EQ{H(ST )} where, typically, we may have H(S) = (S - K)+. For this
purpose, in what follows we adapt a procedure from Chapter 7 in Lamberton and Lapeyre
(1997).
Recall first from Theorem 2.5 that a general absolutely continuous measure transformation
from P to Q transforms the P -local characteristics into Q-local characteristics of the
form (t t ,ht (y)m(dy)). Recalling furthermore (63) with (62) and (64), it is easily seen
that, under the measure Q corresponding to the above local characteristics, the discounted
value of St satisfies
dSt = St- -t t dt + t dw
Q
t +  (Yt)dNt , (129)
where we have put t = E  (y)ht(y)m(dy) and t = t t . Using the exponential formula
(17) to integrate (129), that is of the form of (14) with the representation (15), one immediately
finds that, for a given initial asset price S0, the value v0(S0) of the claim H(ST ) is
given by
v0(S0) = EQ
H S0 exp -
T
0
t t +
2
t
2
dt +
T
0
t dwQ
t
NT
n=1
1 +  (Yn) .
(130)
Next let
V (S0) := EQ
H S0 exp -
T
0
2
t
2
dt +
T
0
t dwQ
t (131)
so that, for H(S) = (S - K)+, the V (S0) is given by the Black­Scholes formula, i.e.,
V (S0) = BS(S0). With the use of V (S0) we can now write
v0(S0) = EQ
V S0 exp -
T
0
t t dt
NT
n=1
1 +  (Yn)
=

k=0
EQ
V S0 exp -
T
0
t t dt
k
n=1
1 +  (Yn)
Hk
k!
e-H
, (132)
where, due to the local characteristics under Q, we have H =
T
0 ss ds and where the
expectation is with respect to the joint distribution of the Yn that in Assumption 6.1 were
supposed to be independent. This latter expectation can be explicitly computed in special
Ch. 5: Jump-Diffusion Models 207
cases, in more complicated cases one has to use simulations. Again, for the actual computations,
the infinite sum has to be truncated at a sufficiently large positive integer.
We close this section by mentioning that in Glasserman and Kou (1999), for the term
structure models of simple forwards in the jump-diffusion setup described therein, the
authors study the pricing of some derivative securities after having characterized arbitragefree
dynamics. The derivative prices are also used to investigate what types of patterns in
implied volatilities are produced through jumps.
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Chapter 6
HYPERBOLIC PROCESSES IN FINANCE
BO MARTIN BIBBY
Department of Mathematics and Physics, The Royal Veterinary and Agricultural University,
Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark
MICHAEL SRENSEN
Department of Statistics and Operations Research, Institute of Mathematical Sciences,
University of Copenhagen, Universitetsparken 5, DK-2100 Kbenhavn , Denmark
Contents
Abstract 212
1. Hyperbolic and related distributions 213
1.1. The generalized hyperbolic distribution 213
1.2. The generalized inverse Gaussian distribution 222
1.3. Statistical inference 226
2. Lévy processes 227
3. Stochastic differential equations 230
3.1. Diffusion models 231
3.2. Statistical inference for diffusion processes 233
3.3. Ornstein­Uhlenbeck processes 235
3.4. Compound processes 236
4. Stochastic volatility models 238
Acknowledgment 242
Appendix 243
References 244
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
212 B.M. Bibby and M. Srensen
Abstract
Distributions that have tails heavier than the normal distribution are ubiquitous in finance.
For purposes such as risk management and derivative pricing it is important to use relatively
simple models that can capture the heavy tails and other relevant features of financial
data. A class of distributions that is very often able to fit the distributions of financial data
is the class of generalized hyperbolic distributions. This has been established in numerous
investigations, see, e.g., Eberlein ad Keller (1995), Bibby and Srensen (1997), Hurst
(1997), Eberlein, Keller and Prause (1998), Rydberg (1999), Küchler et al. (1999), Jiang
(2000), and Barndorff-Nielsen and Shephard (2001c). The class of generalized hyperbolic
distributions includes the standard hyperbolic distributions, the normal inverse Gaussian
distributions, the scaled t-distributions and the variance-gamma distributions. The use of
scaled t-distributions in finance was studied by Praetz (1972) and Blattberg and Gonedes
(1974), while Madan and Seneta (1990) introduced the variance-gamma distributions in
the financial literature. The normal distribution appears as a limit of generalized hyperbolic
distributions. The tail behaviour of the generalized hyperbolic distributions thus span
a range from Gaussian tails via exponential tails to the power tails of the t-distributions.
In Section 1 we present the generalized hyperbolic distributions and their most important
properties. We also discuss the generalized inverse Gaussian distributions which play
an important role in the theory of generalized hyperbolic distributions and processes. This
class of distributions is also of interest in its own right as a model of positive quantities in
finance. Its right-hand tail behaviour spans a range from exponential decrease to a Pareto
tail. In the following sections we present a number of stochastic process models for which
the marginal distributions or the distributions of increments (or both) are generalized hyperbolic.
The models are increasingly complex. They are thus able to fit an increasing number
of the stylized features of financial data. The well established features of financial data are
for instance reviewed in Barndorff-Nielsen (1998) and Rydberg (2000). In Section 2 we
discuss Lévy process models, while in Section 3 we discuss models defined by stochastic
differential equations. These include classical diffusion models and Ornstein­Uhlenbeck
models driven by Lévy processes as well as superpositions of such models. In the final
Section 4 we present generalized hyperbolic stochastic volatility models.
Ch. 6: Hyperbolic Processes in Finance 213
1. Hyperbolic and related distributions
In this section we present the generalized hyperbolic distributions and describe their most
important properties. We will also discuss the generalized inverse Gaussian distributions
which play an important role in the theory of generalized hyperbolic distributions and
processes. As mentioned earlier, this class of distributions is also of independent interest
as a model of positive quantities in finance. We will present a few examples of how well
these distributions fit financial data.
1.1. The generalized hyperbolic distribution
The generalized hyperbolic distributions were introduced by Barndorff-Nielsen (1977) and
include, among others, the hyperbolic distributions, the normal-inverse Gaussian (NIG)
distributions, the scaled t-distributions and the variance-gamma distributions. We shall
discuss these sub-classes in more detail later. First we present the generalized hyperbolic
distributions and their properties.
A generalized hyperbolic distribution has five parameters. If X follows a generalized
hyperbolic distribution we write
X  H(,,,,).
The probability density function of a generalized hyperbolic distribution is given by
(/)

2K( )

K-1/2( 2 + (x - )2)
( 2 + (x - )2/)1/2-
 e(x-)
, x  R, (1)
where  2 = 2 -2, and K is the modified Bessel function of the third kind with index .
Definitions and results concerning Bessel functions are collected in an appendix.
The parameter domain for the class of generalized hyperbolic distributions is given by
 0,  > 0, 2
> 2
, if  > 0,
 > 0,  > 0, 2
> 2
, if  = 0,
 > 0,  0, 2
2
, if  < 0.
In all cases   R. If  = 0 or 2 = 2 the generalized hyperbolic density in (1) is defined
as the limit expression obtained by using (A.5). Note that if  is equal to zero, the
distribution is symmetric.
The class of generalized hyperbolic distributions is closed under affine transformation.
That is, if X  H(,,,,) and Y is defined as Y = aX + b, for some positive a, then
we have that
Y  H ,

a
,

a
,a,a + b . (2)
214 B.M. Bibby and M. Srensen
From (2) we also see that the parameter  is invariant under affine transformations of a
generalized hyperbolic random variable.
From (A.3) it follows that the mode points for the generalized hyperbolic distribution
are solutions to the equation
x - 
2 + (x - )2

K-3/2( 2 + (x - )2)
K-1/2( 2 + (x - )2)
=


. (3)
If  = 0, it follows immediately that the distribution is unimodal with mode point . If
 3
2 , the ratio of the modified Bessel functions in (3) increases monotonically from 0
to 1, and therefore the distribution is unimodal. See Blsild (1978) for further discussion
of features of the generalized hyperbolic density function.
The Laplace transform of the generalized hyperbolic distribution is given by
L(z) = ez

   K(z)
 
z  K( )
, | + z| < , (4)
where  2
z = 2 - ( + z)2. From (A.3) we get that
EX =  +
K+1( )
 K( )
, (5)
and
VarX =
K+1( )
 K( )
+
22
 2
K+2( )
K( )
-
K2
+1( )
K2
( )
. (6)
Expressions for the skewness and kurtosis involve modified Bessel functions in a rather
complicated way and can be found in Barndorff-Nielsen and Blsild (1980).
Sometimes it is useful to reparametrize the generalized hyperbolic density in terms of
the parameters , , , , and , where  = / and  =  . Using this parametrization,
the generalized hyperbolic density has the form,



2K()

K-1/2(

1 + 2 1 + ((x - )/)2)
( 1 + ((x - )/)2/

1 + 2)1/2-
 e(x-)/
, x  R. (7)
The parameters , , and  are invariant under affine transformations of a random variable
following the generalized hyperbolic distribution. More precisely, the result equivalent to
(2) is that Y  H(,,,a,a+ b). From this result we see that  is a scaling parameter
and  is a location parameter. In Figure 1 generalized hyperbolic densities are drawn for
different values of , , and . In all cases the mean value is 0 and the variance is 1. The
tail behaviour of the distributions is more easily seen in Figure 2, where the logarithm of
the same densities are plotted.
Ch. 6: Hyperbolic Processes in Finance 215
Fig. 1. Generalized hyperbolic densities with mean 0 and variance 1 for different values of the parameters , ,
and .
Fig. 2. The logarithm of generalized hyperbolic densities with mean 0 and variance 1 for different values of the
parameters , , and .
216 B.M. Bibby and M. Srensen
We shall now consider the important special cases of the generalized hyperbolic distribution
mentioned earlier. The hyperbolic distributions is the subclass obtained when  is
equal to 1. With  equal to 1 in (1), we get the following expression for the density of a
hyperbolic distribution,

2K1( )
exp - 2 + (x - )2 + (x - ) , x  R. (8)
From (8) we see that the logarithm of the density of a hyperbolic distribution is a hyperbola,
which should be compared to the parabolic log-density of the normal distribution. The
name of the hyperbolic distribution stems from this observation. In fact, the definition of
the hyperbolic distributions was inspired by the empirical finding by the founding father
of the physics of wind blown sand, Brigadier R.A. Bagnold, that the log-density of the
distribution of the logarithm of the grain size of natural sand deposits looks more like a
hyperbola than like a parabola, as had previously been assumed by geomorphologists, see
Bagnold (1941).
For the hyperbolic distributions Equation (3), which determines the mode points of the
generalized hyperbolic distribution, simplifies to
x - 
2 + (x - )2
=


,
implying that the distribution is unimodal with mode point
x =  +


.
Letting  tend to zero and using (A.5), we get the asymmetric Laplace distribution as a
special case of the hyperbolic distribution, that is,
2 - 2
2
e(x-)-|x-|
, x  R.
The normal distribution can also be obtained as a limit case of the hyperbolic distribution.
Letting ,   in such a way that /  2, we get, using (A.6), the normal density:
1

22
e
- 1
22 (x-)2
, x  R.
According to Barndorff-Nielsen et al. (1985) we have that the skewness (1) and the
kurtosis (2) for large values of  and small values of / satisfy that
(1,2)  3,32
,
Ch. 6: Hyperbolic Processes in Finance 217
where
 =
/

1 + 
and  =
1

1 + 
.
Based on this observation Barndorff-Nielsen et al. (1985) suggested that the parameters
 and  are natural measures of asymmetry and "kurtosis" for the hyperbolic distribution.
Note that they are invariant under location-scale transformations. The parameters  and 
vary in the so-called shape triangle defined by
(,)  R2
| 0 || <  < 1 . (9)
Note that the normal and the (possibly skew) Laplace distributions are obtained as limit
distributions when   1 and   0, respectively. In Figure 3 hyperbolic log density
functions are plotted for different values of  and  in the shape triangle.
In Figure 4 a histogram based on 2666 observations of the daily returns of IBM-stocks
(returns are increments on a logarithmic scale of the stock prices) in the period from 1 January
1990 to 20 March 2000 is given. Each point indicates the mid-point of the top of a
column in the histogram. The best generalized hyperbolic, hyperbolic,and normal densities
are superimposed on the histogram. The parameter values corresponding to the generalized
Fig. 3. Hyperbolic log densities with mean 0 and variance 1 for different values of the parameters  and 
(-0.8,-0.6,... ,0.8 for  and 0.0,0.25,... ,1.0 for ). The log densities are placed at the corresponding values
of  and .
218 B.M. Bibby and M. Srensen
Fig. 4. A histogram of 2666 daily IBM-stock returns. Superimposed are the best fitting generalized hyperbolic,
hyperbolic, and normal densities. The parameter values corresponding to the generalized hyperbolic density are
 = 5.174,  = 0.0048,  = 0.0262,  = 0.0002, and  = -1.933. The parameter values corresponding to the
hyperbolic density are  = 82.26,  = 3.725,  = 0.0060, and  = -0.0007.
hyperbolic density are  = 5.174,  = 0.0048,  = 0.0262,  = 0.0002, and  = -1.933.
For the hyperbolic density the parameter values are  = 82.26,  = 3.725,  = 0.0060,
and  = -0.0007. In Figure 5 the logarithms of the same histogram points and the same
densities are plotted.
Log-histograms and log-densities are very useful when the interest is focussed on tail
behaviour. From Figures 4 and 5 it is evident that a heavy-tailed distribution such as a generalized
hyperbolic or hyperbolic distribution provides a good fit to the data, and certainly
a much better fit than the normal distribution, in particular in the tails. A plot like Figure 5,
which emphasizes differences in tail behaviour, reveals that the extreme tails of the histogram
are a bit heavier than those of the fitted generalized hyperbolic distribution. There
is no reason to be overly concerned about this minor discrepancy, because, first, it should be
remembered that it is measured on a logarithmic scale, and secondly, the two log-histogram
points in the extreme left tail are based on only 1 and 2 observations, respectively, while
each of the two points in the extreme right tail represents 2 observations.
The normal-inverse Gaussian (NIG) distributions is the subclass obtained for  equal
to -1
2 . The density of the normal-inverse Gaussian distribution is given by


e

K1( 2 + (x - )2)
2 + (x - )2
 e(x-)
, x  R. (10)
Ch. 6: Hyperbolic Processes in Finance 219
Fig. 5. The logarithm of the histogram in Figure 4 of 2666 daily IBM-stock returns. Superimposed are the logarithms
of the best fitting generalized hyperbolic, hyperbolic, and normal densities. The parameter values are as in
Figure 4.
If the distribution of X has density function (10), we write
X  NIG(,,,).
If we let  tend to zero, it follows from (A.5) that the NIG-distribution converges to the
Cauchy distribution with location parameter  and scale parameter .
The Laplace transform of a NIG-distribution is especially simple:
L(z) = ez+( -z)
, | + z| < , (11)
where  2
z = 2 - ( + z)2. Expressions for the mean and variance are also simple in the
case of a NIG-distribution:
EX =  +


, VarX =
2
 3
.
The skewness is 32 -5 and the kurtosis is 32(2 + 42) -7. Although these expressions
are quite simple, it is also for the NIG-distributions informative to use the shape
triangle, which can be defined in complete analogy with that for the hyperbolic distributions,
see, e.g., Rydberg (1997). In Figure 6 NIG log-density functions are drawn for different
values of  and  in the shape triangle defined in the same way as for the hyperbolic
distribution.
220 B.M. Bibby and M. Srensen
Fig. 6. Normal-inverse Gaussian log densities with mean 0 and variance 1 for different values of the parameters 
and  (-0.8,-0.6,... ,0.8 for  and 0.0,0.25,... ,1.0 for ). The log-densities are placed in the shape triangle
at the corresponding values of  and .
Finally, but not least, the class of normal-inverse Gaussian distributions is closed under
convolution when the parameters  and  are fixed, that is if X1 and X2 are independent
so that Xi  NIG(,,i,i), i = 1,2, then we have that
X1 + X2  NIG(,,1 + 2,1 + 2). (12)
Only two subclasses of the generalized hyperbolic distributions are closed under convolution.
The other class with this important property is the class of variance-gamma (VG)
distributions, which is obtained when  is equal to 0. This is only possible when  > 0 and
 > ||. The variance-gamma distributions (with  = 0) were introduced in the financial
literature by Madan and Seneta (1990). Another and perhaps more natural name for the
full class is the normal-gamma (NG) distributions. The density function is given by
 2

 ()(2)-1/2
|x - |-1/2
K-1/2 |x - | e(x-)
, x  R, (13)
where denotes the gamma-function. If X follows a variance-gamma distribution, we
write
X  VG(,,,).
Ch. 6: Hyperbolic Processes in Finance 221
The reader is reminded that the parameter domain is  > 0,  > || 0 and   R. The
Laplace transform of a VG-distribution is simple:
L(z) = ez 
z
2
, | + z| < , (14)
where again  2
z = 2 - ( + z)2. From (14) (or from (5) and (6)) it easily follows that
EX =  +
2
 2
, VarX =
2
 2
1 + 2


2
.
The class of variance-gamma distributions is closed under convolution when  and  are
fixed. If X1 and X2 are independent random variables such that Xi  VG(i,,,i),
i = 1,2, then we have that
X1 + X2  VG(1 + 2,,,1 + 2). (15)
This convolution property follows from (14).
By (A.6), the tails of a VG-distribution decrease as |x|-1 e-|x|+x when x  .
The logarithm of the densities of variance-gamma distributions are plotted for different
values of  in Figure 7. In all cases  = 0, the mean is zero, and the variance is one. From
this figure appears a disadvantage of the class of VG-distributions. The probability density
is very peaked at the centre for  < 1, while for  1 the tail-behaviour does not fit the tails
found in typical financial data like those in Figure 5 as well as other generalized hyperbolic
distributions like for instance the NIG-distribution.
We will finally consider the subclass of the generalized hyperbolic distributions that
is obtained when  = ||, or equivalently  = 0. This is only possible when  < 0 and
 > 0. It is convenient to introduce the reparametrization  = -2. For  = 0 we obtain
the density function


2(-1)/2 (/2)

K(+1)/2(|| 2 + (x - )2)
( 2 + (x - )2/||)(+1)/2
 e(x-)
, x  R, (16)
where  > 0,  > 0,   R and   R. A natural name for this distribution is the asymmetric
scaled t-distribution, as will soon be clear. From (A.6) it follows that when  is positive, the
left-hand tail decreases as |x|-(/2+1) e2x, while the right-hand tail decreases as x-(/2+1).
When  is negative, the behaviour of the two tails is interchanged. The expectation exists
provided  > 2, and the variance exists when  > 4. More generally, the n-th moment
exists when  > 2n. The Laplace transform of the distribution given by (16) is
ez (-z(z + 2))/2K/2(-z(z + 2))
(/2)2/2-1
(17)
222 B.M. Bibby and M. Srensen
Fig. 7. The logarithm of the densities of variance-gamma distributions with  = 0, mean 0, and variance 1 for
different values of the parameter .
with domain -2 < z 0 when  > 0 and 0 z < -2 when  < 0. When  = 0, the
domain is the set {0}, and we obtain the density function
(( + 1)/2)


 (/2)(1 + ((x - )/)2)(+1)/2
, x  R,
which is the well-known density of the scaled t-distribution with  degrees of freedom.
1.2. The generalized inverse Gaussian distribution
The second class of distributions, that we consider in this section, is the class of generalized
inverse Gaussian (GIG) distributions. The GIG-distributions are described by three
Ch. 6: Hyperbolic Processes in Finance 223
parameters and defined on the positive half axis. The generalized inverse Gaussian density
is of the form
(/)
2K( )
 x-1
 exp -
1
2
2
x-1
+  2
x , x > 0. (18)
The parameter domain is given by
 > 0,  0, if  < 0,
 > 0,  > 0, if  = 0,
 0,  > 0, if  > 0.
The class of generalized inverse Gaussian distributions was first proposed in 1946 by Étienne
Halphen, who used it to model the distribution of the monthly flow of water in hydroelectric
stations, see Seshardi (1997). The class was rediscovered by Sichel (1973) who
used it to construct mixtures of Poisson distributions and by Barndorff-Nielsen (1977) who
used it to construct the class of generalized hyperbolic distributions, but also realized its
broad usefulness and initiated an in depth study of the class. We shall return to the relation
to the generalized hyperbolic distributions later. The generalized inverse Gaussian distributions
were briefly mentioned by Goog (1953) as an intermediate between Pearson's curves
of Type III and V. The class of generalized inverse Gaussian distributions was investigated
extensively in Jrgensen (1982).
Using (A.5) we see that for  > 0 and  > 0 the gamma distribution emerges as limit
distribution when  tends to zero, that is we get the following density for positive  and  ,
( 2/2)
()
 x-1
 e 2x/2
, x > 0.
Similarly, the inverse gamma distribution with density given by
(2/2)
(-)
 x-1
 e(2/2)/x
, x > 0,
is obtained when  tends to zero for  < 0 and  > 0. This distribution has a tail of the
Pareto type. Finally, for  = -1
2 we get the inverse Gaussian distribution with density
function given by


2x3
 e- (x-/ )2/(2x)
, x > 0.
224 B.M. Bibby and M. Srensen
The generalized inverse Gaussian distributions are unimodal with mode point given by



 - 1 + ( - 1)2 + 2 2
 2
if  > 0,
2
2(1 - )
if  = 0.
If X has a generalized inverse Gaussian distribution, we write X  GIG(,, ). In Figure
8 generalized inverse Gaussian densities are plotted for different values of  and
 =  . In all cases the variance is 1.
The Laplace transform of the GIG(,, )-distribution is
L(z) =
K( 1 - 2z/ 2)
K()(1 - 2z/ 2)/2
(19)
for  > 0 and  > 0. The domain of L is z <  2/2 when  0 and z  2/2 when  < 0.
In the cases  = 0 or  = 0, the Laplace transform is obtained from (19) by (A.5). For
 = 0,
L(z) = 1 -
2z
 2
,
z <
 2
2
,
Fig. 8. Generalized inverse Gaussian densities with variance 1 for different values of the parameters  and
 =  .
Ch. 6: Hyperbolic Processes in Finance 225
which is the well-known Laplace transform of the gamma-distribution. For  = 0 we ob-
tain
L(z) =
2K(

-22z)
(-)(-2z/2)/2
, z 0.
For positive values of  and  the moments of X are given by
EXj
=


j
K+j ()
K()
, j = 1,2,.... (20)
When either  or  is zero, the moments of X are also known and are obtained as limits of
(20). The variance of X is given by
VarX =


2
K+2()
K()
-
K2
+1()
K2
()
. (21)
In Figure 9 a histogram of 307 monthly observations of interest rates in the period from
June 1964 to December 1989 is given along with a fitted generalized inverse Gaussian
density corresponding to the parameter values  = 0.2693,  = 11.23, and  = -7.0707.
More precisely, the data are annualized monthly yields of U.S. one-month Treasury bills.
The same data set was studied in Chan et al. (1992).
There is the following important relationship between the generalized hyperbolic distribution
and the generalized inverse Gaussian distribution, which was, in fact, how the genFig.
9. A histogram of 307 monthly interest rates. The generalized inverse Gaussian density with parameters
 = 0.2693,  = 11.23, and  = -7.0707 is superimposed.
226 B.M. Bibby and M. Srensen
eralized hyperbolic distribution was originally derived in Barndorff-Nielsen (1977). The
generalized hyperbolic distribution is a normal variance­mean mixture where the mixing
distribution is generalized inverse Gaussian. What is meant by this is that if
X|W = w  N( + w,w),
and W  GIG(,, ), then the marginal distribution of X will be generalized hyperbolic,
X  H(,,,,), where 2 = 2 + 2. This property provides a possible interpretation
of non-Gaussian stochastic variation described by a generalized hyperbolic distribution.
As special cases we have that the normal-inverse Gaussian distribution appears when
the mixing distribution is an inverse Gaussian distribution, and the variance-gamma distribution
emerges as a normal variance­mean mixture where the mixing distribution is a
gamma distribution. This explains the names of the distributions. The asymmetric scaled
t-distribution is a normal variance­mean mixture with an inverse gamma mixing distribution.
As a special case we get the well-known result that the t-distribution is a normal
variance mixture ( = 0) with an inverse gamma mixing distribution.
The mixing result implies that there is the following simple relationship between the
Laplace transform, LX, of the generalized hyperbolic distribution H(,,,,) and that
of the GIG(,, 2 - 2)-distribution, LW :
LX(z) = ez
 LW z +
1
2
z2
.
Barndorff-Nielsen and Halgreen (1977) showed that generalized inverse Gaussian distributions
are infinitely divisible. Using that the generalized hyperbolic distributions are
normal variance-mean mixtures with generalized inverse Gaussian mixing distributions,
they also proved that generalized hyperbolic distributions are infinitely divisible. Halgreen
(1979) showed that generalized hyperbolic distributions and generalized inverse Gaussian
distribution are even self-decomposable. In the following section, the properties of infinite
divisibility and self-decomposability will turn out to be important because they allow the
construction of certain hyperbolic stochastic process models.
1.3. Statistical inference
Inference for the parameters when dealing with independent and identically generalized
hyperbolic or generalized inverse Gaussian distributed observations should be based on the
likelihood function. The C-program HYP described in Blsild and Srensen (1992) can be
used for maximum likelihood estimation in the situation where independent and identically
(possibly multi-dimensional) hyperbolic distributed observations are considered. The
program HYP also has the facility of basing the inference on the multinomial likelihood
function obtained by only observing the number of observations in given intervals. More
precisely, if I1,...,Ik are disjoint intervals with union the entire real line and yj denotes
Ch. 6: Hyperbolic Processes in Finance 227
the number of observations in Ij , j = 1,...,k, then the multinomial log-likelihood function
is given by
(,,,) =
k
j=1
yj logpj , (22)
where pj is the probability that a hyperbolic distributed random variable takes a value
in Ij , that is,
pj =
Ij

2K1( )
exp - 2 + (x - )2 + (x - ) dx, j = 1,...,k.
(23)
Inference based on grouped observations from other distributions can of course be carried
out in a similar way using (22) and the equivalent of (23). Küchler et al. (1999) note
that if the observations are not independent then inference based on the multinomial likelihood
function for grouped observations will be more robust to effects of the dependence
than inference based on the original likelihood function for independent observations.
2. Lévy processes
A homogeneous Lévy process X is a stochastic process with X0 = 0 and with the property
that its increments over non-overlapping time intervals are independent. Moreover, the
increment, Xt+s - Xs, over any time interval of length t has the same distributions as Xt .
The homogeneous Lévy processes are also called processes with independent, stationary
increments or additive processes. The mathematical theory of Lévy processes can be found
in Bertoin (1996) or Sato (1999). An example of a Lévy process that is well-known from,
for instance, the Black­Scholes­Merton option pricing theory is the Brownian motion (or
Wiener process), where the increments are normally distributed.
For every generalized hyperbolic distribution there exists a homogeneous Lévy process
X such that the probability distribution of the value of the process, Xt , at a fixed time point
t is that particular generalized hyperbolic distribution. A thorough review of the theory of
these generalized hyperbolic Lévy processes and their application in finance can be found
in Eberlein (2001), see also Prause (1999) and Eberlein and Raible (2001). The distributions
that can appear as the distribution of the instantaneous value of a homogeneous Lévy
process are exactly those that have the property called infinite divisibility. As mentioned
in Section 1 the generalized hyperbolic distributions are infinitely divisible. Usually, the
distribution of the value Xs at a time point s different from t will not be generalized hyperbolic.
However, in the case of the NIG and VG distributions, the convolution properties
(12) and (15) imply that the value of the Lévy process will be NIG-distributed, respectively
VG-distributed, at all time points. This makes the NIG and VG Lévy processes more
228 B.M. Bibby and M. Srensen
natural generalized hyperbolic Lévy processes than the other generalized hyperbolic Lévy
processes. Simulation of the NIG Lévy process was studied in Rydberg (1997).
A generalized hyperbolic Lévy processes can be written in the form
Xt = t + Zt,
where Zt is a pure jump martingale with infinitely many small jumps in every finite time
interval, however small. The behaviour of Zt is reflected in the so-called Lévy measure,
see (27) and the discussion following this formula. The Lévy measure of the generalized
hyperbolic distribution is
q(x) =



ex
|x|

0
exp(-|x| 2y + 2)
2y(J2
 (

2y) + Y2
 (

2y))
dy + e-|x|
if  0,
ex
|x|

0
exp(-|x| 2y + 2)
2y(J2
-(

2y) + Y2
-(

2y))
dy if  < 0.
(24)
Here J and Y denote Bessel functions of the first and second kind, respectively, see the
appendix. The Lévy measure was essentially found by Halgreen (1979), see also Prause
(1999). For the NIG-distribution this expression simplifies to
q(x) = -1
|x|-1
K1 |x| ex
, (25)
where K1 is a modified Bessel function of the third kind. The behaviour near zero is particularly
important, so the following expansion for generalized hyperbolic distributions
(Raible, 2000) is useful:
x2
q(x) =


+
 + 1/2
2
|x| +


x + o |x| (26)
as x  0. We see that for every generalized hyperbolic distribution the Lévy measure has
infinite mass in every neighbourhood of the origin. The process Zt is given by
Zt =
t
0 R\{0}
x X
(du,dx) - q(x)dudx , (27)
where the integer-valued random measure X is defined by
X
(dt,dx) =
s>0
1{ Xs=0}(s, Xs)(dt,dx).
Here a denotes the Dirac measure at a, and Xs = Xs - Xs- is the jump of the process
X at time s (for most time points Xs = 0). Integrals of the type (27) are treated in,
Ch. 6: Hyperbolic Processes in Finance 229
e.g., Jacod and Shiryaev (1987) or Protter (1990). The random measure X is Poissonian
with intensity measure q(x)dx dt. This implies that for any closed interval A that does not
contain the origin, the number of jumps in the time interval [0,t] with a size that belongs
to A, i.e.,
NA
t = X
[0,t],A ,
is a Poisson process with intensity A q(x)dx, which is a finite number. In particular, NA
t is
Poisson distributed with mean value t A q(x)dx. As the boundary of the interval A tends
to zero, the mean value goes to infinity, cf. (26). It is interesting to note that a generalized
hyperbolic Lévy process has no continuous Brownian motion component and has infinitely
many jumps on every time interval.
The generalized hyperbolic Lévy processes do, however, have a nice relation to the
Brownian motion. Let B be a standard Brownian motion, and let (t) be a Lévy process
for which the distribution of (1) is a generalized inverse Gaussian distribution. Then the
process
Xt = t + (t) + B(t) (28)
is a generalized hyperbolic Lévy process. Because the increments of  are generalized inverse
Gaussian distributed and hence can only be positive, the process  is increasing and
can thus be interpreted as a time that increases with a randomly varying speed. A process
 with this property is called a subordinator, and the construction (28) is called subordination.
The randomly increasing time  has been interpreted as an operational time or a
business time reflecting, for instance, the volume of trade at an exchange. Some times a lot
is happening at the exchange and the business time increases rapidly. At other times the
exchange is tranquil and the business time goes only slowly. That the distribution of X1
is generalized hyperbolic follows because this distribution is a variance-mean mixture of
normal distributions where the mixing distribution is the generalized inverse Gaussian distribution,
see Section 1.2. The fact that a Lévy process  exists such that (1) is generalized
inverse Gaussian distributed follows because these distributions are infinitely divisible, as
mentioned in Section 1. In the case of a NIG-distribution, the construction by subordination
can be done in the following simple way (Barndorff-Nielsen, 1998). Let (Ut,Vt) be a
two-dimensional standard Brownian motion starting at (0,0) and with drift vector (, ),
where  > 0. Let (t) denote the first time the second component V attains the value
t > 0 with  > 0. Then {(t): t > 0} is an inverse Gaussian Lévy process, and
Xt = t + U(t)
is a NIG-Lévy process. Specifically, Xt is NIG(,,t,t) distributed, where  =
2 +  2.
Construction of financial models by subordination was first proposed by Praetz (1972)
who used a scaled t-distribution to model stock returns and obtained a good fit to weekly
230 B.M. Bibby and M. Srensen
returns from the Sydney Stock Exchange. This is a particular example of a generalized
hyperbolic distribution where the mixing distribution is an inverse gamma distribution, see
Section 1.1. Praetz attributed the mixing of normals to the change in activity at the exchange.
Clark (1973) and Epps and Epps (1976) found that there is a dependency between
trading volume and the variance of returns, but did not suggest generalized hyperbolic
models. These finding have been confirmed by Ané and Geman (2000). In Madan and
Seneta (1990), Madan and Lime (1991) and Madan and Chang (1996) the so-called variance
gamma model is introduced and studied as a model for share market returns. This
model is the generalized hyperbolic Lévy process with a gamma mixing distribution. For
a discussion of the subordination approach in finance, see, e.g., Hurst, Platen and Rachev
(1997).
The use of generalized hyperbolic Lévy processes to model the prices of stocks and other
assets and the corresponding theory of option pricing has been thoroughly investigated by
Eberlein and Keller (1995), Keller (1997), Eberlein, Keller and Prause (1998) and Eberlein
and Prause (2002). Eberlein and Jacod (1997) proved that the set of equivalent martingale
measures is large and that the corresponding price range is the entire non-arbitrage interval.
A theory of the term structure of interest rates based on the hyperbolic Lévy process was
developed in Eberlein and Raible (1999). A useful review can be found in Eberlein (2001).
For the processes discussed in this section, estimation based on observations at equidistant
discrete time points is as easy as estimation for independent generalized hyperbolic
distributions, because the increments of the process between the observation times are
independent. Usually one would use a Lévy process for which the increments are generalized
hyperbolic and then estimate the parameters, for instance by means of the computer
program mentioned in Section 1.3. A simple check of the fit of the model to the data
can be made as follows. If, for instance, the data are daily observations, then it should
be checked that the distributions calculated from the estimated model of the increments
over a number of suitably chosen longer time spans fit the corresponding increments calculated
from the data. For the NIG and VG Lévy processes these distributions are simply
given by the formulae (12) and (15). For an example of this procedure, see Eberlein
(2001).
3. Stochastic differential equations
In this section we present various methods for constructing diffusion processes with generalized
hyperbolic and generalized inverse Gaussian marginal distributions. A diffusion
process is the solution of a stochastic differential equation driven by a Wiener process.
Estimation of parameters based on discrete-time observations of a diffusion process is
considered too. Furthermore, we consider Ornstein­Uhlenbeck type processes driven by
Lévy processes and models given as sums of processes defined by stochastic differential
equations.
Ch. 6: Hyperbolic Processes in Finance 231
3.1. Diffusion models
We consider a one-dimensional diffusion process {Xt} and suppose that it is the unique
weak solution to the stochastic differential equation
dXt = b(Xt;)dt + (Xt ;)dWt, (29)
where (x;) is positive for all x in the state space (l,r) (- l < r ) and all  in
some p-dimensional parameter space . We will focus on ergodic diffusions and denote
the density of the corresponding invariant probability measure by  .
Diffusion processes with a specific marginal distribution are typically constructed by
determining drift b and diffusion coefficient  so that the invariant distribution is of the
required type. This method will result in the appropriate marginal distribution for large
values of t or for all t provided that the initial distribution is equal to the invariant distribution
(i.e., X0   ). Under mild conditions we have the following relationship between
the drift, diffusion coefficient, and the density of the invariant distribution,
2b(x;) - v (x;) = v(x;)
 (x)
 (x)
, l < x < r,   , (30)
where v denotes the squared diffusion coefficient, v(x;) = 2(x;).
Using (30), Bibby and Srensen (2001) discussed a method for constructing diffusion
processes with a prescribed marginal (invariant) distribution. Letting the drift be given by
b(x;) =
1
2
v(x;)
d
dx
log v(x;)f (x) ,
where f is a function that is integrable on the interval (l,r), it was shown under some
regularity conditions that the diffusion process given by (29) has invariant density  proportional
to f , irrespective of the choice of the function v. Bibby and Srensen (2001) also
considered the special case where
v(x;) = 2
f (x),
2
> 0,   [0,1],
in particular the situation where the invariant density was hyperbolic. This led to the following
stochastic differential equation,
dXt =
1
2
2
(1 - )f (Xt)-
 (Xt
- )
2 + (Xt - )2
dt + f (Xt)-/2
dWt, (31)
where f is proportional to the hyperbolic density function given by (8), that is
f (x) = exp - 2 + (x - )2 + (x - ) .
232 B.M. Bibby and M. Srensen
Note that the drift is towards the mode point of the hyperbolic distribution,  + / . The
diffusion process given by (31) was successfully used to describe the logarithm of the price
of VW-stocks after a linear trend had been subtracted.
In Bibby and Srensen (1997) the special case where  = 1 was considered in the situation
of a hyperbolic invariant density. Note that this results in a diffusion process with no
drift, that is the solution to the stochastic differential equation given by
dXt =  exp
1
2
 2 + (Xt - )2 -
1
2
(Xt - ) dWt . (32)
It turns out that this is an example of a local martingale which is not a martingale. Also
the hyperbolic diffusion process given as the solution of (32) was fitted successfully to
the logarithm of stock-prices (minus a linear trend) in Bibby and Srensen (1997). The
construction leading to the hyperbolic diffusion (31) can obviously be made similarly for
any generalized hyperbolic distribution. In the special case  = 1, this was done in Rydberg
(1999), where the corresponding NIG-diffusion was fitted successfully to stock prices
(minus a linear trend).
In Küchler et al. (1999) a hyperbolic diffusion process with constant diffusion coefficient
was discussed. This corresponds to letting the function v be equal to a constant 2, or to
 = 0 in (31), and gives the following stochastic differential equation,
dXt =
1
2
2
 - 
Xt - 
2 + (Xt - )2
dt +  dWt . (33)
The hyperbolic diffusion process given by (33) was first proposed in Barndorff-Nielsen
(1978).
For values of  between the two extremes 0, corresponding to stationarity being obtained
by pure reversion, and 1, where stationarity is obtained by pure diffusion, both these effects
are present to varying degrees.
Srensen (1997b) considers the construction of diffusion processes with a generalized
inverse Gaussian invariant distribution. If v is a positive function, then the solution to the
stochastic differential equation
dXt = v(Xt )v (Xt) +
1
2
v(Xt )2
( - 1)X-1
t -
 2
2
+
1
2
2
X-2
t dt
+ v(Xt )dWt (34)
will have a generalized inverse Gaussian invariant density given by (18) under suitable
regularity conditions on v. The focus in Srensen (1997b) is on the special case where
v(x) = x for constants  0 and  > 0. With this choice of diffusion coefficient, the
diffusion process is the solution to the stochastic differential equation given by
dXt = 1X2-1
t - 2X2
t + 3X
2(-1)
t dt + X
t dWt, (35)
Ch. 6: Hyperbolic Processes in Finance 233
where
1 =
1
2
2
( - 1) + 2
, 2 =
1
4
( )2
, 3 =
1
4
()2
.
Note that if  = 1
2 and 3 = 0, then the diffusion process is the solution to
dXt = (1 - 2Xt)dt +  Xt dWt , (36)
that is the Cox­Ingersoll­Ross process (CIR-process) used in finance to model short term
interest rates, see Cox, Ingersoll Jr. and Ross (1985).
A completely different way of constructing hyperbolic diffusion models was proposed
in Jensen and Pedersen (1999). These authors consider processes given by Xt = h(Yt ),
where Y is a stationary Ornstein­Uhlenbeck process:
dYt = -Yt dt +  dWt
with  > 0 and  > 0. Suppose F is the distribution function of a given probability distribution,
and let  denote the distribution function of the standard normal distribution.
If 2 = 2 and h(y) = F-1((y)), then the distribution of Xt will have the distribution
function F. If, in particular, F is the distribution function of a generalized hyperbolic distribution,
we obtain a generalized hyperbolic diffusion process. Unfortunately, there is no
explicit expression for the distribution function of a generalized hyperbolic distribution. An
advantage of this approach is that there is an expression for the transition density involving
the function h. Since the distribution function of a generalized hyperbolic distribution,
and hence h, can be calculated numerically, it is relatively easy to calculate the likelihood
function, which is usually not the case for diffusion models. A disadvantage is that the drift
and diffusion coefficients of the diffusion process X are not explicit functions.
3.2. Statistical inference for diffusion processes
Inference for discretely observed diffusion processes is made difficult by the fact that the
likelihood function is generally not tractable. In recent years many different methods have
been proposed to overcome this obstacle. We will here briefly discuss the methods most
commonly used in connection with financial data. For an excellent overview of a wide
variety of procedures for estimating parameters based on discretely observed diffusions,
see H. Srensen (2000).
Approximate likelihood methods are considered by Pedersen (1995), At-Sahalia (2002),
and Poulsen (1999). In Pedersen (1995) it is shown that the likelihood function can be calculated
to any given precision using simulations and the Euler approximation in a clever
way. Unfortunately, the method is very computer intensive. Honoré (1997) successfully
applied the Pedersen method to the CKLS-model for interest rates (proposed by Chan et
al. (1992)). In At-Sahalia (2002) an analytical approximation to the likelihood function
234 B.M. Bibby and M. Srensen
based on a truncated Hermite expansion is developed. Poulsen (1999) obtained an approximation
to the likelihood function by numerically solving the Chapman­Kolmogorov
forward equations. He used his method to fit the CKLS-model to interest rate data. Asymptotic
results for the maximum likelihood estimator based on discrete time observations of
a diffusion model were derived in Dacunha-Catelle and Florens-Zmirou (1986).
Inference for diffusion processes based on martingale estimating functions is considered
in Bibby and Srensen (1995, 1996, 1997). For observations Xt1 ,Xt2 ,...,Xtn the martingale
estimating functions introduced in Bibby and Srensen (1995, 1996) are of the form
Gn() =
n
i=1
gi(Xti-1 ;) Xti - E (Xti |Xti-1 )
+
n
i=1
hi(Xti-1 ;) Xti - E (Xti |Xti-1 )
2
- Var (Xti |Xti-1 ) . (37)
Note that in analogy with the unknown score function, Gn is a sum of functions of consecutive
pairs of observations, and Gn is a martingale with respect to the natural filtration.
The conditional expectations in (37) can easily be calculated using simulations, and an estimator
for the parameter  is then obtained by solving the equation Gn() = 0. In Bibby
and Srensen (1995) the resulting estimator is shown to be consistent and asymptotically
normal as the number of observations tends to infinity. An optimal choice of the functions
gi and hi as well as simpler approximately optimal functions that are useful in practice
are given in Bibby and Srensen (1995, 1996). As mentioned earlier the hyperbolic diffusion
process given by (32) was fitted to the log-prices of stocks after a linear trend had
been subtracted in Bibby and Srensen (1997). The parameters in this hyperbolic diffusion
model were estimated using the martingale estimating function
Kn() =
n
i=1
˙v(Xti-1 ;)
(ti - ti-1)v(Xti-1 ;)3
(Xti - Xti-1 )2
- E (Xti - Xti-1 )2
|Xti-1 ,
where v is the squared diffusion coefficient and a dot denotes differentiation with respect to
the parameter . This is an approximately optimal modification of (37) taking into account
that the diffusion has no drift.
Kessler and Srensen (1999) considered martingale estimating functions based on eigenfunctions
of the infinetisimal generator of the diffusion process. The advantage of such
martingale estimating functions is that they are adapted to concrete models and are easy to
calculate in cases where the eigenfunctions are explicitly known. Unfortunately this is not
often the case.
It is usually easy to obtain an estimator from a simple estimating function of the form
Fn() =
n
i=1
f (Xti ;),
Ch. 6: Hyperbolic Processes in Finance 235
where the function f satisfies that
r
l f (x,) (x)dx = 0 with  denoting the density
of the invariant probability measure. Such simple estimating functions were studied by
Hansen and Scheinkman (1995), Kessler (2000), and Jacobsen (2001). The advantage of
these estimating functions is that they are indeed simple and fast to work with because it
is straightforward to explicitly find functions f with the property needed. The main disadvantages
are that only parameters appearing in the invariant density can be estimated using
simple estimating functions and that the estimators may be far from efficient because the
dependence structure in the data is ignored. An improved version of the simple estimating
function where each term in the sum depends on a pair of consecutive observations was
considered by Hansen and Scheinkman (1995) and Jacobsen (2001). Optimality questions
were treated in Kessler (2000) and Jacobsen (2001). For the improved version it is also not
possible to estimate all parameters, see the discussion in Hansen and Scheinkman (1995).
A review of estimating function inference for diffusion models can be found in Srensen
(1997a) and Bibby, Jacobsen and Srensen (2002).
Indirect inference procedures based on auxiliary models and extensive simulations were
proposed by Gouriéroux, Monfort and Renault (1993) and Gallant and Tauchen (1996).
These procedures have gained some popularity in the finance literature under the name
of the efficient method of moments. However, the quality of the estimators depend on the
choice of the auxiliary model, which is not a straightforward matter.
Finally, Bayesian MCMC-methods have been applied to diffusion models by Eraker
(2001) and Elerian, Chib and Shepard (2001). In these methods, the likelihood function is
calculated in a way similar to that in Pedersen (1995).
3.3. Ornstein­Uhlenbeck processes
A stochastic process X is called a process of the Ornstein­Uhlenbeck type, if it satisfies a
stochastic differential equation of the form
dXt = -Xt dt + dZt, (38)
where  > 0 and where the driving process Z is a homogeneous Lévy process. It is not
difficult to see that
Xt = e-t
X0 +
t
0
e-(t-s)
dZs. (39)
If X is stationary and square integrable, the autocorrelation function of X is
(u) = exp(-u). (40)
When the process Z is the standard Wiener process, the solution X is the usual OrnsteinUhlenbeck
process. Ornstein­Uhlenbeck type processes have been studied by Wolfe
(1982), Sato and Yamazato (1982, 1984) and Sato, Watanabe and Yamazato (1994); see
236 B.M. Bibby and M. Srensen
also Jurek and Vervaat (1983), Jurek and Mason (1993), and Barndorff-Nielsen, Jensen
and Srensen (1998). A necessary and sufficient condition for (38) to have a stationary
solution is that E(log(1 + |Z(1)|)) < .
For every generalized hyperbolic distribution there exists a stationary OrnsteinUhlenbeck
type process such that for all t 0 the distribution of Xt is the given generalized
hyperbolic distribution. The same is true for all generalized inverse Gaussian distributions.
This is because these distributions have the property called self-decomposability, as discussed
in Section 1. The Lévy process driving the NIG Ornstein­Uhlenbeck type process
was studied by Barndorf-Nielsen (1998), while the process driving the symmetric variancegamma
Ornstein­Uhlenbeck type process, was found by Jiang (2000). For symmetric distributions,
the driving Lévy process is, in the case of the NIG Ornstein­Uhlenbeck process,
the sum of a NIG Lévy process and a compound Poisson process, while for the variancegamma
Ornstein­Uhlenbeck process, it is simply a compound Poisson process.
As for most ordinary diffusion processes, the likelihood function is usually not explicitly
available for processes of the Ornstein­Uhlenbeck type. Since these processes are Markov
processes, a simple and natural approach to statistical inference goes via estimating functions
based on conditional moments defined in analogy with those discussed in Section 3.2.
3.4. Compound processes
Quite often, the exponentially decreasing autocorrelation function (40) is too simple to fit
financial data. However, models with a much more flexible covariance structure are easily
obtained by summing independent Ornstein­Uhlenbeck type processes, as was proposed
by Barndorff-Nielsen, Jensen and Srensen (1998). The process
Xt = X(1)
t +  + X(m)
t , (41)
where the processes X(i)
t , i = 1,...,m, are independent Ornstein­Uhlenbeck type
processes given by
dX(i)
t = -iX(i)
t dt + dZ(i)
t (42)
for independent Lévy processes Z(i)
t , i = 1,...,m, has an autocorrelation function of the
form
(u) = 1 exp(-1u) +  + m exp(-mu), (43)
where i is proportional to the variance of X
(i)
t , and 1 +  + m = 1. A much better fit
to financial data than that obtained by (40) can often obtained even for m = 2. Examples
can be found in Barndorff-Nielsen, Jensen and Srensen (1998) and Barndorff-Nielsen and
Shephard (2001c).
For every generalized hyperbolic distribution and for every generalized inverse Gaussian
distribution there exists a stationary process X of the form (41), (42) such that for all t 0
Ch. 6: Hyperbolic Processes in Finance 237
the distribution of Xt is that particular distribution. Again this is because these distributions
are self-decomposable, see Barndorff-Nielsen, Jensen and Srensen (1998). More
complex types of superpositions of Ornstein­Uhlenbeck type processes were investigated
in Barndorff-Nielsen (2001).
The construction (41) can be made for diffusion models with linear drift and non-linear
diffusion coefficient too, see Bibby, Skovgaard and Srensen (2002). As an example, suppose
we want a stationary stochastic process with autocorrelation function (43) for given
values of 1,...,m and 1,...,m, and such that the marginal distribution of Xt is a
gamma distribution with shape parameter  and scale parameter . This can be obtained
by defining m independent processes as the stationary solutions to
dX(i)
t = -i X(i)
t - i dt + 2iX(i)
t dW(i)
t , (44)
i = 1,...,m. Each of the processes, X(i)
t , is a CIR-process, (36), which is a particular
example of the generalized inverse Gaussian diffusions given by (35). Since X(i)
t is gamma
distributed with shape parameter i and scale parameter , it follows that Xt defined by
(41) has the required gamma distribution, and since the autocorrelation function of X
(i)
t is
exp(-iu), the autocorrelation function of the sum Xt is given by (43). This construction
will come in handy in Section 4, where processes of the type (41) will be used as models
for stochastic volatility.
Empirical autocorrelations that might be interpreted as an indication of long range dependence,
may often alternatively be approximated very well by autocorrelation functions
of the type (43). However, if a model with genuine long range dependence is desirable, a
NIG-process of this type can be constructed as follows.
Let X(i), i = 1,2,..., be a sequence of independent NIG Ornstein­Uhlenbeck processes
with NIG-parameters (,,0,i), where
i  i-1-2(1-H)
,
for some H  (0,1), and all with the same value of the drift parameter . Barndorff-Nielsen
(1998) showed that the process
Xt =

i=1
X
(i)
t/i, (45)
which is stationary and well-defined as a mean-square limit, has as its marginal distribution
the NIG distribution with parameters (,,0,), where  = 
i=1 i. Moreover, its
autocorrelation function r(u) satisfies
r(u)  L(u)u-2(1-H)
,
for some slowly varying function L. Thus if 1
2 < H < 1, the process X exhibits long range
dependence with exponent H . The construction of long range dependent processes by a
238 B.M. Bibby and M. Srensen
sum of the type (45) is similar to a construction proposed by Cox (1984). Almost the same
construction was used in Barndorff-Nielsen, Jensen and Srensen (1990). The construction
(45) can also be applied to a sequence of independent stationary NIG-diffusions given as
solutions of stochastic differential equations defined in analogy to (31).
Likelihood inference for the various compound processes considered here is complicated
by the fact that the likelihood function is not explicitly available. A feasible alternative is
provided by prediction-based estimating functions, see M. Srensen (2000).
4. Stochastic volatility models
A generalization of the Black­Scholes model for the logarithm of an asset price
dXt =  + 2
dt +  dWt, (46)
that takes into account the empirical finding that the volatility 2 varies randomly over
time is a stochastic volatility process:
dXt = ( + vt )dt +

vt dWt. (47)
Here the volatility vt is a stochastic process that cannot be observed directly. If the data are
observations at the time points i, i = 0,1,2,...,n, then the returns Yi = Xi - X(i-1)
can be written in the form
Yi =  + Si + SiAi, (48)
where
Si =
i
(i-1)
vt dt, (49)
and where the Ais are independent, standard normal distributed random variables. If the
integrated volatility Si is independent of Ai, and if it is generalized inverse Gaussian
distributed, then the distribution of the return Yi is generalized hyperbolic. This follows
from the representation of the generalized hyperbolic distributions as variance­mean mixtures
of normal distributions mentioned in Section 1.2. Unfortunately, no continuous time
process v with the property that the integrated volatility (49) is exactly generalized inverse
Gaussian distributed is presently known. Therefore we will instead consider models where
the volatility process v is stationary with vt generalized inverse Gaussian distributed. For
small values of , the distribution of Si will then be close to a generalized inverse Gaussian
distribution, and hence the distribution of Yi will be close to a generalized hyperbolic distribution.
Thus we obtain models that are not exactly generalized hyperbolic, but which
have marginal distribution with much the same tail properties when  is not too large.
Ch. 6: Hyperbolic Processes in Finance 239
When  tends to infinity, the distribution of -1/2(Yi -  - Si) =

Si/Ai tends
to a normal distribution with mean zero and variance equal to the mean volatility, E(vt ),
provided that the process v is ergodic. This is in accordance with the empirical finding that
the distribution of returns over short periods have heavy tails and are well approximated by
generalized hyperbolic distributions, whereas the distribution of returns over long periods
is close to a normal distribution. Limit theorems relating, for small , the distribution of
Yi to the generalized hyperbolic distribution obtained by assuming that Si is exactly generalized
inverse Gaussian distributed are given in Genon-Catalot, Jeantheau and Larédo
(1998). A rather different type of discrete time stochastic volatility models with exactly
generalized hyperbolic distributed returns was proposed in Banrdorff-Nielsen (1997). It
should be noted that stochastic volatility models can be interpreted as being obtained by
subordination. Here the operational time or business time is the integral of the volatility
process (t) =
t
0 vs ds, which can be interpreted as discussed in Section 2.
A simple specification of the volatility process v is to assume that it is one of the stationary
and ergodic generalized inverse Gaussian diffusions defined in Section 3 as the solution
of (35). A particularly simple choice is to assume that v is the stationary CIR-model given
by (36), for which vt is gamma-distributed so that a variance-gamma stochastic volatility
model is obtained. This model was proposed by Hull and White (1988) and was considered
further by Heston (1993). Its advantage is that analytically it is relatively tractable. For instance,
all moments and mixed moments can be found explicitly, see, e.g., M. Srensen
(2000). A problem is that because of the linear drift, the autocorrelation function is an exponential
function, whereas it is a well-established empirical fact that the autocorrelation
function of the volatility process decreases more slowly than a single exponential function.
Under relatively weak regularity conditions a diffusion model has an exponentially
decreasing autocorrelation function. A sufficient condition is that it is -mixing, for which
simple conditions are given in Jeantheau and Larédo (2000). For this reason, stochastic
volatility models with a diffusion volatility process can usually not fit the autocorrelation
of the volatility process well.
In applications where the autocorrelation of the volatility process is important, a solution
is to use the construction in Section 3.4, i.e., to define the volatility process as the sum
vt = v
(1)
t +  + v
(m)
t , (50)
where v(1)
t ,...,v(m)
t are independent CIR-processes, with v(i)
t defined like the process X(i)
given by (44). Also in this case a variance-gamma model is obtained, which is exactly as
analytically tractable as the variance-gamma model just discussed, but the autocorrelation
structure of the volatility process (50) is given by (43) and is thus very flexible. This approach
is studied for more general diffusion models in Bibby and Srensen (2002).
It has been found empirically that for equities a fall in the price is associated with an
increase in the future volatility. This phenomenon is referred to as leverage, Black (1976)
and Nelson (1991). Stochastic volatility models of the form (47), where the Wiener process
driving the price process is independent of the volatility process, as we have so far assumed,
cannot deal with leverage, because for such a model the future fluctuations of the volatility
240 B.M. Bibby and M. Srensen
are independent of the present price. We can, however, easily generalize the model to allow
for the leverage phenomenon. Again we let the volatility process v be given by (50), and
denote the Wiener process driving the kth CIR-process v(k)
t by B(k). Then we define the
log-price process by
dXt = ( + vt )dt +

vt dWt, (51)
where W is the standard Wiener process
Wt =
Wt + Bt
1 + 2
with   R and
Bt =
B
(1)
t +  + B
(m)
t

m
. (52)
A lengthy calculation shows that for  = 0 the covariance between Yi and Y2
i+j (j 1) is

1 + 2
1

m
m
k=1
bk e-kj
.
Here
bk = 2 ek
1 - e-k 2

-3/2
k E v
(k)
1 v1 > 0,
where  is the shape parameter of the gamma distribution of the volatility, and k is the
speed of reversion of the kth volatility component. We see that the correlation between
Yi and Y2
i+j is negative if  < 0, which is exactly what we wanted. For  = 0 there is
no leverage effect as expected. Note that the effect decreases as j tends to infinity. The
decrease is of the same type as that of the autocorrelation function (43), but with different
weights. It is thus very flexible and can in particular be slow.
Barndorff-Nielsen and Shephard (2001b, c) proposed to model the volatility process v as
an Ornstein­Uhlenbeck type process, i.e., a solution to the stochastic differential equation
(38). Such a process can be chosen stationary with a generalized inverse Gaussian marginal
distribution, as discussed in Section 3.3. Processes of this type have the advantage that the
drift is linear and the coefficient in front of the driving Lévy process is constant, which,
analogous to the situation for the classical Wiener-driven Ornstein­Uhlenbeck process,
implies an unusual analytic tractability. For instance the integrated volatility, which is a
key quantity in finance, has the simple structure
t
s
vs ds = -1
(Zt - Zs) - (vt - vs) ,
Ch. 6: Hyperbolic Processes in Finance 241
where s < t, and where Z is the driving Lévy process. This relation implies, for instance,
that stochastic volatility processes of this type can be simulated as accurately as the volatility
process can be simulated. This is because the random variables Si, given by (49), are
simple functions of the processes Z and v. An efficient method of simulating OrnsteinUhlenbeck
type processes is based on results by Rosínski (1991) and Rosínski (2001), see
the exposition in Barndorff-Nielsen and Shephard (2001b).
Barndorff-Nielsen and Shephard (2001a) have studied the distributional properties of
integrated Ornstein­Uhlenbeck type processes in detail. For the Ornstein­Uhlenbeck type
volatility process with inverse Gaussian marginal distributions they found that while the
integrated volatility process is not distributed exactly as the inverse Gaussian distribution,
its tails have the same behaviour as this distribution. This implies that the returns will have
the expected NIG tail behaviour.
For an Ornstein­Uhlenbeck type volatility process v, the autocorrelations of the discrete
time processes Si and Y2
i have the following simple form. Here Si is given by (49), while
Yi denotes the return given by (48).
cor(Si,Si+j ) = d exp -(j - 1) , (53)
and
cor Y2
i ,Y2
i+j = c exp -(j - 1) , (54)
where
1 d =
[1 - exp(-)]2
2[exp(-) - 1 + ]
c =
[1 - exp(-)]2
6[exp(-) - 1 + ] + 2()2(/)2
0,
with  and  denoting the mean and variance of the volatility vt . Therefore, as discussed
in Barndorff-Nielsen and Shephard (2001c), S and Y2 are constrained ARMA(1,1)
processes with common autoregressive parameter, and with the moving average root being
stronger for S than for Y2. The ARMA structure implies that the return process Y is
weak GARCH(1,1) in the sense of Drost and Nijman (1993). Note that the formulae (53)
and (54) also hold for the stochastic volatility model discussed above, where the volatility
process is a CIR-diffusion. Hence for this model, the processes S and Y2 have the same
ARMA structure.
Barndorff-Nielsen and Shephard (2001c) also proposed a model with a Lévy-driven
Ornstein­Uhlenbeck volatility process that allows for the leverage phenomenon. The logprice
is modelled by
dXt = ( + vt)dt +

vt dWt +  dZt, (55)
242 B.M. Bibby and M. Srensen
where Zt = Zt - E(Zt) is the centered version of the Lévy process Z that drives the
volatility process. This model has properties similar to those of the model with leverage
discussed above (when m = 1). It is not a generalized hyperbolic model in the sense of
the other stochastic volatility models in this section because of the term  dZt . It is not
clear to what extend the model is approximately hyperbolic. A complication is that the
log-price process is a diffusion with jumps rather than a classical diffusion process driven
by a Wiener process.
As already mentioned in Section 3.3, the autocorrelation function of an OrnsteinUhlenbeck
type process decreases exponentially, which, as also mentioned earlier, is faster
than what is typically found in financial data. Volatility processes of the form (50), where
v(1)
t ,...,v(m)
t are independent, stationary Ornstein­Uhlenbeck type processes such that
the marginal distribution of v is a generalized inverse Gaussian distribution, have a much
more flexible autocorrelation structure. That such a volatility process exists was discussed
in Section 3.4. Stochastic volatility models of this type often provide a much better fit to financial
data. An example of this is given in Barndorff-Nielsen and Shephard (2001c). Also
models where the volatility process is a sum of independent Ornstein­Uhlenbeck processes
are analytically tractable.
Statistical inference for stochastic volatility models cannot easily be based on the likelihood
function as it is not explicitly available and quite hard to simulate. Harvey, Ruiz and
Shephard (1994) proposed a pseudo-likelihood method based on a Gaussian approximation
that allowed them to apply the Kalman filter. More recently, likelihood based methods
for stochastic volatility models have been proposed by Kim, Shephard and Chib (1998),
and simulation based Bayesian methods using Markov chain Monte Carlo have been developed
by Elerian, Chib and Shephard (2001) and Eraker (2001). A new and quite simple
way of obtaining an approximate likelihood function for stochastic volatility models,
which seems very promising, has been proposed by H. Srensen (2001). The method takes
advantage of the fact that lag-k conditional densities are relatively easy to obtain by simulation
for stochastic volatility models. Other methods are the indirect inference methods
of Gouriéroux, Monfort and Renault (1993), Galant and Tauchen (1996), and Gallant and
Long (1997). The prediction-based estimating functions of M. Srensen (2000) can be
applied to all models discussed in this section, while the estimators proposed by GenonCatalot,
Jeantheau and Larédo (1999) based on limit results (where the time between observations
goes to zero) in Genon-Catalot, Jeantheau and Larédo (1998) are developed for
volatility processes of the diffusion type. Recently methods based on realized volatility
have been proposed, see Gloter (1999) and Banrdorff-Nielsen and Shephard (2002). Surveys
that discuss the literature on stochastic volatility models up to 1995 can be found in
Ghyseles, Harvey and Renault (1996) and Shephard (1996).
Acknowledgment
We are grateful to Ole E. Barndorff-Nielsen for his several useful comments on an earlier
version of this chapter. The research of Michael Srensen was supported by MaPhySto, The
Ch. 6: Hyperbolic Processes in Finance 243
Centre for Mathematical Physics and Stochastics, funded by a grant from The Danish National
Research Foundation, and by the European Commission through the DYNSTOCH
Network under the Human Potential Programme. Michael Srensen was also supported
by the Centre for Analytical Finance and the Danish Mathematical Finance Network, both
financed by the Danish Social Science Research Council. The data were put at our disposal
by the Centre for Analytical Finance.
Appendix
In this appendix a few definitions and results concerning Bessel functions are collected.
The modified Bessel function of the third kind with index   R can be defined by the
following integral representation,
K(x) =
1
2

0
u-1
e-x(u+u-1)/2
du, x > 0.
The modified Bessel function has the following properties:
K-(x) = K(x), (A.1)
K+1(x) =
2
x
K(x) + K-1(x), (A.2)
K(x) = -

x
K(x) - K-1(x). (A.3)
For  = n + 1/2, n = 0,1,2,..., we have that
Kn+1/2(x) =

2x
e-x
1 +
n
i=1
(n + i)!
(n - i)!i!
(2x)-i
. (A.4)
For small values of the argument it holds that
K(x)  ()2-1
x,
x  0, if  > 0. (A.5)
Similarly, we have for large values of the argument that
K(x) =

2x
 e-x
 1 +
42 - 1
8x
+
(42 - 1)(42 - 9)
2!(8x)2
+
(42 - 1)(42 - 9)(42 - 25)
3!(8x)3
+  . (A.6)
244 B.M. Bibby and M. Srensen
The Bessel function of the first kind with index   R can be defined for x > 0 by
J(x) =
1


0
cos x sin(u) - u du -
sin()


0
e-x sinh(u)-u
du.
For  > -1
2 we have
J(x) =
2(x/2)

 ( + 1/2)
1
0
(1 - u2
)-1/2
cos(xu)du, x  R,
where denotes the gamma function.
The Bessel function of the second kind with index   R can be defined for x > 0 by
Y(x) =
1


0
sin x sin(u) - u du -
1


0
eu
+ e-u
cos() e-x sinh(u)
du.
The function Y(x) is often alternatively denoted N(x) and is sometimes called Weber's
function. The relationship between J(x) and Y(x) is
Y(x) =
J(x)cos() - J-(x)
sin()
.
In connection with the NIG-distribution, it is useful to know that
J1/2(x) =
2
x
sin(x) and Y1/2(x) = -
2
x
cos(x).
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Chapter 7
STABLE MODELING OF MARKET AND CREDIT VALUE AT RISK
SVETLOZAR T. RACHEV
Department of Statistics and Applied Probability, University of California, Santa Barbara, USA
Institute of Statistics and Mathematical Economics, University of Karlsruhe, Germany
e-mail: rachev@lsoe-4.wiwi.uni-karlsruhe.de
EDUARDO S. SCHWARTZ
Anderson School of Management, University of California, Los Angeles, USA
IRINA KHINDANOVA
Colorado School of Mines
Contents
Abstract 250
1. Introduction 251
2. "Normal" modeling of VaR 253
2.1. VaR for a single asset 253
2.2. Portfolio VaR 254
3. A finance-oriented description of stable distributions 255
3.1. Parameters and properties of stable distributions 255
3.2. Estimation of parameters of stable distributions 259
3.2.1. Tail estimation 259
3.2.2. Entire-distribution modeling 262
3.2.3. Tail estimation: Fast Fourier transform method 263
4. VaR estimates for stable distributed financial returns 264
4.1. In-sample evaluation of VaR estimates 264
4.2. Forecast-evaluation of VaR estimates 279
5. Stable modeling and risk assessment for individual credit returns 283
6. Portfolio credit risk for independent credit returns 287
7. Stable modeling of portfolio risk for symmetric dependent credit returns 290
8. Stable modeling of portfolio risk for skewed dependent credit returns 296
9. One-factor model of portfolio credit risk 299
10. Credit risk evaluation for portfolio assets 300
11. Portfolio credit risk 305
11.1. Independent credit risks 305
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
250 S.T. Rachev et al.
11.2. Symmetric dependent credit risks 305
11.3. Skewed dependent credit risks 307
12. Conclusions 309
Appendix A. Stable modeling of credit returns in figures 311
Appendix B. Tables 317
Appendix C. OLS credit risk evaluation for portfolio assets in figures 320
Appendix D. GARCH credit risk evaluation for portfolio assets in figures 324
Acknowledgments 326
References 326
Abstract
The chapter examines the use of stable Paretian distributions in modeling market and credit
Value at Risk (VaR). The in-sample- and forecast-evaluations show that stable market VaR
modeling outperforms the "normal" modeling for high values of the VaR confidence level.
The chapter also develops a new technique for estimating correlation, constructs a new
method for simulating portfolio values, and assesses portfolio VaR in various cases of credit
instruments' distributions: independent, symmetric dependent, and skewed dependent.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 251
1. Introduction
One of the most important tasks of financial institutions is evaluating the exposure to market
and credit risks. Market risks arise from variations in prices of equities, commodities,
exchange rates, and interest rates. Credit risks refer to potential losses that might occur because
of a change in the counterparty's credit quality such as a rating migration or a default.
The dependence on market and credit risks can be measured by changes in the portfolio
value, or profits and losses. A commonly used methodology for estimation of risks is the
Value at Risk (VaR). In the text below, the market VaR implies the VaR measurements
associated with market risks and the credit VaR means the VaR linked to credit risks.
A VaR measure is the highest possible loss over a certain period of time at a given
confidence level. For example, if the daily VaR for a given portfolio of assets is reported to
be $2 million at the 95 percent confidence level, it means that, without abrupt changes in
the market conditions, one-day losses will exceed $2 million 5 percent of the time.
Formally, a VaR = VaRt, is defined as the upper bound of the one-sided confidence
interval:
Pr P() < -VaR = 1 - c, (1)
where c is the confidence level and P() = Pt() is the relative change (return) in the
portfolio value over the time horizon .
Pt() = P(t + ) - P(t),
where P(t) = logS(t), S(t) is the portfolio value at t, the time period is [t,T ], with
T - t = , and t is the current time.
The essence of the VaR computations is estimation of low quantiles in the portfolio
return distributions. The VaR techniques suggest different ways of constructing the portfolio
return distributions. The traditional methods are the parametric method, historical
simulation, Monte Carlo simulation, and stress-testing. One of the parametric approaches,
the variance­covariance method, is based on the normal assumption for the distribution
of financial returns. However, financial data often violate the normality assumption. The
empirical observations exhibit "fat" tails and excess kurtosis. The historical method does
not impose distributional assumptions but it is not reliable in estimating low quantiles of
P with a small number of observations in the tails. The performance of the Monte Carlo
method depends on the quality of distributional assumptions on the underlying risk factors.
A well-known methodology of constructing credit portfolio return distributions is the
CreditMetricsTM product of J.P. Morgan.1 It is based on the rating transition model of
Jarrow, Lando and Turnbull (1997) and assumptions that joint credit quality changes are
driven by joint movements of firms' assets values.
1 See Gupton, Finger and Bhatia (1997).
252 S.T. Rachev et al.
The existing methods do not provide satisfactory evaluation of VaR. The main drawback
is inadequate approximation of distributional forms of portfolio returns. Given the nature
(heavy tails, excess kurtosis, and skewness2) of empirical financial data, the stable Paretian
distributions seem to be the most appropriate distributional models.3
The chapter examines the use of stable Paretian distributions in modeling market and
credit VaR. The stable distributions are described by four parameters:  tail index,  skewness,
 location, and  scale. Modeling with such parameters will depict fat tails and
skewness of distributions. Empirical analysis reported here confirms that, indeed, stable
modeling captures heavy-tailedness and asymmetry of financial returns, and, therefore,
produces more accurate risk estimates. The in-sample- and forecast-evaluations show that
stable market VaR modeling outperforms the "normal" modeling for high values of the VaR
confidence level. The stable distributions possess the additivity property: a linear combination
of independent stable (or jointly stable) random variables with stability index  is
again a stable random variable with the same . The additivity property provides analytic
formulas for parameters of portfolio returns. In the case of independent instruments, the
formulas are simple and can be used for estimating portfolio risk without simulations. An
analyst can employ "independent" risk measurements as lower bounds of portfolio risk
estimates. A symmetric stable random variable can be interpreted as a transformation of
a normal random variable. Based on this property, a new technique is developed here for
estimating correlation. A stable random variable can be decomposed into the "symmetry"
and "skewness" parts. Building on this feature, we construct a new method for simulating
a distribution of portfolio values. We apply this method for portfolio risk evaluation in
various cases of credit instruments' distributions: independent, symmetric dependent, and
skewed dependent.
The remainder of the chapter is organized as follows. In Section 2 we discuss computation
of VaR using the variance­covariance method, which is based on the normality
assumption for the distribution of financial returns. Section 3 provides a finance-oriented
description of stable distributions. In Section 4 we estimate the market VaR measurements
employing normal and stable modeling of financial returns.4 Section 5 investigates stable
modeling of credit returns and discusses risk assessment for individual credit instruments.
Section 6 considers portfolio risk estimation for independent portfolio assets and derives
lower bounds for risk measurements. Sections 7 and 8 present, respectively, evaluation of
portfolio risk in two cases of dependent portfolio instruments': symmetric and skewed.
Section 9 describes a main framework of the one-factor model. Section 10 discusses credit
risk evaluation for portfolio assets. Section 11 explains portfolio credit risk estimation.
Section 12 states conclusions.
2 Skewness is most pronounced in distributions of value changes of credit instruments. For references, see
Gupton, Finger and Bhatia (1997), Federal Reserve System Task Force on Internal Credit Risk Models (1998),
Basle Committee on Banking Supervision (1999).
3 Cheng and Rachev (1995), Chobanov et al. (1996), Fama (1965), Gamrowski and Rachev (1994, 1995a, b),
Mandelbrot (1962, 1963a, b, 1967), McCulloch (1996), Mittnik and Rachev (1991, 1993a, b), Mittnik, Rachev
and Chenyao (1996), Mittnik, Rachev and Paolella (1998).
4 See also Gamrowski and Rachev (1996).
Ch. 7: Stable Modeling of Market and Credit Value at Risk 253
2. "Normal" modeling of VaR
From the definition of VaR = VaRt, in Equation (1), the VaR values are obtained from the
probability distribution of portfolio value returns:
1 - c = FP (-VaR) =
-VaR
fP
(x)dx,
where FP (x) = Pr(P x) is the cumulative distribution function (cdf) of portfolio
returns in one period, and fP (x) is the probability density function (pdf) of P .5
If the changes in the portfolio value are characterized by a parametric distribution, VaR
can be computed using the distribution parameters. In this section we review "normal"
modeling ­ a parametric method based on the normal distribution assumption. It is often
called the variance­covariance method. We describe applications of the methodology for
computing VaR of a single asset and portfolio VaR.
2.1. VaR for a single asset
Assume that a portfolio consists of a single asset, which depends only on one risk factor.
Traditionally, in this setting, the distribution of asset returns is assumed to be the univariate
normal distribution, identified by two parameters: the mean, , and the standard deviation,
. The problem of calculating VaR is then reduced to finding the (1 - c)-th percentile
of the standard normal distribution z1-c:
1 - c =
X
g(x)dx
=
z1-c
(z)dz
= N(z1-c), with X
= z1-c + ,
where (z) is the standard normal density function, N(z) is the cumulative normal distribution
function, X is the portfolio return, g(x) is the normal distribution function for returns
with mean  and standard deviation , and X is the lowest return at a given confidence
level c.
In many applications investors assume that the expected return  equals 0. This assumption
is based on the conjecture that the magnitude of  is substantially smaller than
the magnitude of the standard deviation  and, therefore, can be ignored. Then we have
X
= z1-c and, therefore, VaR = -Y0X
= -Y0z1-c,
where Y0 is the initial portfolio value.
5 If fP (x) does not exist, then VaR can be obtained from the cdf FP .
254 S.T. Rachev et al.
2.2. Portfolio VaR
If a portfolio consists of many assets, the computation of VaR is performed in several steps.
Portfolio assets are decomposed into "building blocks", which depend on a finite number of
risk factors. Exposures of the portfolio securities are combined into risk categories. Then,
the total portfolio risk is obtained by aggregating risk factors and their correlations. We
denote:
* Xp is the portfolio return in one period,
* N is the number of assets in the portfolio,
* Xi is the i-th asset return in one period ( = 1), Xi = P(1) = Pi(1)-Pi(0), where Pi
is the log-spot price of asset i, i = 1,...,N. More generally, Xi can be the risk factor
that enters linearly6 in the portfolio return.
* wi is the i-th asseťs weight in the portfolio, i = 1,...,N.
The portfolio return is
Xp =
N
i=1
wiXi.
In matrix notation,
Xp = wT
X,
where
w = (w1,w2,...,wN )T
, X = (X1,X2,...,XN )T
.
Then the portfolio variance is
V (Xp) = wT
w =
N
i=1
w2
i ii +
N
i=1
N
j=1
i=j
wiwj ij ij ,
where ii is the variance of returns on the i-th asset, i is the standard deviation of returns
on the i-th asset, ij is the correlation between the returns on the i-th and the j-th assets,
 is the covariance matrix,  = [ij ], 1 i N, 1 j N.
If all portfolio returns are jointly normally distributed, the portfolio return, as a linear
combination of normal variables, is also normally distributed. The portfolio VaR based on
the normal distribution assumption is
VaR = -Y0z1-c(XP ),
6 If the risk factor does not enter linearly (as in a case of an option), then a linear approximation is used.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 255
where (Xp) is the portfolio standard deviation (the portfolio volatility),
(Xp) = V (Xp).
Thus, risk can be represented by a combination of linear exposures to normally distributed
factors. Hence, estimation of risk reduces to evaluation of the covariance matrix of
portfolio risk factors (in the simplest case, individual asset returns).
The simplicity of normal modeling explains its common use for VaR computation despite
the fact that financial data often violate the normality assumption. We conjecture that
stable distributions are more adequate distributional models. In the following sections we
analyze the stable modeling of market and credit VaR. We begin the analysis with providing
a finance-oriented description of stable distributions.
3. A finance-oriented description of stable distributions
In this part we describe parameters and some finance-oriented properties of stable distributions.
We also examine methods of estimating parameters of stable laws.
3.1. Parameters and properties of stable distributions
A random variable R is said to be stable7 if for any a > 0 and b > 0 there exist constants
c > 0 and d  R such that
aR1 + bR2
d
= cR + d,
where R1 and R2 are independent copies of R and
d
= denotes the equality in distribution.
In general, stable distributions do not have closed form expressions for the density and
distribution functions. Stable random variables (R) are commonly described by their characteristic
functions:
R() = E exp(iR) = exp -
||
1 - i sign()tan

2
+ i , if  = 1,
R() = E exp(iR) = exp -|| 1 + i
2

sign()ln + i , if  = 1,
where  is the index of stability, 0 <  2,  is the skewness parameter, -1  1, 
is the scale parameter,  0, and  is the location parameter,   R. To indicate the
dependence of a stable random variable R on its parameters, we write R  S(,,). If
7 Often R is called -stable or Pareto stable or Pareto­Lévy-stable (for  < 2).
256 S.T. Rachev et al.
the index of stability  = 2, then the stable distribution reduces to the Gaussian distribution.
In empirical studies, the modeling of financial return data is done typically with stable
distributions having 1 <  < 2.8 Stable distributions are unimodal and the smaller  is, the
stronger the leptokurtic feature of the distribution (the peak of the density becomes higher
and the tails are heavier). Thus, the index of stability can be interpreted as a measure of
kurtosis. When  > 1, the location parameter  measures the mean of the distribution. If
the skewness parameter  = 0, the distribution of R is symmetric and the characteristic
function is
R() = E exp(iR) = exp -
||
+ i .
If  > 0, the distribution is skewed to the right. If  < 0, the distribution is skewed to the
left. Larger magnitudes of  indicate stronger skewness. If  = 0 and  = 0, then the stable
random variable R is called symmetric -stable (ss). The scale parameter (the volatility)
 allows any stable random variable R to be expressed as R = R0, where R0 has a
unit scale parameter, and the same index of stability  and skewness parameter  as R.
The scale parameter generalizes the definition of standard deviation. The stable analog of
variance is the variation:  = .
In VaR estimations we are interested in investigating the behavior of the distributions in
the tails. The tails of the stable (non-Gaussian) distributions have a power decay and are
characterized by the following properties:
lim
+

P(R > ) = k
1 + 
2

and
lim
+

P(R < -) = k
1 - 
2

,
where
k =
1 - 
(2 - )cos(/2)
, if  = 1, k =
2

, if  = 1.9
The p-th absolute moment, E|R|p =

0 P(|R|p > x)dx, is
* finite if p <  or  = 2, and
* infinite otherwise.
8 The financial returns modeled with -stable laws exhibit finite means but infinite variances.
9 Note that, in contrast to the normal case, the tails of the non-Gaussian (Pareto) stable distributions are much
fatter, which will be an important issue in estimating VaR.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 257
Thus, the second moment of any non-Gaussian stable distribution is infinite.
Stable distributions possess the additivity property: a linear combination of independent
stable random variables with stability index  is again a stable random variable with the
same .10
Example. If R1,R2,...,Rn are independent stable random variables with stability index
, Ri  S(i,i,i), then R = n
i=1 wiRi is a stable random variable with the same 
and parameters:
(a) if  = 1,
 = |w1|1

+  + |wn|n
 1/
,
 =
sign(wi)1(|w1|1) +  + sign(wn)n(|wn|n)
(|w1|1) +  + (|wn|n)
,
 = w11 +  + wnn;
(b) if  = 1,
 = |w1|1 +  + |wn|n,
 =
sign(w1)1|w1|1 +  + sign(wn)n|wn|n
|w1|1 +  + |wn|n
,
 = w11 +  + wnn -
2

w1 ln|w1|11 +  + wn ln|wn|nn .
Since the Pareto-stable distributions have infinite variances, one cannot estimate risk
by variance and dependence by correlations. We shall introduce variance- and covariancesimilar
notions for stable laws. These notions are based on the multivariate assumptions of
stable distributions.
A random vector R of dimension d is stable if for any a > 0 and b > 0 there exist c > 0
and a d-dimensional vector D such that
aR1 + bR2
d
= cR + D,
where R1 and R2 are independent copies of R.
If a random vector is stable with  > 1, then it means that all components of the vector
are stable with the same index of stability and any linear combination (for example,
portfolio returns) is again stable.11
10 This property is shared only by normal and stable laws, and is the main advantage of the use of stable laws for
portfolio returns.
11 We shall model the dependence structure of the vector of returns (R1,... ,Rd ) of a portfolio by assuming that
(R1,...,Rd ) is an -stable vector.
258 S.T. Rachev et al.
The characteristic function of a d-dimensional vector is given by:
(a) if  = 1,
R() = R(1,2,...,d)
= E exp iT
R
= exp -
Sd
T
s

1 - isign T
s tan

2
 (ds) + iT
 ,
(b) if  = 1,
R() = exp -
Sd
T
s 1 + i
2

sign T
s ln T
s  (ds) + iT
 ,
where  is a bounded nonnegative measure on the unit sphere Sd, s is the integrand unit
vector (s  Sd ) and  is the shift vector. The measure  is named a spectral measure. Let
H be the distribution function of  . Then, the characteristic function in polar coordinates
is as follows
(a) if  = 1,
R() = exp -||

0

0
...
2
0
cos(,)

× 1 - sgn cos(,) tan

2
dH() + iT
 ,
(b) if  = 1,
R() = exp -||

0

0
...
2
0
cos(,)

× 1 - sgn cos(,)
2

ln  cos(,) dH() + iT
 ,
where for  given by its polar coordinates, ( sin 1 sind-1, sin 1 sin d-2 ×
cosd-1, sin 1 sind-3 cosd-2,..., cos1), we denote
cos(,) =
d-1
i=1
sini sini +
d-2
i=1
sin i sini cosd-1 cosd-1
+  + cos1 cos1.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 259
If  > 1, then  is the mean vector,  = ER. The scale parameter of a linear combination
of the components of a stable vector R satisfies the relation:

wT
R = 
(w1R1 +  + wdRd) =
Sd
wT
s

 (ds).
Viewing R = (R1,...,Rd) as the vector of individual returns in a portfolio with weights
w1,...,wd, (wTR) will be the portfolio risk-measure. As we defined above,  = 
is the variation, the stable equivalent of variance. Similarly to the traditional interpretation
of covariance as an indicator of dependence, one can use the covariation to estimate the
dependence between two ss distributions:
[R1;R2] =
1

(w1R1 + w2R2)
w1 w1=0; w2=1
=
Sn
s1s
-1
2  (ds),
where (R1,R2) is a ss vector (1 <  ) and x k = |x|k sgn(x) (signed power). The matrix
of covariations [Ri;Rj ], 1 i d, 1 j d, determines the dependence structure
among the individual returns in the portfolio.
3.2. Estimation of parameters of stable distributions12
We shall examine the methods of estimating the stable parameters and their applicability
in VaR computations, where the primary concern is the tail behavior of distributions. It has
been proposed that it is more useful to evaluate directly the tail index (the index of stability)
instead of fitting the whole distribution. The latter method is claimed to negatively affect
the estimation of the tail behavior by its use of "center" observations. We shall describe
both approaches: tail estimation and entire-distribution modeling. We suggest a method,
which combines the two techniques: it is designed for fitting the overall distribution with
greater emphasis on the tails.
3.2.1. Tail estimation
Tail estimators for the index of stability  are based on the asymptotic Pareto tail behavior
of stable distributions.13 We shall consider the following estimators of tail thickness: the
Hill, the Pickands, and the modified unconditional Pickands.14
12 For additional references on estimation of the four parameters of stable univariate laws, see Chobanov et
al. (1996), Gamrowski and Rachev (1994, 1995a, b), Klebanov, Melamed and Rachev (1994), Kozubowski and
Rachev (1994), McCulloch (1996), Mittnik and Rachev (1991), Rachev and SenGupta (1993). For the multivariate
case estimation of: the spectral measure, the index of stability, the covariation and tests for dependence of stable
distributed returns, see Cheng and Rachev (1995), Gamrowski and Rachev (1994, 1995a, b, 1996), Heathcote,
Cheng and Rachev (1995), Mittnik and Rachev (b), Rachev and Xin (1993).
13 See Section 3.1.
14 For details on the Hill, Pickands, and the modified unconditional Pickands estimators, see Mittnik, Paolella
and Rachev (1998c) and references therein.
260 S.T. Rachev et al.
The Hill estimator15 is described by
^Hill =
1
1
k
k
j=1 ln(Xn+1-j:n) - lnXn-k:n
,
where Xj:n denotes the j-th order statistic of sample X1,...,Xn;16 the integer k points
where the tail area "starts". The selection of k is complicated by a tradeoff: it must be
adequately small so that Xn-k:n is in the tail of the distribution; but if it is too small, the
estimator is not accurate. The disadvantage of the estimator is the condition to explicitly
determine the order statistic Xn-k:n. It is proved that, for stable Paretian distributions,
the Hill estimator is consistent and asymptotically normal. Mittnik, Paolella and Rachev
(1998c) found that, the small sample performance of ^Hill does not resemble its asymptotic
behavior, even for n > 10 000 (see Figure 117). It is necessary to have enormous data series
in order to obtain unbiased estimates of , for example, with  = 1.9, reasonable estimates
are produced only for n > 100 000 (see Figure 218). Alternatives to the Hill estimator are
the Pickands and the modified unconditional Pickands estimators.
The "original" Pickands estimator19 takes the form
^Pick =
ln2
ln(Xn-k+1:n - Xn-2k+1:n) - ln(Xn-2k+1:n - Xn-4k+1:n)
, 4k < n.
The Pickands estimator requires choice of the optimal k, which depends on the true
unknown . Mittnik and Rachev (1996) proposed a new tail estimator named "the modified
unconditional Pickands (MUP) estimator", ^MUP. An estimate of  is obtained by applying
the nonlinear least squares method to the following system:
k2 = X2,X-1
1 k1 + ,
where
X1 =
Xn-k+1:n
X-2
n-k+1:n
Xn-2k+1:n
X-2
n-2k+1:n
, X2 =
Xn-3k+1:n
X-2
n-3k+1:n
Xn-4k+1:n
X-2
n-4k+1:n
,
15 Hill (1975).
16 Given a sample of observations X1,... ,Xn, we rearrange the sample in increasing order X1:n  Xn:n,
then the j-th order statistic is equal to Xj:n.
17 In Figure 1, the true value of  is 1.9, the sample size is n = 10000; the x-axis shows values of k from
1 to n/2 = 5000. Notice that the estimator for ^ = ^(k(n),n) is unbiased when limn(k(n)/n)  0. So,
unbiasedness of the estimator requires very small values of k. However, for a small value of k, the variance of the
estimator is large. A close look at the estimator ^(k,n) suggests value of ^ around 2.2, whereas  = 1.9.
18 In Figure 2, the true  is again 1.9, the sample size is n = 500,000, k = 1,... ,n/2 = 250,000. One can see
that, for very small values of k,   1.9.
19 Pickands (1975).
Ch. 7: Stable Modeling of Market and Credit Value at Risk 261
Fig. 1. Hill estimator for 10 000 standard stable observations with index  = 1.9.
Fig. 2. Hill estimator for 500 000 standard stable observations with index  = 1.9.
262 S.T. Rachev et al.
k1 =
k - 1
2k - 1
, and k2 =
3k - 1
4k - 1
.
Mittnik, Paolella and Rachev (1998c) found that the optimal k for ^MUP is far less dependent
on  than in the case of either the Hill or Pickands estimators. Studies demonstrated
that ^MUP is approximately unbiased for   [1.00,1.95) and nearly normally distributed
for large sample sizes. The MUP estimator appears to be useful in empirical analysis.
3.2.2. Entire-distribution modeling
We shall describe the following methods of estimating stable parameters with fitting the
entire distribution: quantile approaches, characteristic function (CF) techniques, and maximum
likelihood (ML) methods.
Fama and Roll (1971) suggested the first quantile approach based on observed properties
of stable quantiles. Their method was designed for evaluating parameters of symmetric stable
distributions with index of stability  > 1. The estimators exhibited a small asymptotic
bias. McCulloch (1986) offered a modified quantile technique, which provided consistent
and asymptotically normal estimators of all four stable parameters, for   [0.6,2.0] and
  [-1,1]. The estimators are derived using functions of five sample quantiles: the 5%,
25%, 50%, 75%, and 95% quantiles. Since the estimators do not consider observations in
the tails (below the 5% quantile and above the 95% quantile), the McCulloch method does
not appear to be suitable for estimating parameters in VAR modeling.
Characteristic function techniques are built on fitting the sample CF to the theoretical CF.
Press (1972a, b) proposed several CF methods: the minimum distance, the minimum r-th
mean distance, and the method of moments. Koutrouvelis (1980, 1981) developed the iterative
regression procedure. Kogon and Williams (1998) modified the Koutrouvelis method
by eliminating iterations and limiting the estimation to a common frequency interval.20 CF
estimators are consistent and under certain conditions are asymptotically normal.21
Maximum likelihood methods for estimating stable parameters differ in a way of computing
the stable density. DuMouchel (1971) evaluated the density by grouping data and
applying the fast Fourier transform to "center" values and asymptotic expansions ­ in
the tails. Mittnik, Rachev and Paolella (1998) calculated the density at equally spaced
grid points via a fast Fourier transform of the characteristic function and at intermediate
points ­ by linear interpolation. Nolan (1998a) computed the density using numerical
approximation of integrals in the Zolotarev integral formulas for the stable density.22
DuMouchel (1973) proved that the ML estimator is consistent and asymptotically normal.
In Section 4 we analyze applicability of the ML method in VAR estimations.
20 For additional references, see Arad (1980), Feuerverger and McDunnough (1981), Mittnik, Rachev and
Paolella (1998), Paulson, Holcomb and Leitch (1975).
21 Heathcote, Cheng and Rachev (1995).
22 For additional references, see Mittnik et al. (1997).
Ch. 7: Stable Modeling of Market and Credit Value at Risk 263
3.2.3. Tail estimation: Fast Fourier transform method
Tail estimation using the Fourier Transform (FT) method is based on fitting the characteristic
function in a neighborhood of the origin t = 0. Here we use the classical tail estimate:
P X -
1
a
P |X|
1
a
K
a
a
0
1 - Re fX(t) dt, for all a > 0,
where Re{fX(t)} is the real part of the characteristic function fX(t) and the constant K =
1/(1 - sin 1) < 1/7. Precise estimation of the characteristic function guarantees accurate
tail estimation, which leads to an adequate evaluation of VaR.
Suppose that the distribution of returns r is symmetric--stable,23 that is: the characteristic
function of r is given by
fr(t) = E eirt
= eit-|ct|
.
If  > 1,24 then, given observations r1,...,rn, we estimate  by the sample mean
 = r = 1
n
n
i=1 ri. For large values of n, the characteristic function of observations
Ri = ri - r approaches fR(t) = e-|ct|
. Consider the empirical characteristic function of
the centered observations: ^fR,n(t) = 1
n
n
k=1 eiRkt . Because the theoretical characteristic
function, fR(t), is real and positive, we have that
^fR,n(t) = Re
1
n
n
k=1
eiRkt
=
1
n
n
k=1
cos(Rkt).
Now the problem of estimating  and c is reduced to determining ^ and ^c such that
M
0
^fR,n - fR(t, ^, ^c) =
M
0
1
n
n
k=1
cos(Rkt) - e-(^ct)^
dt
is minimal, where M is a sufficiently large value.
The realization of the FT method is performed in the following steps:
Step 1. Given the asset returns r1,...,rn, compute the centered returns Ri = ri - r, i =
1,...,n, where r = 1
n
n
i=1 ri.
Step 2. Construct the sample characteristic function
^f (tj ) =
1
n
n
k=1
cos(Rktj ),
23 Empirical evidence suggests that  does not play a significant role for VAR estimation.
24 As we have already observed, in all financial return data, fitting an -stable model results in  > 1, which
implies existence of the first moment.
264 S.T. Rachev et al.
where tj = j 
 , j = 1,...,,  is the maximal value of t,  is the number of grid
points on (0,].25
Step 3. Do the search for best ^ and ^c such that

j=1
1
n
n
k=1
cos(Rktj ) - e-(^ctj )^
is minimal.
4. VaR estimates for stable distributed financial returns
In this section we consider a stable VaR model, which assumes that the portfolio return
distribution follows a stable law. We derive "stable" VaR estimates and analyze their properties
applying in-sample and forecast evaluations. We use "normal" VaR measurements
as benchmarks for investigating characteristics of "stable" VaR measurements.
We conduct analysis for various financial data sets:
* the Yen/British Pound (BP) exchange rate,
* the BP/US$ exchange rate,
* the Deutsche Mark (DM)/BP exchange rate,
* the S&P 500 index,
* the DAX30 index,
* the CAC40 index,
* the Nikkei 225 index,
* the Dow Jones Commodities Price Index (DJCPI).
A short description of the data is given in Table 1.
4.1. In-sample evaluation of VaR estimates
In this part we evaluate stable and normal VaR models by examining distances between the
VaR estimates and the empirical VaR measures.
By a formal definition of VaR in Equation (1), VaR estimates, VaRt, , are such that
Pr Pt() < -VaRt,  1 - c, (2)
where c is the confidence level, Pt() is the relative change in the portfolio value over
the time horizon , i.e., Pt () = Rt, is the portfolio return at moment t over the time
horizon  and t is the current time.
25 For computation purposes, we have chosen  = 20 and  = 10000. In the realization of the FT method
we selected the following grid steps ht: if 0 t 1, ht = 20/50000: if t > 1, ht = 20/1000. In order
to emphasize the tail behavior, we refined the mesh near t = 0 and named that approach FT-Tail (FTT): if
0 t 0.1, ht = 20/100000; if 0.1 t 1.0, ht = 20/10000; if t > 1, ht = 20/1000. The numerical
results are reported in Section 4.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 265
Table 1
Financial data series
Series Source Number of observations Time period Frequency
Yen/BP Datastream 6285 1.02.74­1.30.98 Daily (D)
BP/US$ D. Hindanov 6157 1.03.74­1.30.98 D
DM/BP Datastream 6285 1.02.74­1.30.98 D
S&P 500 Datastream 7327 1.01.70­1.30.98 D
DAX30 Datastream 8630 1.04.65­1.30.98 D
CAC40 Datastream 2756 7.10.87­1.30.98 D
Nikkei 225 Datastream 4718 1.02.80­1.30.98 D
DJCPI Datastream 5761 1.02.76­1.30.98 D
For the purpose of testing VaR models financial regulators advise to choose a time horizon
of one day, so we take  = 1. In the text below, if the time horizon is not stated explicitly,
it is assumed to equal one day. At each time t, an estimate VaRt is obtained using lw
recent observations of portfolio returns Rt-1,Rt-2,...,Rt-lw:
VaRt = VaR(Rt-1,Rt-2,...,Rt-lw). (3)
The lw parameter is called the window length. In this subsection, VaR is estimated employing
the entire sample of observations, i.e., lw = N, where N is the sample size. Hence,
we do not point out the present time t.
We obtain "stable" ("normal") VaR measurements at the confidence level c in two steps:
(i) fitting empirical data by a stable (normal) distribution,
(ii) calculating a VaR as the negative of the (1 - c)-th quantile of a fitted stable (normal)
distribution.
"Stable" fitting is implemented using three methods: maximum likelihood (ML), Fourier
Transform (FT), and Fourier Transform-Tail (FTT).26 Estimated parameters of densities
and corresponding confidence intervals are presented in Table 2. In the FT and FTT fitting
we assume that distributions of returns are symmetric, i.e., the skewness parameter  is
equal to zero. Since the index of stability  > 1 for our data series, the location parameter
 is approximated by the sample mean. The ML estimates were computed applying the
STABLE program by J.P. Nolan.27 The confidence intervals (CI) for the FT and FTT parameter
estimates were derived using a bootstrap method with 1000 replications.28 Empirical
analysis showed that a set of 1000 replications is:
(i) satisfactory for constructing 95% CI;
(ii) insufficient for obtaining reliable 99% CI.
26 Evaluation of parameters of stable distributions is provided in Section 3.2.
27 The STABLE program is described in Nolan (1997).
28 For references on bootstrapping, see Heathcote, Cheng and Rachev (1995); for discussion on CI based on ML
parameter estimates, see Nolan (1998a).
266 S.T. Rachev et al.
Table 2
Parameters of stable and normal densitiesa
Series Normal Stable
Mean Standard Method    
deviation
Yen/BP -0.012 0.649 ML 1.647 -0.170 -0.023 0.361
FT 1.61 -0.018 0.34
[1.57,1.66] [-0.095,0.015] [0.33,0.36]
[1.55,1.68] [-0.178,0.025] [0.33,0.37]
FTT 1.50 -0.018 0.32
[1.46,1.55] [-0.131,0.034] [0.31,0.34]
[1.44,1.64] [-0.261,0.070] [0.31,0.39]
BP/US$ 0.006 0.658 ML 1.582 0.038 0.007 0.349
FT 1.57 0.006 0.33
[1.53,1.65] [-0.096,0.045] [0.32,0.36]
[1.51,1.75] [-0.393,0.065] [0.32,0.47]
FTT 1.45 0.006 0.31
[1.41,1.51] [-0.134,0.070] [0.30,0.33]
[1.40,1.62] [-0.388,0.097] [0.30,0.47]
DM/BP -0.012 0.489 ML 1.590 -0.195 -0.018 0.256
FT 1.60 -0.012 0.24
[1.54,1.75] [-0.064,0.013] [0.23,0.26]
[1.53,1.75] [-0.165,0.022] [0.23,0.27]
FTT 1.45 -0.012 0.23
[1.41,1.55] [-0.114,0.038] [0.22,0.26]
[1.40,1.77] [-0.402,0.061] [0.22,0.40]
S&P 500 0.032 0.930 ML 1.708 0.004 0.036 0.512
FT 1.82 0.032 0.54
[1.78,1.84] [-0.013,0.057] [0.53,0.54]
[1.77,1.84] [-0.062,0.067] [0.53,0.55]
FTT 1.60 0.032 0.48
[1.56,1.65] [-0.066,0.078] [0.47,0.49]
[1.54,1.66] [-0.120,0.095] [0.46,0.50]
DAX30 0.026 1.002 ML 1.823 -0.084 0.027 0.592
FT 1.84 0.026 0.60
[1.81,1.88] [-0.015,0.050] [0.59,0.60]
[1.80,1.89] [-0.050,0.057] [0.58,0.62]
FTT 1.73 0.026 0.57
[1.69,1.77] [-0.031,0.061] [0.56,0.58]
[1.68,1.79] [-0.124,0.073] [0.56,0.59]
CAC40 0.028 1.198 ML 1.784 -0.153 0.027 0.698
FT 1.79 0.028 0.70
[1.73,1.85] [-0.050,0.088] [0.68,0.73]
[1.71,1.87] [-0.174,0.103] [0.67,0.74]
a The CIs right below the estimates are the 95% CIs, the next CIs are the 99% CIs.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 267
Table 2
(Continued)
Series Normal Stable
Mean Standard Method    
deviation
FTT 1.76 0.028 0.69
[1.71,1.84] [-0.053,0.091] [0.67,0.72]
[1.69,1.87] [-0.394,0.101] [0.66,0.77]
Nikkei 0.020 1.185 ML 1.444 -0.093 -0.002 0.524
225 FT 1.58 0.02 0.59
[1.53,1.64] [-0.127,0.102] [0.57,0.62]
[1.52,1.67] [-0.421,0.130] [0.57,0.69]
FTT 1.30 0.02 0.49
[1.26,1.47] [-0.451,0.316] [0.47,0.69]
[1.05,1.67] [-1.448,0.860] [0.47,1.10]
DJCPI 0.006 0.778 ML 1.569 -0.060 0.003 0.355
FT 1.58 0.006 0.35
[1.53,1.66] [-0.026,0.100] [0.34,0.37]
[1.52,1.67] [-0.140,0.120] [0.33,0.39]
FTT 1.49 0.006 0.33
[1.44,1.55] [-0.160,0.062] [0.32,0.36]
[1.44,1.69] [-0.396,0.100] [0.32,0.46]
In our experiments, sets of 1000 replications generated:
(i) 95% CI for  and  whose bounds coincided up to two decimal points; 95% CI for 
with slightly varying bounds;
(ii) varying 99% CI, with insignificant variation of left limits.
VaR measurements were calculated at confidence levels c = 99% and c = 95%. The
99% (95%) VaR was determined as the negative of the 1% (5%) quantile. For calculating
stable quantiles we used our program, built on the Zolotarev integral representation form
of the cumulative distribution function. The 99% and 95% VaR estimates are reported in
Tables 3 and 4, respectively. Biases of stable and normal VaR measurements are provided
in Table 5.29
We accompany our computations with plots of:
* daily price levels,
* daily returns,
* fitted empirical, normal, and stable densities with the ML, FT, and FTT estimated para-
meters,
* daily empirical, normal, and stable VAR* estimates at the 99% and 95% confidence
levels.30
29 Biases are computed by subtracting the empirical VAR from the model VAR estimates.
30 The VAR* numbers are the negative values of the VAR estimates, VAR = -VAR.
268 S.T. Rachev et al.
Combined plots of densities and VaR estimation are displayed in Figures 3­10. In order
to illustrate that confidence intervals for the FT parameter estimates are sufficiently narrow,
we show stable densities and VaR measures at boundary values of confidence intervals for
^Yen,FT and ^Yen,FT in Figures 11­14.
As Figures 3­10 demonstrate, the VaR estimates obtained at confidence level c = 95%
seem to belong to the area between the "tail" and the "center". The VaR at level c = 99%
is really in the tail area. Hence, we compare performance of stable and normal models
separately for the cases c = 95% and c = 99%.
In general, the stable modeling (ML, FT, and FTT) provided evaluations of the 99%
VaR greater than the empirical 99% VaR (see Figures 3­10, Tables 3 and 4). It underestimated
the sample 99% VaR in the applications of two methods: FT ­ for the CAC40,
S&P 500, and DAX30 indices, and ML ­ for the DAX30 index. Biased downwards stable
VaR estimates were closer to the true VaR than the normal estimates (see Table 5).
Among the methods of stable approximation, the FT method provided more accurate VaR
estimates for 7 data sets (see Table 5). For all analyzed data sets, the normal modeling underestimated
the empirical 99% VaR. Stable modeling provided more accurate 99% VaR
estimates: mean absolute bias31 under the stable (FT) method is 42% smaller than under
the normal method.
At 95% confidence level, the stable VaR estimates were lower than the empirical VaR
for all data sets. The normal VaR measurements exceeded the true VaR, except the Yen/BP
exchange rate series (see Table 6). For the exchange rate series (Yen/BP, BP/US$, and
DM/BP), the normal method resulted in more exact VaR estimates. For the S&P 500,
DAX30, CAC40, and DJCPI indices, stable methods underestimated VaR, though the estimates
were closer to the true VaR than the normal estimates. Mean absolute biases under
stable and normal modeling are of comparable magnitudes.
In-sample examination of VaR models showed:
* the stable modeling generally results in conservative and accurate 99% VaR estimates,
which is preferred by financial institutions and regulators,32
* the normal approach leads to overly optimistic forecasts of losses in the 99% VaR esti-
mation,
* from a conservative point of view, the normal modeling is acceptable for the 95% VaR
estimation,
* the stable models underestimate the 95% VaR. In fact, the stable 95% VaR measurements
are closer to the empirical VaR than the normal 95% VaR measurements.
The next step in evaluating VaR models is analysis of their forecasting characteristics.
31 Let bm,s be a bias of a VaR estimate: bm,s = VaRm,s - VaREmpirical,s. The mean absolute bias equals
MABm = ( 8
s=1 |bm,s|)/8, where m denotes normal, stable-ML, stable-FT, and stable-FTT methods, and s a
series.
32 In the 99% VaR estimation for data series from Table 1, mean absolute bias under the stable modeling was
42% smaller than under the normal modelling.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 269
Fig. 3. VAR estimation for the DM/BP exchange rate.
270 S.T. Rachev et al.
Fig. 4. VAR estimation for the Yen/BP exchange rate.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 271
Fig. 5. VAR estimation for the BP/US$ exchange rate.
272 S.T. Rachev et al.
Fig. 6. VAR estimation for the CAC40 index.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 273
Fig. 7. VAR estimation for the Nikkei 225 index.
274 S.T. Rachev et al.
Fig. 8. VAR estimation for the S&P 500 index.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 275
Fig. 9. VAR estimation for the DAX30 index.
276 S.T. Rachev et al.
Fig. 10. VAR estimation for the DJCPI index.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 277
Fig. 11. Stable fitting at limiting values of a confidence interval for alpha.
Fig. 12. VAR estimation at limiting values of a confidence interval for alpha.
278 S.T. Rachev et al.
Fig. 13. Stable fitting at limiting values of a confidence interval for sigma.
Fig. 14. VAR estimation at limiting values of a confidence interval for sigma.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 279
Table 3
Empirical, normal, and stable 99% VaR estimatesa
Series 99% VaR
Empirical Normal Stable
ML FT FTT
Yen/BP 1.979 1.528 2.247 2.112 2.494
[1.968, 2.252] [2.276, 2.736]
[1.919, 2.415] [2.230, 2.836]
BP/US$ 1.774 1.526 2.221 2.200 2.668
[2.014, 2.412] [2.436, 2.925]
[1.956, 2.593] [2.358, 3.029]
DM/BP 1.489 1.149 1.819 1.520 1.996
[1.190, 1.712] [1.792, 2.211]
[1.179, 1.742] [1.700, 2.329]
S&P 500 2.293 2.131 2.559 2.200 2.984
[2.117, 2.358] [2.757, 3.243]
[2.106, 2.470] [2.700, 3.336]
DAX30 2.564 2.306 2.464 2.375 2.746
[2.260, 2.502] [2.557, 2.949]
[2.240, 2.569] [2.523, 2.997]
CAC40 3.068 2.760 3.195 3.019 3.144
[2.753, 3.364] [2.788, 3.504]
[2.682, 3.520] [2.700, 3.841]
Nikkei 225 3.428 2.737 4.836 3.842 6.013
[3.477, 4.254] [5.190, 6.701]
[3.367, 4.453] [4.658, 19.950]
DJCPI 2.053 1.804 2.446 2.285 2.603
[1.955, 2.423] [2.382, 2.870]
[1.916, 2.474] [2.288, 3.035]
a The CIs right below the estimates are the 95% CIs, the next CIs are the 99% CIs.
4.2. Forecast-evaluation of VaR estimates
In this section we investigate the forecasting properties of stable and normal VaR modeling
by comparing predicted VaR with observed returns.
We test the null hypothesis that Equation (1) for a time horizon of 1 day ( = 1) holds
at any time t:
Pr[Pt < -VaRt] = 1 - c, (4)
where Pt is the relative change (return) in the portfolio value, i.e., Pt = Rt is the portfolio
return at moment t, VaRt is the VaR measure at time t, c is the VaR confidence level,
280 S.T. Rachev et al.
Table 4
Empirical, normal, and stable 95% VaR estimatesa
Series 95% VaR
Empirical Normal Stable
ML FT FTT
Yen/BP 1.103 1.086 1.033 0.968 0.995
[0.926,1.047] [0.937,1.132]
[0.911,1.186] [0.911,1.329]
BP/US$ 1.038 1.077 0.981 0.944 0.986
[0.898,1.072] [0.917,1.158]
[0.876,1.599] [0.895,1.588]
DM/BP 0.806 0.816 0.772 0.687 0.748
[0.652,0.749] [0.695,0.894]
[0.641,0.894] [0.678,1.418]
S&P 500 1.384 1.497 1.309 1.308 1.319
[1.275,1.361] [1.265,1.423]
[1.265,1.411] [1.246,1.503]
DAX30 1.508 1.623 1.449 1.451 1.452
[1.415,1.500] [1.405,1.521]
[1.402,1.533] [1.395,1.650]
CAC40 1.819 1.943 1.756 1.734 1.734
[1.653,1.837] [1.647,1.845]
[1.621,1.944] [1.616,2.288]
Nikkei 225 1.856 1.929 1.731 1.666 1.840
[1.570,1.839] [1.582,2.512]
[1.558,2.280] [1.500,5.022]
DJCPI 1.066 1.274 1.031 0.994 1.011
[0.888,1.047] [0.944,1.188]
[0.870,1.200] [0.915,1.615]
a The CIs right below the estimates are the 95% CIs, the next CIs are the 99% CIs.
t is the current time, t  [1,T ], and T is the length of the testing interval. The test is performed
by checking whether Pr[Rt < -VaRt] is reasonably close to 1 - c, where VaRt is
the estimate of VaRt . Recall that VaRt is computed using the last lw observations.33
Let bt be the indicator function 1{Rt < -VaRt }, 1 t T . If Equation (4) holds, then
bt = 1 Rt < -VaRt =
1, probability = 1 - c,
0, probability = c.
33 See Equation (3).
Ch. 7: Stable Modeling of Market and Credit Value at Risk 281
Table 5
Biases of normal and stable 99% VaR estimates
Series 99% VaRm - 99% VaREmpirical
Normal Stable
ML FT FTT
Yen/BP -0.451 0.268 0.133 0.515
BP/US$ -0.248 0.447 0.426 0.894
DM/BP -0.340 0.330 0.031 0.507
S&P 500 -0.162 0.266 -0.093 0.691
DAX30 -0.258 -0.100 -0.189 0.182
CAC40 -0.308 0.127 -0.049 0.076
Nikkei 225 -0.691 1.408 0.414 2.585
DJCPI -0.249 0.393 0.232 0.550
Mean absolute bias 0.338 0.416 0.196 0.750
Table 6
Biases of normal and stable 95% VaR estimates
Series 95% VaRm - 95% VaREmpirical
a
Normal Stable
ML FT FTT
Yen/BP -0.017 -0.070 -0.135 -0.108
BP/US$ 0.039 -0.057 -0.094 -0.052
DM/BP 0.010 -0.034 -0.119 -0.058
S&P 500 0.113 -0.075 -0.076 -0.065
DAX30 0.115 -0.059 -0.057 -0.056
CAC40 0.124 -0.063 -0.085 -0.085
Nikkei 225 0.073 -0.125 -0.190 -0.016
DJCPI 0.208 -0.035 -0.072 -0.055
Mean absolute bias 0.087 0.065 0.104 0.070
a m denotes normal, stable-ML, stable-FT, and stable-FTT methods.
Let us denote by E the number of exceedings (Rt < -VaRt )34 over the testing interval
[1,T ]. If Equation (4) is valid, then the variable E = T
t=1 bt has a binomial distribution.
We can formulate a testing rule: reject the null hypothesis at level of significance x if
E
t=0
T
t
(1 - c)t
cT -t x
2
or
E
t=0
T
t
(1 - c)t
cT -t
1 -
x
2
.
34 In nominal levels, an exceeding implies a case when actual losses exceeded the predicted losses.
282 S.T. Rachev et al.
Table 7
Admissible VaR exceedings and exceeding frequencies
VaR confidence Length of a testing Admissible VaR Admissible VaR
level, c interval, T exceedings, E frequencies, E/T (%)
Significance level, x Significance level, x
5% 1% 5% 1%
95% 500 [17,33] [14,36] [3.40,6.60] [2.80,7.20]
1500 [61,89] [56,94] [4.07,5.93] [3.73,6.27]
99% 500 [2,8] [0,10] [0.40,1.60] [0.00,2.00]
1500 [9,21] [6,23] [0.60,1.40] [0.40,1.53]
For large T and sufficiently high VaR confidence levels, the binomial distribution can be
approximated by the normal distribution. Hence, the testing rule for large T is: reject the
null hypothesis at level of significance x if
E < T (1 - c) - z1-x/2 T (1 - c)c or E > T (1 - c) + z1-x/2 T (1 - c)c,
where zp is the p% standard normal quantile. The bounds of admissible VaR exceedings
E and exceedings frequencies, E/T , for testing at level of significance 5% and 1% are
provided in Table 7.
We examined forecasting properties of stable and VaR models for data series described
in Table 1. In testing procedures we considered the following parameters:
* window lengths lw = 260 observations (data over 1year) and lw = 1560 observations
(data over 6 years),
* lengths of testing intervals T = 500 days and T = 1500 days.
Evaluation results are reported in Tables 8 and 9. We indicate by the bold font the numbers,
which are outside of acceptable ranges.
From Table 8 we can see that normal models for the 99% VaR computations commonly
produce numbers of exceedings above the acceptable range, which implies that normal
modeling significantly underestimates VaR (losses). At window length of 260 observations,
stable modeling is not satisfactory. It provided permissible number of exceptions
only for the BP/US$ and DJCPI series. At sample size of 1560 and testing interval of
500 observations, exceedings by the stable-FT method are outside of the admissible interval
for the S&P 500, DAX30, and CAC40 indices. Testing on the longer interval with
T = 1500 showed that numbers of "stable" exceptions are within permissible range. Table
8 demonstrates that increasing the window length from 260 observations to 1560 observations
reduces the number of stable-FT exceedings. In contrast, extending the window
length for normal models does not decrease E, in some cases, even elevates it. Results
illustrate that stable modeling outperforms normal modeling in the 99% VaR estima-
tions.
The 95% VaR normal estimates (except the DAX30 series), obtained using 260 observations,
are within the permissible range. Increasing the window length generally worsens the
Ch. 7: Stable Modeling of Market and Credit Value at Risk 283
Table 8
99% VaR exceedings
Series Length of a testing 99% VaR exceedings
interval, T Window length = 260 obs. Window length = 1560 obs.
Normal FT Normal FT
E E/T (%) E E/T (%) E E/T (%) E E/T (%)
Yen/BP 500 15 3.00 13 2.60 10 2.00 2 0.40
1500 40 1.67 34 2.27 45 3.00 21 1.40
BP/US$ 500 10 2.00 5 1.00 1 0.20 0 0.00
1500 26 1.73 13 0.86 17 1.33 5 0.33
DM/BP 500 18 3.60 14 2.80 17 3.40 8 1.60
1500 45 3.00 33 2.20 50 3.33 19 1.27
S&P 500 500 17 3.40 13 2.60 25 5.00 13 2.60
1500 35 2.33 27 1.80 28 1.87 14 0.93
DAX30 500 21 4.20 14 2.80 19 3.80 18 3.60
1500 41 2.73 29 1.93 25 1.67 20 1.33
CAC40 500 16 3.20 14 2.80 14 2.80 13 2.60
1500 34 2.27 29 1.93 17 1.63 19 1.27
Nikkei 225 500 15 3.00 14 2.80 13 2.60 7 1.40
1500 31 2.07 23 1.53 26 1.73 10 0.67
DJCPI 500 12 2.40 7 1.40 15 3.00 10 2.00
1500 29 1.93 15 1.00 28 1.87 17 1.13
normal VaR measurements. The stable-FT method provided sufficient 95% VaR estimates
for the Yen/BP and BP/US$ exchange rates and the CAC40 and Nikkei 225 indices.
A study of the predictive power of VaR models suggests that:
* the normal modeling significantly underestimates 99% VaR,
* the stable method results in reasonable 99% VaR estimates,
* 95% normal measurements are in the admissible range for the window length of 260
observations. Increasing lw to 1560 observations might deteriorate the precision of the
estimates.
5. Stable modeling and risk assessment for individual credit returns
Recall that the stable distributions are characterized by four parameters: -tail index,
-skewness, -location, and -scale. Modeling with such parameters allows for heavy tails
and skewness of the distributions. Our empirical analysis confirms that, indeed: (i) credit
returns exhibit asymmetry and heavy-tails; (ii) stable modeling captures these features of
the returns.
284 S.T. Rachev et al.
Table 9
95% VaR exceedings
Series Length of a testing 95% VaR exceedings
interval, T Window length = 260 obs. Window length = 1560 obs.
Normal FT Normal FT
E E/T (%) E E/T (%) E E/T (%) E E/T (%)
Yen/BP 500 35 7.00 38 7.60 27 5.40 31 6.2
1500 94 6.27 104 6.93 109 7.27 122 8.13
BP/US$ 500 33 6.60 45 9.00 10 2.00 17 3.40
1500 73 4.87 96 6.40 46 3.07 57 3.80
DM/BP 500 32 6.40 38 7.60 29 5.80 37 7.40
1500 89 5.93 114 7.60 105 7.00 139 9.27
S&P 500 500 34 6.80 39 7.80 43 8.60 47 9.40
1500 79 5.27 98 6.53 62 4.13 69 4.60
DAX30 500 47 9.40 50 10 42 8.40 45 9.00
1500 98 6.53 109 7.27 62 4.13 79 5.27
CAC40 500 32 6.40 34 6.80 31 6.20 32 6.40
1500 81 5.40 87 5.80 51 4.90 82 5.47
Nikkei 225 500 37 7.40 40 8.00 28 5.60 33 6.60
1500 85 5.67 90 6.00 68 4.53 87 5.80
DJCPI 500 29 5.80 35 7.00 37 7.40 46 9.20
1500 70 4.67 93 6.20 77 5.13 108 7.20
The "assets" used in the study are the Merrill Lynch indices of the US government and
corporate bonds with maturities from one to 10 years and credit ratings from "BB" to
"AAA". Returns on indices are modeled as stable-distributed: Ri  Si (i,i,i), where
i = 1,...,21. Some analysis of the indices is provided in Table 10. Daily returns series
are illustrated on Figure 15 and in Appendix A. The benchmark for assessment of the
stable model properties is the "normal" model, i.e., approximation of returns by normal
distributions. By categorization of stable distributions, a normal distribution has a tail index
 = 2 and a symmetric distribution has a skewness parameter  = 0. Values of  < 2
indicate thicker tails than the tails of the normal distribution. In general, as  is smaller, the
tails are heavier and the peak of the density is higher. If  < 0, the distribution is skewed
to the left. If  > 0, the distribution is skewed to the right. Larger absolute magnitudes
of  point to stronger skewness. The stable and normal parameter estimates for the bond
indices are presented in Table 10. For all 17 considered indices, the tail index  is less than
two, which reveals heavy-tailedness, and the skewness parameter  is below zero, which
implies skewness to the left. The fitted empirical, stable, and normal densities of indices
are displayed in Figure 16 and in Appendix A.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 285
Table 10
Normal and stable parameter estimates of bond indices
Indexa Rating or Maturity Normal Stable
issuer (year) Mean St. dev. Tail index Skewness Location Scale
   
G102 US gov-t 1-3 0.026 0.096 1.696 -0.160 0.029 0.055
G202 US gov-t 3-5 0.030 0.204 1.739 -0.134 0.036 0.122
G302 US gov-t 5-7 0.032 0.275 1.781 -0.134 0.032 0.169
G402 US gov-t 7-10 0.033 0.352 1.808 -0.172 0.033 0.218
C1A1 AAA 1-3 0.027 0.096 1.654 -0.080 0.053 0.027
C2A1 AAA 3-5 0.029 0.175 1.695 -0.112 0.029 0.099
C3A1 AAA 5-7 0.032 0.249 1.710 -0.116 0.031 0.145
C4A1 AAA 7-10 0.032 0.319 1.739 -0.155 0.031 0.190
C1A2 AA 1-3 0.028 0.099 1.686 -0.105 0.027 0.056
C2A2 AA 3-5 0.029 0.177 1.722 -0.111 0.029 0.104
C3A2 AA 5-7 0.032 0.250 1.757 -0.121 0.032 0.150
C4A2 AA 7-10 0.033 0.325 1.778 -0.148 0.033 0.198
C1A3 A 1-3 0.028 0.098 1.688 -0.135 0.027 0.056
C2A3 A 3-5 0.030 0.180 1.702 -0.122 0.029 0.104
C3A3 A 5-7 0.032 0.255 1.743 -0.133 0.033 0.151
C4A3 A 7-10 0.033 0.333 1.753 -0.167 0.033 0.199
C1A4 BBB 1-3 0.029 0.112 1.653 -0.113 0.029 0.054
C2A4 BBB 3-5 0.032 0.183 1.662 -0.042 0.033 0.096
C3A4 BBB 5-7 0.034 0.249 1.690 -0.125 0.035 0.140
C4A4 BBB 7-10 0.035 0.316 1.694 -0.136 0.035 0.180
H0A1 BB 1-3 0.027 0.185 1.686 -0.252 0.042 0.104
a Each index set, except H0A1, includes 2418 daily observations from 3.13.90 to 7.29.99. Source of index series:
Merrill Lynch, used with permission.
In order to assess riskiness of the individual credit series and properties of stable modeling
in the credit risk evaluation, the 99% and 95% Value at Risk (VaR) measurements
were computed. The stable and normal VaR estimates are reported in Table 11. Normal
VaR measurements are given for comparison purposes. The differences between empirical
and modeled VaR are given in Appendix B, Table B.1. The VaR evaluation for the bond
indices is illustrated on Figures 17 and in Appendix A. Results of VaR estimations lead to
the following conclusions:35
Since credit returns have skewed and heavy-tailed distributions, VaR measurements provide
more adequate indication of risk than symmetric measurements (standard deviation
or, in case of stable distributions, scale parameter) do.
35 This section computes "in-sample" VaR. Hence, the conclusions discuss in-sample VaR properties.
286 S.T. Rachev et al.
Fig. 15. H0A1 daily returns.
Fig. 16. Stable and normal fitting of the HOA1 index.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 287
Fig. 17. VAR estimation for the HOA1 index.
* the stable modeling produces conservative and accurate 99% VaR estimates, which is
preferred by financial institutions and regulators. "Conservative" VaR estimates exceed
empirical VaR, which implies that forecasts of losses were greater than observed losses,
* the stable modeling underestimates the 95% VaR,
* the normal modeling gives overly optimistic forecasts of losses in the 99% VaR esti-
mation,
* the normal modeling is acceptable for the 95% VaR estimation.
The stable modeling for high values of the VaR confidence level is superior because it
adequately describes heavy tails and skewness in the data. Our empirical analysis demonstrates
advantages of stable modeling in evaluation of riskiness of single credit returns
series. The next step is to examine properties of stable modeling in evaluation of portfolio
risk.
6. Portfolio credit risk for independent credit returns
Suppose that a portfolio includes n credit assets. Then, the portfolio return is given by
RP = n
i=1 wiRi, where Ri is the return on the i-th asset, wi is the weight of the i-th
asset, i = 1,...,n, n
i=1 wi = 1. The modeling in this section assumes that distributions
288 S.T. Rachev et al.
Table 11
Empirical, normal, and stable VaR estimates for bond indices
Index 99% VaR estimates 95% VaR estimates
Empirical Normal Stable Empirical Normal Stable
G102 0.242 0.198 0.275 0.127 0.132 0.119
G202 0.518 0.446 0.576 0.303 0.306 0.283
G302 0.739 0.609 0.747 0.412 0.421 0.399
G402 0.928 0.785 0.932 0.545 0.545 0.518
C1A1 0.238 0.196 0.284 0.129 0.130 0.119
C2A1 0.437 0.377 0.509 0.244 0.258 0.236
C3A1 0.687 0.548 0.734 0.369 0.378 0.353
C4A1 0.883 0.712 0.931 0.480 0.494 0.467
C1A2 0.237 0.201 0.285 0.132 0.134 0.125
C2A2 0.443 0.382 0.505 0.254 0.261 0.244
C3A2 0.663 0.550 0.689 0.373 0.380 0.355
C4A2 0.870 0.722 0.890 0.482 0.501 0.474
C1A3 0.237 0.207 0.286 0.135 0.134 0.125
C2A3 0.469 0.390 0.530 0.260 0.267 0.248
C3A3 0.705 0.560 0.719 0.376 0.386 0.361
C4A3 0.893 0.741 0.949 0.487 0.514 0.485
C1A4 0.262 0.231 0.290 0.124 0.155 0.119
C2A4 0.478 0.392 0.511 0.243 0.268 0.228
C3A4 0.711 0.545 0.741 0.361 0.375 0.343
C4A4 0.862 0.702 0.960 0.467 0.486 0.451
H0A1 0.557 0.403 0.570 0.258 0.277 0.245
of Ri are: (i) independent -stable and (ii) characterized by the same index of stability,
Ri  S(Ri ,Ri ,0),36 i = 1,...,n. The additivity property of independent stable random
variables provides analytic formulas for parameters of portfolio returns RP . The formulas
lead to estimates of portfolio parameters and risk without simulations. In practice, the
"independent" risk measurements are lower bounds of portfolio risk.
By the additivity property of stable distributions, a linear combination of independent
stable random variables is again a stable random variable. Therefore, RP = n
i=1 wiRi
follows a stable law:
RP  S(RP ,RP ,0),
36 We assume that a > 1 (this assumption is always supported by the empirical studies) and the mean Ri
= 0.
If Ri
= 0, we "center" the Ri observations: R
i = Ri - Ri
.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 289
where  is the tail index, RP is the scale parameter RP is the skewness parameter,
RP =
n
i=1
|wi|Ri

1/
,
RP =
n
i=1[sign(wi)Ri (|wi|Ri )]
n
i=1(|wi|Ri )
.
Thus, the distribution of the portfolio returns is characterized by three parameters: tail
index (index of stability) , skewness RP , and scale RP . The parameter  is exogenous.
Estimation of RP and RP can be implemented in three steps:
Step 1: Find estimates of Ri and Ri by stable fitting sets of Rit , t = 1,...,T, i =
1,...,n.
Step 2: Evaluate portfolio parameters RP and RP :
^RP =
n
i=1
|wi|^Ri

1/
, (5)
^RP =
n
i=1[sign(wi) ^Ri (|wi|^Ri )]
n
i=1(|wi|^Ri )
. (6)
Having estimates of parameters of the portfolio credit risk, ^RP and ^RP , the portfolio
VaR is the negative of the appropriate quantile of the RP -distribution.
As an illustration of the method, portfolio risk is estimated for equally weighted returns
on indices of the investment grade corporate bonds: C1A1, C2A1, C3A1, C4A1, C1A2,
C2A2, C3A2, C4A2, C1A3,C2A3, C3A3, C4A3, C1A4, C2A4, C3A4, and C4A4.37 Description
of indices is given in Table 10 of Section 5. By assumption, the indices are
(i) characterized by the same tail index  and (ii) independent. Fix  at 1.708, the average
of the  values for the bond return series (see Table 10), and recalculate other stable
parameters: Ri , Ri , and Ri . New estimates are reported in Table 12. The condition requiring
the same tail index  for all analyzed series does not appear to be very restrictive:
new parameter estimates (given in Table 12) do not differ much from the previous parameter
estimates (reported in Table 10); the new stable VaR estimates (see Table B.2 in
Appendix B) are close to the initial stable VaR measurements (Table 11).
The  estimates are all small. Further on, we shall assume  = 0. Portfolio parameters
following formulas (1), (2) are ^UP = 0.659, ^UP = -0.125. Thus, RP 
37 A digit after letter "C" denotes the maturity band: 1 ­ from 1 to 3 years, 2 ­ from 3 to 5 years, 3 ­ from 5 to
7 years, 4 ­ from 7 to 10 years; a digit after letter "A" denotes credit rating: 1 ­ "AAA", 2 ­ "AA", 3 ­ "A", 4 ­
"BBB".
290 S.T. Rachev et al.
Table 12
Stable fitting of the bond indices with fixed 
Bond indices Maturity Stable parameters at  = 1.708
(years)
  
C1A1 1-3 -0.084 0.027 0.054
C2A1 3-5 -0.111 0.029 0.099
C3A1 5-7 -0.116 0.031 0.144
C4A1 7-10 -0.146 0.031 0.188
C1A2 1-3 -0.107 0.027 0.057
C2A2 3-5 -0.105 0.029 0.103
C3A2 5-7 -0.098 0.033 0.148
C4A2 7-10 -0.128 0.032 0.194
C1A3 1-3 -0.144 0.027 0.057
C2A3 3-5 -0.120 0.030 0.104
C3A3 5-7 -0.125 0.032 0.149
C4A3 7-10 -0.151 0.032 0.196
C1A4 1-3 -0.118 0.029 0.054
C2A4 3-5 -0.045 0.033 0.098
C3A4 5-7 -0.128 0.035 0.141
C4A4 7-10 -0.143 0.035 0.180
S1.708(0.659,-0.125,0). The portfolio c% VaR is calculated as the negative of the (1 -c)th
quantile of the RP -distribution. For the analyzed portfolio, the 99% VaR equals 3.518
and the 95% VaR equals 1.757. As credit returns typically have the non-negative dependence
structure, the assumption of independence for single credit returns results in the lowest
VaR measurement, the lower bound for portfolio VaR estimates. The upper bound of
the portfolio VaR measurements is given by the non-diversified VaR, the sum of the standalone
VaR values.38 For our portfolio, the non-diversified stable 99% VaR is 9.813 and the
non-diversified stable 95% VaR is 4.733. Analysis in Section 5 showed the 99% stable VaR
estimates slightly exceed the empirical 99% VaR, whereas the 95% stable VaR evaluation
underestimates the empirical 95% VaR. Therefore, 9.813 is a biased upwards estimate of
the portfolio non-diversified 99% VaR and 4.733 is a biased downwards measurement of
the portfolio non-diversified 95% VaR.
7. Stable modeling of portfolio risk for symmetric dependent credit returns
In this section we suppose that distributions of credit returns are symmetric -stable and
dependent. We interpret a symmetric random variable as a transformation of a normal ran-
38 The stand-alone VaR is the VaR for the individual asset.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 291
dom variable. Based on this interpretation, we develop a new methodology for correlation
estimation. We apply the methodology for portfolio risk assessment.
We evaluate portfolio risk by determining portfolio VaR: (i) simulating a distribution of
the RP = n
i=1 wiRi values; (ii) finding a certain quantile of the RP distribution, say, the
1% quantile, which corresponds to the 99% VaR confidence level. The aim of simulations
is to project possible portfolio return values RP at time T + 1 given: (i) observations
of individual returns over time: Ri1,Ri2,...,RiT , i = 1,...,n; (ii) weights of portfolio
assets w1,...,wn. The simulations must account for dependence among individual credit
returns Ri, i = 1,...,n. A traditional approach of quantifying dependence is to calculate
the covariance matrix. Under the -stable assumption for distributions of Ri , computation
of the covariance matrix is impossible.
We suggest a new method for deriving the dependence (association) structure. The
method assumes that Ri are symmetric strictly stable: Ri  SRi
(Ri ,0,0). A symmetric
-stable (SS) random variable can be interpreted as a random rescaling transformation
of a normal random variable (see Property 1 below). If a collection of SS variables is
obtained by applying a similar transformation to dependent normal variables, the dependence
structure among variables will remain. Thus, the dependence among SS random
variables can be explained by the dependence among underlying normal random variables.
Property 1.39 Assume that:
(i) G is a normal random variable with a zero mean:
G  S2(G,0,0) = N 0,22
G ,
(ii) Y is a symmetric -stable random variable,  < 2:
Y  S(Y ,0,0),
(iii) S is a positive 
2 -stable random variable:
S  S/2
2
Y
2
G
cos

4
2/
,1,0 ,
(iv) S and G are independent.
Then, the symmetric -stable random variable Y can be represented as a random rescaling
transformation of the normal random variable G:
Y = S1/2
G.
Simulations of the portfolio return values RP can be divided into two fragments:
39 Property 1 is a slightly modified version of Proposition 1.3.1 in Samorodnitsky and Taqqu (1994).
292 S.T. Rachev et al.
(i) generating individual returns Ri with the same dependence structure as the Ri's. We
derive the dependence among Ri supposing that Ri  SRi
(Ri ,0,0). Based on Property
1, Ri can be expressed as a transformation of a normal random variable:
Ri = S
1/2
i Gi, (7)
where
Gi  S2(Gi ,0,0) = N 0,22
Gi
, (8)
Si  SRi
/2
2
Ri
2
Gi
cos

4
2/Ri
,1,0 , (9)
Si is independent of Gi, i = 1,...,n.
Random rescaling transformations of normal variables Gi into Ri preserve the dependence
structure. Hence, the dependence among Ri can be explained by the dependence
among Gi, i = 1,...,n. Based on this property, we generate dependent normal variables
Gi, maintaining the initial dependence,40 then, we generate Ri = S
1/2
i Gi, where
Si is a simulated value of Si ;
(ii) computing RP = n
i=1 wiRi.
The simulations are performed according to the following algorithm:41
Step 1: Estimate stable parameters of Ri: Ri , Ri , Ri .42
Step 2: "Center" the Ri observations: R
i = Ri - Ri . Further on, we shall assume Ri =
0 and consider R
i as Ri : Ri  SRi
(Ri ,0,0), i = 1,...,n.
Step 3: Assume: (i) Ri can be decomposed according to expressions (7)­(9); (ii) the covariance
matrix of (Gi)1 i n is equal to the covariance matrix of truncated (Ri)1 i n.
Evaluate the covariance matrix of (Gi)1 i n at time T + 1, T +1 = {cij,T +1|T }, i =
1,..., n,j = 1,...,n, using exponential weighting:
c2
i,T +1|T = (1 - )
K
k=0
k
R2
i,T -k, (10)
c2
ij,T +1|T = (1 - )
K
k=0
k
Ri,T -kRj,T -k, (11)
40 Variables Gi, which enter formulas (1) and (8), are not observable. We suppose the dependence structure of
Gaussian variables (Gi)1 i n is "inherited" from the dependence structure of truncated values of stable variables
(Ri)1 i n. Because we believe that the "outliers" are very important for the description of the dependence
structure, we take the truncation value for Ri sufficiently large.
41 The algorithm is implemented in the Mercury Software Package for Market Risk (VaR). See Rachev et al.
(1999).
42 This section assumes Ri
= 0.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 293
where T + 1|T denotes a forecast for time T + 1 conditional on information up to
time T ;  is a decay factor, 0 <  < 1; K is a number of observations' lags. Exponential
weighting (6), (7) allows to account for volatility and correlation clustering (GARCH
effects).43 Formulas (6), (7) can be expressed in recursive (GARCH-type) form:44
c2
i,T +1|T = c2
i,T |T -1 + (1 - )R2
i,T ,
c2
ij,T +1|T = c2
ij,T |T -1 + (1 - )Ri,T Rj,T .
Step 4: Generate a value of the multivariate normal random variable G = (G1,G2,...,
Gn) with the covariance matrix T +1.
Step 5: Simulate values of stable random variables
Si  SRi
/2
22
Ri
c2
i
cos

4
2/Ri
,1,0 , i = 1,...,n.
Step 6: Compute Ri = S
1/2
i Gi, i = 1,...,n.
Step 7: Calculate RP = n
i=1 wiRi.
Step 8: Repeat Steps 4­7 a large number of times to form an RP -distribution.
Obtain a portfolio VaR measurement as the negative of a specified quantile of the RP -
distribution.
We evaluate portfolio risk for equally weighted returns on indices of the investment
grade corporate bonds: C1A1, C2A1, C3A1, C4A1, C1A2, C2A2, C3A2, C4A2, C1A3,
C2A3, C3A3, C4A3, C1A4, C2A4, C3A4, and C4A4. Description of indices is given in Table
10 of Section 5. We impose an assumption that returns on these indices are symmetric-stable.
We compute the 99% and 95% VaR measurements in two procedures: (i) simulation
of portfolio returns following the above described algorithm; (ii) calculation of the
99% (95%) VaR as the negative of the 1% (5%) quantile. In step 3 of the portfolio returns
simulations, derivation of the covariance matrix T +1, we used different truncation
points and decay factor values. In order to estimate accuracy of simulations, we calculate
the Kolmogorov Distance (KD) and Anderson­Darling (AD) statistics:
KD = sup
x
Fe(x) - Fs(x) ,
AD = sup
x
|Fe(x) - Fs(x)|

Fe(x)(1 - Fe(x))
,
where Fe(x) is the empirical cumulative density function (cdf) and Fs(x) is the simulated
cdf. The computation results are summarized in Table 13.
43 An exponential weighting methodology follows the RiskMetrics' exponentially weighted moving average
model. See Longerstaey and Zangari (1996).
44 Formulas are adapted from Longerstaey and Zangari (1996).
294 S.T. Rachev et al.
Table 13
Portfolio VaR for symmetric dependent credit returns
Decay Truncation Portfolio VaR Kolmogorov Andersonfactor
 points (%) distance Darling
99% VaR 95% VaR
0.85 10-90 7.508 4.886 3.880 0.086
5-95 7.777 5.153 3.736 0.093
No 8.286 5.346 4.859 0.111
0.94 10-90 7.793 5.147 3.556 0.081
5-95 8.076 5.248 4.362 0.104
1-99 8.389 5.434 5.650 0.128
No 8.114 5.252 5.212 0.117
0.975 10-90 8.028 5.036 3.452 0.077
5-95 8.166 5.318 9.085 0.234
1-99 8.469 5.493 5.805 0.130
No 8.516 5.470 7.274 0.167
The 99% VaR estimates in Table 13 are within the 99% VaR range (3.518, 9.813) derived
in Section 6. At each truncation band, increasing the decay factor leads to higher values
of the 99% VaR. Thus, as the decay factor grows, the 99% VaR generally rises. At each
value of the decay factor, in general, reduction of truncated observations produced higher
VaR numbers. We explain the latter observation by positive correlation in tails (concurrent
extreme events). Consideration of a larger number of tail observations results in higher
VaR. The KD and AD statistics, in general, decline with smaller decay factors. We examine
how selection of the decay factor and the truncation method affects estimation of marginal
risks. The marginal risk is a risk added by an asset to the portfolio risk. It is computed
as the difference between the portfolio risk with an analyzed asset and the portfolio risk
without the asset. We report the examination results in Table 14.
The decay factor of 0.85 does not produce cases "Marginal VaR > Stand-alone VaR"
and "Within one maturity band, higher ratings contribute more risk". In sum, the decay
factor = 0.85 results in the lower KD and AD statistics and does not lead to irregular
cases; the no-truncation method better accounts for correlation in tails. Hence, we would
recommend the choice of the decay factor = 0.85 and the no-truncation method.
In Table 15 we report marginal 99% VaR, stand-alone 99% VaR, and diversification
effects at the decay factor of 0.85 and the no-truncation method. Marginal VaR estimates
of Table 15 are consistent with the expectation that, for a given credit rating, bonds with
longer maturities contribute more risk. Having marginal VaR numbers, we can identify
concentration risks. We find that the C4A3 bond index makes the highest addition to the
portfolio 99% VaR: the C4A3 marginal VaR of 0.920 exceeds all other marginal VaR.
Marginal risks for all bond indices are smaller than stand-alone risks, which indicates that,
indeed, diversification reduces risk. From Table 15, we notice that the C4A1 and C3A4
bond indices have highest diversification effects.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 295
Table 14
Marginal risk for symmetric dependent credit returns
Decay factor Truncation (%) Cases: Cases: Higher
Marginal VaR > ratings assets
Stand-alone VaR contribute more risk
0.85 10-90 0 0
5-95 0 0
No 0 0
0.94 10-90 0 0
5-95 0 0
1-99 3 2
No 0 2
0.975 10-90 0 0
5-95 0 4
1-99 2 4
No 3 4
Table 15
Marginal VaR, stand-alone 99% VaR, and diversification effects for bond indices (decay
factor = 0.85, no truncation)
Bond indices Marginal VaR Stand-alone VaR Diversification effect
C1A1 0.199 0.284 0.085
C2A1 0.338 0.509 0.171
C3A1 0.572 0.734 0.162
C4A1 0.713 0.931 0.218
C1A2 0.245 0.285 0.040
C2A2 0.494 0.505 0.011
C3A2 0.575 0.689 0.114
C4A2 0.788 0.890 0.102
C1A3 0.190 0.286 0.096
C2A3 0.403 0.530 0.127
C3A3 0.592 0.719 0.127
C4A3 0.920 0.949 0.029
C1A4 0.185 0.290 0.105
C2A4 0.338 0.511 0.173
C3A4 0.522 0.741 0.219
C4A4 0.803 0.960 0.157
We studied stable modeling of portfolio risk under the assumptions of the independent
and symmetric dependent instruments. In the next section we consider portfolio risk evaluation
in the most general case ­ skewed dependent instruments.
296 S.T. Rachev et al.
8. Stable modeling of portfolio risk for skewed dependent credit returns
We quantify portfolio risk RP by generating a distribution of its possible values and deriving
a portfolio VaR from the constructed distribution of RP . In a case of portfolio assets
with skewed dependent credit returns, simulations of the RP values should reflect the "cumulative"
skewness and maintain the dependence (association) among them. In order to
do that, we decompose single credit returns Ri into two independent parts: the first part
accounts for dependence and the second ­ for skewness. Then, we obtain the portfolio
dependence and skewness components separately aggregating the dependence and skewness
parts of individual credit returns. Simulations of the portfolio credit returns values RP
can be divided into three portions: (i) generation of the portfolio dependence component
maintaining the dependence structure among individual credit returns, (ii) generation of the
portfolio skewness component, and (iii) computation of RP as a sum of the two generated
components. Explanations of our methodology are provided below.
A stable random variable R  S(,,0) can be decomposed (in distribution) into two
independent stable random variables R(1) and R(2):
R
d
= R(1)
+ R(2)
,
where
R(1)
 S(1,1,0), R(2)
 S(2,2,0),
 = 
1 + 
2
1/
, (12)
 =
1
1 + 2
2

1 + 
2
. (13)
Suppose that: (i) R(1) is a symmetric stable variable: 1 = 0; (ii) 1 = 2 = . Then,
formulas (12) and (13) can be reduced to the following expressions:
 = 21/

, (14)
 =
1
2
2. (15)
From Equations (14) and (15), we have

= 2-1/
, 2 = 2.
In sum, a stable random variable R  S(,,0) can be decomposed (in distribution)
into two independent stable random variables: symmetric R(1) and skewed R(2):
R
d
= R(1)
+ R(2)
, (16)
Ch. 7: Stable Modeling of Market and Credit Value at Risk 297
where
R(1)
 S 2-1/
,0,0 , (17)
R(2)
 S 2-1/
,2,0 . (18)
Using methodology (16)­(18), we can divide individual credit returns Ri  SRi
(Ri ,
Ri ,0) into the "dependence" and "skewness" parts. First, we partition Ri into the "symmetry"
and "skewness" fragments:
Ri
d
= R
(1)
i + R
(2)
i ,
where
R(1)
i  SRi
2-1/Ri Ri ,0,0 , R(2)
i  SRi
2-1/Ri Ri ,2Ri ,0 ,
parts R(1)
i and R(2)
i are independent, i = 1,...,n. Second, we suppose: (i) R(1)
i , i = 1,
...,n, are dependent and (ii) R(2)
i , i = 1,...,n, are independent. Consequently, symmetric
terms R(1)
i explain dependence (association) among Ri's and terms R(2)
i account for
skewness of Ri's.
Based on Property 1 (see Section 7), R(1)
i  SRi
(2-1/Ri Ri ,0,0) can be written as a
transformation of a normal random variable:
R(1)
i = S
1/2
i Gi,
where
Gi  S2(Gi ,0,0) = N 0,22
Gi
,
Si  SRi
/2
2-2/Ri 2
Ri
2
Gi
cos

4
2/Ri
,1,0 ,
Si is independent of Gi, i = 1,...,n.
Random rescaling transformations of normal variables Gi into R(1)
i maintain the dependence
structure. Therefore, from the dependence among Gi's we can determine the dependence
among R(1)
i , or the dependence among Ri.
Adding separately the dependence and skewness terms of Ri 's, we obtain the two components
of the portfolio returns RP :
RP = R(1)
P + R(2)
P , (19)
where R
(1)
P = n
i=1 wiR
(1)
i = n
i=1 wiS
1/2
i Gi is the "dependence" component and R
(2)
P =
n
i=1 wiR
(2)
i is the "skewness" component.
298 S.T. Rachev et al.
We simulate the RP values based on decomposition (19): RP = R(1)
P + R(2)
P . The simulations
are executed according to the next algorithm:45
Step 1: Estimate stable parameters of Ri: Ri , Ri , Ri , Ri.
Step 2: "Center" the Ri observations: R
i = Ri - Ri . Further on, we shall assume Ri =
0 and consider R
i as Ri: Ri  SRi
(Ri ,Ri ,0), i = 1,...,n.
Step 3: Evaluate the covariance matrix of normal random variables (Gi)1 i n at time
T + 1, T +1 = {cij,T +1|T }, i = 1,..., n,j = 1,...,n, using exponential weighting:
c2
i,T +1|T = (1 - )
K
k=0
k
R2
i,T -k,
c2
ij,T +1|T = (1 - )
K
k=0
k
Ri,T -kRj,T -k,
where T +1|T denotes a forecast for time T +1 conditional on information up to time T ;
 is a decay factor, 0 <  < 1; K is a number of observations' lags.
Step 4: Generate a value of the multivariate normal random variable G = (G1,G2,...,
Gn) with the covariance matrix T +1.
Step 5: Simulate values of stable random variables
Si  SRi
/2
21-2/Ri 2
Ri
c2
i
cos

4
2/Ri
,1,0 , i = 1,...,n.
Step 6: Compute R
(1)
i = S
1/2
i Gi, i = 1,...,n.
Step 7: Generate R
(2)
i  SRi
(2-1/Ri Ri ,2Ri ,0), i = 1,...,n.
Step 8: Calculate RP = n
i=1 wiR
(1)
i + n
i=1 wiR
(2)
i .
Step 9: Repeat Steps 4­8 a large number of times to form an RP -distribution.
Derive a portfolio VaR estimate as the negative of a chosen quantile of the RP -distri-
bution.
We implement the suggested procedure (Step 1­Step 9) for the risk assessment of the
same portfolio of indices as in Section 7. We suppose that returns on indices are dependent
skewed--stable. The portfolio VaR estimates are presented in Table 16.
The 99% portfolio VaR estimates fall within the 99% VaR range (3.518, 9.813) of Section
6. From Table 16, the VaR magnitude generally: (i) increases when the decay factor
 increases from 0.85 to 0.94; (ii) declines when  changes from 0.94 to 0.975. Thus, the
decay factor  = 0.94 leads to more conservative VaR estimates. The 1%­99% truncation
band appears to produce the lowest KD and AD statistics. Based on our observations, we
45 This algorithm is an extended version of the algorithm in Section 7.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 299
Table 16
Portfolio VaR for skewed dependent credit returns
Decay Truncation Portfolio VaR Kolmogorov Andersonfactor
 points (%) distance Darling
99% VaR 95% VaR
0.85 10-90 4.939 2.904 7.22 0.20
5-95 5.380 3.162 5.64 0.18
No 5.449 3.236 5.43 0.17
0.94 10-90 5.101 3.009 6.53 0.19
5-95 5.456 3.248 5.24 0.17
1-99 5.596 3.363 4.70 0.14
No 5.455 3.231 5.13 0.17
0.975 10-90 5.112 3.021 6.54 0.19
5-95 5.416 3.238 5.34 0.17
1-99 5.471 3.307 4.37 0.14
No 5.298 3.238 5.43 0.15
would recommend to employ  = 0.94 and the 1%­99% truncation band in VaR derivations
under the assumption of skewed dependent credit returns. We computed marginal
VaRs for the same combinations of the decay factor and the truncation band as in Table
16. The marginal VaR estimates were smaller than the corresponding stand-alone VaR
measurements, which supports feasibility of suggested procedure for simulating portfolio
returns.
We have applied stable modeling to the total risk assessment of credit returns. Below we
analyze stable modeling of isolated credit risk.
9. One-factor model of portfolio credit risk
In this section we outline a one-factor model for quantifying portfolio credit risk. The
model is built on two postulations: (i) constituent parts of the credit returns are the creditrisk-free
part and the credit risk premium; (ii) the credit risk spread follows a stable law.
Applying the one-factor model, in the following sections we quantify credit risk for single
instruments and then estimate portfolio credit risk as a cumulative result of stable distributed
individual credit risks.
Similarly to the previous sections, we assume that a portfolio includes n assets. Then,
the portfolio return is given by RP = n
i=1 wiRi , where Ri is the return on the i-th asset,
wi is the weight of the i-th asset, i = 1,...,n, n
i=1 wi = 1. We conjecture that individual
returns Ri depend on one credit-risk-free factor Yi:
Ri = ai + biYi + Ui, (20)
300 S.T. Rachev et al.
where ai and bi are constants, Ui is the residual representing compensation for credit risk
and random noise,46 i = 1,...,n.
Suppose the i-th portfolio instrument is a corporate bond of maturity  with returns Ri.
There are two possible choices for an underlying credit-risk-free factor Yi: (i) returns on
a Treasury bond of the same maturity ; (ii) returns on a -year bond with a credit rating
AAA. Then, the spread Ui = Ri - ai - biYi reflects charges for credit risk. If the j-th
portfolio asset is a swap with a counterparty that has a low credit rating, say BBB, we can
choose, as an underlying factor Yj , returns on a similar swap with a company that has a
credit rating AAA: Rj = aj + bj Yj + Uj , the term Uj accounts for the credit risk of the
BBB-swap.
We impose the following assumptions on the components of model (20):
(i) Credit risk spreads Ui are strictly stable, Ui  SUi
(Ui ,Ui ,0),47 Ui > 1.
(ii) Default-free factors Yi are strictly stable, Yi  SYi
(Yi ,Yi ,0),48 Yi > 1.
(iii) Ui and Yi are independent of each other, i = 1,...,n.
Then, the portfolio return RP can be decomposed into three components:
RP = A + YP + UP ,
where YP expresses aggregate effect of underlying factors, UP represents portfolio credit
risk,
A =
n
i=1
wiai, YP =
n
i=1
wibiYi, UP =
n
i=1
wiUi.
We evaluate the portfolio credit risk UP in two steps: (i) quantifying credit risk of each
asset Ui; (ii) estimating UP as a cumulative result of individual Ui, i = 1,...,n. Section 10
discusses credit risk evaluation for single portfolio assets. Section 11 examines portfolio
credit risk estimation under the assumptions of independent, symmetric dependent, and
skewed dependent credit risks.
10. Credit risk evaluation for portfolio assets
Approximations of the credit risk premium values Ui for portfolio assets can be obtained
using model (20):
Uit = Rti - ^ai - ^biYti, (21)
46 We interpret the yield spread as the credit risk premium and include the noise factor into the credit risk part.
The noise factor could incorporate taxability, liquidity, and other premiums.
47 The shift of Ui is, in fact, incorporated in ai .
48 Yi is the centered return. If the returns of portfolio instruments, Zi, are non-centered, then we take Yit =
Zit - Zi, t = 1,... ,T .
Ch. 7: Stable Modeling of Market and Credit Value at Risk 301
where ^ai and ^bi are the OLS estimates,
^ai =
T
t=1 Y2
it
T
t=1 Rit - T
t=1 Yit
T
t=1 Rit Yit
T T
t=1 Y2
it - ( T
t=1 Yit )2
, (22)
^bi =
T T
t=1 Rit Yit - T
t=1 Yit
T
t=1 Rit
T T
t=1 Y2
it - ( T
t=1 Yit )2
, (23)
i = 1,...,n; t = 1,...,T.
Estimators ^ai and ^bi, given by expressions (22) and (23), are unbiased.49
We analyze credit risk of corporate bonds applying one-factor model (20). Assume that
returns on an index of the US corporate bonds, Ri, are described by returns on a creditrisk-free
factor, Yi , and a credit spread, Ui :
Ri = ai + biYi + Ui,
where ai and bi are constants, i = 1,...,16. We examine returns on the same 16 indices
as in Section 5 (see Table 10): Ri  {RC1A1, RC2A1, RC3A1, RC4A1, RC1A2, RC2A2, RC3A2,
RC4A2, RC1A3, RC2A3, RC3A3, RC4A3, RC1A4, RC2A4, RC3A4, and RC4A4}. We choose, as
corresponding credit-risk-free factors, returns on the indices of US government bonds in
the same maturity band: Yi  {RG1O2, RG2O2, RG3O2, RG4O2}.50 For example, if we consider
returns on the index of bonds with maturity from one to three years, RC1A1, then the
returns on the index of the government bonds with maturity from one to three years, RG1O2,
serve as the underlying credit-risk-free factor. We approximate the percentage return values
of the individual credit risks Ui, following approach (21): (i) run OLS regressions of model
(20), (ii) compute the residuals' series Ui. Coefficients of the OLS regressions are given in
Appendix B, Table B.3. Obtained sets of OLS credit risk premiums Ui are plotted in Figure
18 and in figures of Appendix C. Empirical densities of Ui are shown in Figure 19 and
in Appendix C. We observe that the credit risk spread series Ui exhibit volatility clusters
and heavy tails. Such behavior of the individual returns sets can be captured by stable and
GARCH models.
Stable modeling of the credit risk premiums Ui, entailed values of  < 1.6,   0, and
  0 (see Table 17). These values of parameter estimates indicate that credit risk spreads
of the corporate bonds' indices are fat-tailed and almost symmetric. Table 17 presents the
following  and  values of the credit risks of the bond indices with a maturity band from
one to three years: AAA bonds:  = 1.333 and  = 0.011; AA bonds:  = 1.379 and
 = 0.030; A bonds:  = 1.393 and  = -0.021; BBB bonds:  = 1.412 and  = 0.004.
49 For analysis of asymptotic properties of OLS estimators (22) and (23) under the stable distribution assumption
for the disturbance term, see Götzenberger, Rachev and Schwartz (1999).
50 A digit after letter "G" denotes the maturity band: 1 ­ from 1 to 3 years, 2 ­ from 3 to 5 years, 3 ­ from 5 to 7
years, 4 ­ from 7 to 10 years.
302 S.T. Rachev et al.
Fig. 18. OLS credit risk premium of the C1A1 bond index.
Table 17
Stable and normal fitting of the OLS credit risk premiums of bond indices
OLS credit Maturity Normal Stable
risk of bond (years)
indices Mean Standard    
deviation
C1A1 1-3 0.0 0.045 1.333 0.011 0.000 0.017
C2A1 3-5 0.0 0.075 1.528 -0.089 -0.001 0.033
C3A1 5-7 0.0 0.096 1.590 -0.023 0.000 0.047
C4A1 7-10 0.0 0.116 1.456 -0.026 0.000 0.051
C1A2 1-3 0.0 0.037 1.379 0.030 0.001 0.015
C2A2 3-5 0.0 0.064 1.523 -0.074 0.000 0.029
C3A2 5-7 0.0 0.086 1.591 -0.060 0.000 0.044
C4A2 7-10 0.0 0.110 1.426 0.005 0.001 0.050
C1A3 1-3 0.0 0.038 1.393 -0.021 0.000 0.015
C2A3 3-5 0.0 0.069 1.483 -0.084 0.000 0.029
C3A3 5-7 0.0 0.098 1.519 -0.073 0.000 0.042
C4A3 7-10 0.0 0.124 1.366 -0.017 0.001 0.048
C1A4 1-3 0.0 0.074 1.412 0.004 0.001 0.018
C2A4 3-5 0.0 0.096 1.527 -0.024 0.001 0.033
C3A4 5-7 0.0 0.113 1.552 -0.077 0.000 0.048
C4A4 7-10 0.0 0.1424 1.480 -0.055 0.001 0.055
Ch. 7: Stable Modeling of Market and Credit Value at Risk 303
Fig. 19. Stable and normal fitting of C1A1 OLS-credit-risks.
Plots of the stable and normal fitting of the OLS credit risk spreads Ui are shown on
Figure 19 and in Appendix C. Figures demonstrate that stable modeling well captures
excess kurtosis and heavy tails of the credit risks Ui.
The GARCH approach models clustering of volatilities and fat tails, by expressing the
conditional variance as an explicit function of past information:
Ri,t = ai + biYi,t + Ui,t , (24)
where
Ui,t = i,t i,t , (25)
i,t  N(0,1), (26)
2
i,t = ci +
p
j=1
i,j 2
i,t-j +
q
j=1
i,j U2
i,t-j , (27)
i = 1,...,n; t = 1,...,T.
We shall name model (24)­(27) as a GARCH(p,q)-normal model because it is based
on the normality assumption for the disturbance term. In order to detect GARCHdependencies,
we examine sample autocorrelation and partial autocorrelation functions of
the squared residuals Ui. Visual inspection of the correlograms suggests values of p and q.
304 S.T. Rachev et al.
Fig. 20. Credit risks: OLS and GARCH.
Applying the Box­Jenkins methodology, we find that p = q = 1 is adequate to capture
temporal dependence of volatilities:
2
i,t = ci + i2
i,t-1 + iU2
i,t-1. (28)
Coefficients of model (24)­(26) and (28) with Ri  {RC1A1, RC2A1, RC3A1, RC4A1,
RC1A2, RC2A2, RC3A2, RC4A2, RC1A3, RC2A3, RC3A3, RC4A3, RC1A4, RC2A4, RC3A4, and
RC4A4} and Yi  {RG1O2, RG2O2, RG3O2, RG4O2} are reported in Appendix B, Table B.4.
Densities of the GARCH(1,1)-normal residuals Ui,t = ci + i2
i,t-1 + iU2
i,t-1 ×i,t are
displayed in Figures 20 and in Appendix D. Graphs demonstrate that the GARCH credit
risk series have lower peaks.
In the portfolio context, implementation of the GARCH models is computationally complex
because a number of parameters rapidly increases as the portfolio expands.51 Hence,
we evaluate portfolio credit risk UP based on stable modeling of individual credit risks with
accounting for GARCH effects by exponential weighting of observations.52 In estimation
of UP , we separately investigate cases of independent, symmetric dependent, skewed dependent
credit risks of portfolio instruments.
51 For references on the multivariate GARCH, see Engle and Kroner (1995).
52 An approach of modeling time-varying volatilities by exponential weighting follows the RiskMetrics' exponentially
weighted moving average model described in Morgan (1995).
Ch. 7: Stable Modeling of Market and Credit Value at Risk 305
11. Portfolio credit risk
In this section we follow the one-factor model of Section 9 and evaluate portfolio credit
risk as a cumulative effect of stable distributed individual credit risks. We impose different
assumptions on their distributions: independent, symmetric dependent, and skewed
dependent. We show implementation of the approach on a portfolio of equally weighted
OLS-credit-risk premiums from Section 10.
11.1. Independent credit risks
Suppose credit-risk-premiums are: (i) characterized by the same tail index ; (ii) independent.
Then, by the additivity property of stable variables (see Section 3), the portfolio credit
risk UP = n
i=1 wiUi is stably distributed:
UP  S(UP ,UP ,0),
where  is the tail index, UP is the scale parameter, UP is the skewness parameter,
UP =
n
i=1
|wi|Ui

1/
, (29)
UP =
n
i=1[sign(wi)Ui (|wi|Ui )]
n
i=1(|wi|Ui )
. (30)
Consider a portfolio of equally weighted OLS-credit-risk premiums from Section 10.
Assume credit-risk-premiums are independent and have the same tail index . We take
 = 1.472, the average of the  values for the credit-risk-premium series (see Table 17),
and recompute other stable parameters: Ui , Ui , and Ui . New estimates are reported in
Table 18. Similarly to returns on bond indices, a condition of the same tail index  for all
analyzed credit risk series does not seem to be very restraining: new parameter estimates
(Table 18) do not deviate much from the previous parameter estimates (Table 17).
Since obtained estimates of  are very small, we assume  = 0. We evaluate portfolio
parameters applying formulas (29), (30): ^UP = 0.015, ^UP = -0.038. Thus, UP 
S1.472(0.015,-0.038,0). The 99% (95%) credit VaR is derived as the negative of the 1%
(5%) quantile of the UP - distribution: the 99% (95%) VaR equals 0.125 (0.046). Having
analytic formulas for the UP parameters, we obtained estimates of portfolio credit risk
without simulations.
11.2. Symmetric dependent credit risks
In order to assess portfolio credit risk, we obtain portfolio credit VaR. It is computed in
two steps: (i) simulating a distribution of the UP = n
i=1 wiUi values; (ii) inferring port-
306 S.T. Rachev et al.
Table 18
Stable fitting of the OLS credit risk premiums with fixed 
OLS credit risk of Maturity Stable parameters at  = 1.472
bond indices (years)
  
C1A1 1-3 0.000 0.000 0.018
C2A1 3-5 -0.090 -0.001 0.032
C3A1 5-7 -0.019 0.000 0.045
C4A1 7-10 -0.019 0.001 0.052
C1A2 1-3 0.023 0.001 0.015
C2A2 3-5 -0.072 -0.001 0.029
C3A2 5-7 -0.039 0.000 0.042
C4A2 7-10 -0.004 0.000 0.051
C1A3 1-3 -0.040 0.000 0.015
C2A3 3-5 -0.084 0.000 0.029
C3A3 5-7 -0.067 0.000 0.041
C4A3 7-10 -0.032 0.001 0.049
C1A4 1-3 -0.010 0.001 0.019
C2A4 3-5 0.011 0.001 0.033
C3A4 5-7 -0.071 -0.001 0.046
C4A4 7-10 -0.053 0.001 0.055
folio credit VaR from the simulated UP distribution. This section examines the case of
symmetric individual credit risks Ui: Ui  SUi
(Ui ,0,0), i = 1,...,n.
We simulate UP applying the methodology from Section 7:
(i) generate individual credit risks Ui with the same dependence structure as the Ui's. We
express Ui as a transformation of a normal random variable:
Ui = S
1/2
i Gi,
where
Gi  S2(Gi ,0,0) = N 0,22
Gi
,
Si  SUi
/2
2
Ui
2
Gi
cos

4
2/Ui
,1,0 ,
Si is independent of Gi, i = 1,...,n.
The dependence among Ui can be explained by the dependence among Gi, i =
1,...,n. We form dependent normal variables Gi, preserving the initial dependence.
Next, we generate Ui = S
1/2
i Gi, where Si is a simulated value of Si ;
(ii) calculate UP = n
i=1 wiUi.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 307
Table 19
Portfolio credit VaR for symmetric credit risks
Decay Truncation Portfolio VaR Kolmogorov Andersonfactor
 points (%) distance Darling
99% VaR 95% VaR
0.85 10-90 3.502 1.918 8.071 0.210
5-95 3.710 1.896 8.898 0.228
No 3.396 1.856 7.692 0.199
0.94 10-90 3.594 1.963 7.680 0.200
5-95 3.643 1.941 8.162 0.209
1-99 3.476 1.975 8.847 0.227
No 3.321 1.792 6.736 0.164
0.975 10-90 3.623 1.877 7.578 0.194
5-95 3.435 1.943 9.085 0.234
1-99 3.578 2.004 9.665 0.254
No 3.293 1.739 7.174 0.167
A portfolio credit VaR can be measured from the UP -distribution.
As an illustration of the approach, we estimate credit risk for a portfolio of equally
weighted OLS-credit-risk premiums of bond indices (see Section 10) assuming they are
symmetric.53 The estimation results are presented in Table 19. The portfolio credit VaR
does not demonstrate a certain pattern of dependence on the decay factor. For each decay
factor, reduction of the truncated observations does not seem to affect the portfolio
credit VaR in a particular fashion. The no-truncation method approach led to the smallest
VaR measurements. Possibly, the credit risk residuals of the investment grade indices
have negative correlations in far tails. Taking into account more observations with negative
correlations reduces the VaR estimates. Since the decay factor does not influence the
VaR results in a specific way and the KD and AD statistics are smaller at the no-truncation
approach, in further analysis, we consider the no-truncation method and arbitrarily select
the decay factor of 0.85. Computation of the marginal VaR, stand-alone VaR, diversification
effects for the no-truncation approach and the decay factor = 0.85 is summarized in
Table 20.
From Table 20, highest contributions to portfolio credit risk are made by the C4A4,
C4A3, and C4A2 bond indices: their marginal 99% VaR equal 0.366, 0.295, and 0.296.
The credit risk premium of the C4A1 index displays the largest diversification effect.
11.3. Skewed dependent credit risks
For estimation of portfolio risk for the skewed dependent credit risks, we propose to employ
the approach of Section 8: (i) split individual credit risks Ui into the dependence and
53 The symmetry proposition is plausible: the skewness parameters of credit risks premiums of bond indices are
small (see Table 16).
308 S.T. Rachev et al.
Table 20
Marginal VaR, stand-alone 99% VaR, and diversification effects for credit risk
premiums of bond indices (decay factor = 0.85, no truncation)
Bond indices Marginal VaR Stand-alone VaR Diversification effect
C1A1 0.175 0.191 0.016
C2A1 0.203 0.251 0.048
C3A1 0.162 0.305 0.143
C4A1 0.145 0.441 0.296
C1A2 0.024 0.148 0.124
C2A2 0.153 0.222 0.069
C3A2 0.180 0.290 0.110
C4A2 0.296 0.453 0.157
C1A3 0.013 0.149 0.136
C2A3 0.097 0.244 0.147
C3A3 0.203 0.325 0.122
C4A3 0.295 0.507 0.212
C1A4 0.079 0.168 0.089
C2A4 0.091 0.243 0.152
C3A4 0.142 0.346 0.204
C4A4 0.366 0.457 0.091
skewness parts; (ii) find the portfolio dependence and skewness components by combining
the dependence and skewness parts of single credit risks; (iii) evaluate the portfolio credit
risk as a sum of the dependence and skewness fragments. Details are given below.
We divide individual credit risks Ui  SUi
(Ui ,Ui ,0) into the "dependence" and
"skewness" parts, applying methodology (16)­(18) (see Section 8):
Ui
d
= U
(1)
i + U
(2)
i ,
where
U
(1)
i  SUi
2-1/Ui Ui ,0,0 , U
(2)
i  SUi
2-1/Ui Ui ,2Ui ,0 ,
parts U
(1)
i and U
(2)
i are independent, i = 1,...,n. We assume: (i) U
(1)
i , i = 1,...,n, are
dependent and (ii) U
(2)
i , i = 1,...,n, are independent. Then, symmetric components U
(1)
i
explain dependence (association) among Ui's and components U
(2)
i depict skewness of
Ui's.
By Property 1 (see Section 7), U
(1)
i  SUi
(2-1/Ui Ui ,0,0) can be interpreted as a
transformation of a normal random variable:
U
(1)
i = S
1/2
i Gi,
Ch. 7: Stable Modeling of Market and Credit Value at Risk 309
where
Gi  S2(Gi ,0,0) = N 0,22
Gi
,
Si  SUi
/2
2-2/Ui 2
Ui
2
Gi
cos

4
2/Ui
,1,0 ,
Si is independent of Gi, i = 1,...,n.
Random rescaling transformations of normal variables Gi into U
(1)
i maintain the dependence
structure. Hence, we can derive the dependence among U(1)
i , or the dependence
among Ui, from the dependence among Gi 's.
Combining separately the dependence and skewness terms of Ui's, we obtain the two
components of the portfolio credit risk UP :
UP = U(1)
P + U(2)
P ,
U(1)
P =
n
i=1
wiU(1)
i =
n
i=1
wiS
1/2
i Gi,
U(2)
P =
n
i=1
wiU(2)
i ,
where U(1)
P is the "dependence" component and U(2)
P is the "skewness" component. The
portfolio credit risk can be evaluated as a sum of the dependence and skewness frag-
ments.
We suggested methodologies for portfolio credit risk assessment and demonstrated their
applications on analysis of returns on bond indices. The methodologies can be employed
for risk evaluation of any financial instruments if they have fat-tailed and/or skewed distri-
butions.
12. Conclusions
The Value-at-Risk (VaR) measurements are widely applied to estimate the exposure to market
and credit risks. The traditional approaches to VaR computations ­ the delta method,
historical simulation, Monte Carlo simulation, and stress-testing ­ do not provide satisfactory
evaluation of possible losses. The delta-normal methods do not describe well financial
data with heavy tails. Hence, they underestimate VaR measurements in the tails. The historical
simulation does not produce robust VaR estimates since it is not reliable in approximating
low quantiles with a small number of observations in the tails. The stress-testing
VaR estimates are subjective. The Monte Carlo VaR numbers might be affected by model
misspecification.
310 S.T. Rachev et al.
This work proposes the application of stable distributions in market and credit VaR estimation.
Our empirical analysis verifies that stable modeling well captures skewness and
heavy-tails of market and credit returns and isolated credit risks. The superior fit allows
to derive accurate risk estimates. The in-sample- and forecast-evaluation shows that stable
VaR modeling outperforms the normal modeling for high values of the VaR confidence
level:
* the stable modeling generally produces conservative and accurate 99% VaR estimates,
which is preferred by financial institutions and regulators,
* the normal method leads to overly optimistic forecasts of losses in the 99% VaR estima-
tion,
* the normal modeling is acceptable for the 95% VaR estimation.
Based on the properties of stable distributions, we design new methods for the correlation
estimation and simulating portfolio values. We employ the methods in evaluation of
portfolio and marginal VaR for three cases of the credit returns: independent, symmetric
dependent, and skewed dependent. We suggest a one-factor model of credit risks. Applying
the one-factor model, we quantify credit risk for individual assets and then assess portfolio
credit risk as an aggregate effect of stable distributed individual credit risks.
The stable Paretian model, while sharing the main properties of the normal distribution
leading to the CLT (Central Limit Theorem), provides at the same time superior fit
in modeling market and credit VaR. However, additional research is needed. Future work
is this direction will be construction of models that capture the features of financial empirical
data such as heavy tails, time-varying volatility, and short and long range dependence.54
In order to describe thick tails, one can employ the conditional heteroskedastic
models based on the stable hypothesis.55 ARMA-stable-GARCH models can incorporate
both heavy tails and time-varying volatility.56 The fractional-stable GARCH model can
capture all observed phenomena in financial data: heavy tails, time-varying volatility, and
short- and long-range dependence. An analysis of VaR estimation with ARMA--stable,
ARMA-stable-GARCH, and fractional-stable GARCH models will be provided elsewhere.
54 For some preliminary results see Liu and Brorsen (1995), Mittnik, Rachev and Paolella (1998), Mittnik,
Paolella and Rachev (1997, 1998a, b), Panorska, Mittnik and Rachev (1995).
55 These models are named as ARMA--stable models.
56 For discussion of stable-GARCH models see Panorska, Mittnik and Rachev (1995) and Mittnik, Paolella and
Rachev (1997).
Ch. 7: Stable Modeling of Market and Credit Value at Risk 311
Appendix A. Stable modeling of credit returns in figures
Fig. A.1. G302 daily returns.
Fig. A.2. Stable and normal fitting of the G302 index.
312 S.T. Rachev et al.
Fig. A.3. VaR estimation for the G302 index.
Fig. A.4. C3A2 daily returns.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 313
Fig. A.5. Stable and normal fitting of the C3A2 index.
Fig. A.6. VaR estimation for the C3A2 index.
314 S.T. Rachev et al.
Fig. A.7. C4A2 daily returns.
Fig. A.8. Stable and normal fitting of the C4A2 index.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 315
Fig. A.9. VaR estimation for the C4A2 index.
Fig. A.10. C3A3 daily returns.
316 S.T. Rachev et al.
Fig. A.11. Stable and normal fitting of the C3A3 index.
Fig. A.12. VaR estimation for the C3A3 index.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 317
Appendix B. Tables
Table B.1
Deviations of VaR estimates for bond indices
Index 99% VaRmodel - 99% VaRempirical 95% VaRmodel - 95% VaRempirical
Normal Stable Normal Stable
G102 -0.044 0.033 0.005 -0.008
G202 -0.072 0.058 0.003 -0.020
G302 -0.130 0.008 0.009 -0.013
G402 -0.143 0.004 0.000 -0.027
C1A1 -0.042 0.046 0.001 -0.010
C2A1 -0.060 0.072 0.014 -0.008
C3A1 -0.139 0.047 0.009 -0.016
C4A1 -0.171 0.048 0.014 -0.013
C1A2 -0.036 0.048 0.002 -0.007
C2A2 -0.061 0.062 0.007 -0.010
C3A2 -0.113 0.026 0.007 -0.018
C4A2 -0.148 0.020 0.019 -0.008
C1A3 -0.030 0.049 -0.001 -0.010
C2A3 -0.079 0.061 0.007 -0.012
C3A3 -0.145 0.014 0.010 -0.015
C4A3 -0.152 0.056 0.027 -0.002
C1A4 -0.031 0.028 0.031 -0.005
C2A4 -0.086 0.033 0.025 -0.015
C3A4 -0.166 0.030 0.014 -0.018
C4A4 -0.160 0.098 0.019 -0.016
H0A1 -0.154 0.013 0.019 -0.013
318 S.T. Rachev et al.
Table B.2
Stable VaR estimates for bond indices with fixed 
Bond index 99% VaR 95%
Different  Fixed  = 1.708 Different  Fixed  = 1.708
C1A1 0.284 0.257 0.119 0.116
C2A1 0.509 0.494 0.236 0.233
C3A1 0.734 0.732 0.353 0.351
C4A1 0.931 0.979 0.467 0.471
C1A2 0.285 0.273 0.125 0.123
C2A2 0.505 0.517 0.244 0.245
C3A2 0.689 0.747 0.355 0.360
C4A2 0.890 1.003 0.474 0.485
C1A3 0.286 0.277 0.125 0.124
C2A3 0.530 0.523 0.248 0.247
C3A3 0.719 0.763 0.361 0.365
C4A3 0.949 1.022 0.485 0.491
C1A4 0.290 0.260 0.119 0.116
C2A4 0.511 0.471 0.228 0.224
C3A4 0.741 0.716 0.343 0.340
C4A4 0.960 0.934 0.451 0.447
Table B.3
Coefficients of OLS regressions
Dependent variable Variables Coeff. Dependent variable Variables Coeff.
RC1A1
C 0.004723
RC1A3
C 0.003887
RG102 0.882424 RG102 0.946025
RC2A1
C 0.006183
RC2A3
C 0.005709
RG202 0.770132 RG202 0.816271
RC3A1
C 0.005550
RC3A3
C 0.005051
RG302 0.835640 RG302 0.853295
RC4A1
C 0.003735
RC4A3
C 0.003806
RG402 0.847226 RG402 0.877039
RC1A2
C 0.003357
RC1A4
C 0.006401
RG102 0.951165 RG102 0.874032
RC2A2
C 0.005733
RC2A4
C 0.009603
RG202 0.808308 RG202 0.760162
RC3A2
C 0.004730
RC3A4
C 0.008296
RG302 0.853315 RG302 0.804311
RC4A2
C 0.004118
RC4A4
C 0.007725
RG402 0.868154 RG402 0.803091
Ch. 7: Stable Modeling of Market and Credit Value at Risk 319
Table B.4
GARCH-normal coefficients
Dependent Variables Coeff. Std. errors Variance equation
variable Variables Coeff. Std. errors
RC1A1 C 0.003581 0.000525 C 2.50E-05 2.02E-06
RG102 0.937996 0.004895 ARCH(1) 0.116681 0.003541
GARCH(1) 0.885367 0.002486
RC2A1 C 0.004948 0.000981 C 7.67E-05 5.76E-06
RG202 0.838944 0.003466 ARCH(1) 0.130119 0.005720
GARCH(1) 0.870004 0.004714
RC3A1 C 0.004199 0.001331 C 0.000152 1.66E-05
RG302 0.893949 0.003650 ARCH(1) 0.130479 0.003970
GARCH(1) 0.866746 0.002716
RC4A1 C 0.004014 0.001539 C 0.000355 3.22E-05
RG402 0.887583 0.003538 ARCH(1) 0.153744 0.008756
GARCH(1) 0.830941 0.008452
RC1A2 C 0.002746 0.000411 C 4.77E-06 7.97E-07
RG102 0.946016 0.003830 ARCH(1) 0.096428 0.002594
GARCH(1) 0.914737 0.002586
RC2A2 C 0.004229 0.000885 C 1.34E-05 2.38E-06
RG202 0.890123 0.003547 ARCH(1) 0.056718 0.002501
GARCH(1) 0.943510 0.001441
RC3A2 C 0.002970 0.001078 C 0.000609 4.66E-05
RG302 0.894861 0.003899 ARCH(1) 0.289805 0.017996
GARCH(1) 0.669240 0.015835
RC4A2 C 0.003420 0.001329 C 0.000302 2.42E-05
RG402 0.918195 0.003240 ARCH(1) 0.180168 0.009135
GARCH(1) 0.817444 0.007086
RC1A3 C 0.002271 0.000421 C 7.06E-06 9.92E-07
RG102 1.003079 0.003215 ARCH(1) 0.137045 0.003494
GARCH(1) 0.887812 0.002061
RC2A3 C 0.005204 0.000664 C 2.01E-05 3.21E-06
RG202 0.903683 0.002247 ARCH(1) 0.124285 0.004417
GARCH(1) 0.905287 0.002271
RC3A3 C 0.005840 0.001114 C 0.000223 2.20E-05
RG302 0.915408 0.003059 ARCH(1) 0.253670 0.007199
GARCH(1) 0.777935 0.004480
RC4A3 C 0.004076 0.001308 C 0.000792 3.16E-05
RG402 0.942102 0.002728 ARCH(1) 0.401945 0.015612
GARCH(1) 0.639974 0.007830
RC1A4 C 0.002450 0.00570 C -3.27E-06 5.58E-07
RG102 1.036861 0.003468 ARCH(1) 0.101918 0.001997
GARCH(1) 0.945209 0.000666
RC2A4 C 0.007017 0.000839 C 5.77E-05 3.97E-06
RG202 0.879618 0.003199 ARCH(1) 0.231563 0.006013
GARCH(1) 0.841086 0.002770
RC3A4 C 0.007452 0.001276 C 3.99E-05 7.17E-06
RG302 0.893645 0.003645 ARCH(1) 0.101316 0.003132
GARCH(1) 0.907304 0.002295
RC4A4 C 0.007402 0.001393 C 0.000194 1.72E-05
RG402 0.887104 0.002892 ARCH(1) 0.179030 0.005716
GARCH(1) 0.840838 0.003809
320 S.T. Rachev et al.
Appendix C. OLS credit risk evaluation for portfolio assets in figures
Fig. C.1. OLS credit risk premium for the C1A2 bond index.
Fig. C.2. Stable and normal fitting of C1A2 OLS-credit-risks.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 321
Fig. C.3. OLS credit risk premium for the C1A3 bond index.
Fig. C.4. Stable and normal fitting of C1A3 OLS-credit-risks.
322 S.T. Rachev et al.
Fig. C.5. OLS credit risk premium for the C3A3 bond index.
Fig. C.6. Stable and normal fitting of C3A3 OLS-credit-risks.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 323
Fig. C.7. OLS credit risk premium for the C1A4 bond index.
Fig. C.8. Stable and normal fitting of C1A4 OLS-credit-risks.
324 S.T. Rachev et al.
Appendix D. GARCH credit risk evaluation for portfolio assets in figures
Fig. D.1. C2A1 credit risks: OLS and GARCH.
Fig. D.2. C3A1 credit risks: OLS and GARCH.
Ch. 7: Stable Modeling of Market and Credit Value at Risk 325
Fig. D.3. C4A1 credit risks: OLS and GARCH.
Fig. D.4. C1A2 credit risks: OLS and GARCH.
326 S.T. Rachev et al.
Acknowledgments
We thank C. Marinelli of Columbia University and, especially, B. Racheva-Iotova of Bravo
Consulting Group for computational assistance. We also thank Kristina Tetereva of University
of Karlsruhe, Germany, for providing data.
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Chapter 8
MODELLING DEPENDENCE WITH COPULAS
AND APPLICATIONS TO RISK MANAGEMENT
PAUL EMBRECHTS, FILIP LINDSKOG and ALEXANDER MCNEIL
Department of Mathematics, ETHZ, CH-8092 Zürich, Switzerland
web: www.math.ethz.ch/finance
Contents
1. Introduction 331
2. Copulas 332
2.1. Mathematical introduction 332
2.2. Sklar's Theorem 334
2.3. The Fréchet­Hoeffding bounds for joint distribution functions 335
2.4. Copulas and random variables 336
3. Dependence concepts 341
3.1. Linear correlation 341
3.2. Perfect dependence 342
3.3. Concordance 343
3.4. Kendalĺs tau and Spearman's rho 345
3.5. Tail dependence 348
4. Marshall­Olkin copulas 351
4.1. Bivariate Marshall­Olkin copulas 352
4.2. A multivariate extension 354
4.3. A useful modelling framework 355
5. Elliptical copulas 357
5.1. Elliptical distributions 357
5.2. Gaussian copulas 360
5.3. t-copulas 361
6. Archimedean copulas 365
6.1. Definitions 365
6.2. Properties 367
6.3. Kendalĺs tau revisited 370
6.4. Tail dependence revisited 371
 Research of the second author was supported by Credit Suisse Group, Swiss Re and UBS AG through RiskLab,
Switzerland.
 The third author acknowledges financial support from Swiss Re.
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
330 P. Embrechts et al.
6.5. Multivariate Archimedean copulas 373
7. Modelling extremal events in practice 377
7.1. Insurance risk 377
7.2. Market risk 380
References 383
Ch. 8: Modelling Dependence with Copulas 331
1. Introduction
Integrated Risk Management (IRM) is concerned with the quantitative description of risks
to a financial business. Whereas the qualitative aspects of IRM are extremely important, in
the present contribution we only concentrate on the quantitative ones. Since the emergence
of Value-at-Risk (VaR) in the early nineties and its various generalisations and refinements
more recently, regulators and banking and insurance professionals have build up a huge
system aimed at making the global financial system safer. Whereas the steps taken no
doubt have been very important towards increasing the overall risk awareness, continuously
questions have been asked concerning the quality of the safeguards as constructed.
All quantitative models are based on assumptions vis-a-vis the markets on which they
are to be applied. Standard hedging techniques require a high level of liquidity of the
underlying instruments, prices quoted for many financial products are often based on
"normal" conditions. The latter may be interpreted in a more economic sense, or more
specifically referring to the distributional (i.e., normal, Gaussian) behaviour of some underlying
data. Especially for IRM, deviations from the "normal" would constitute a prime
source of investigation. Hence the classical literature is full of deviations from the socalled
random walk (Brownian motion) model and heavy tails appear prominently. The
latter has for instance resulted in the firm establishment of Extreme Value Theory (EVT)
as a standard tool within IRM. Within market risk management, the so-called stylised
facts of econometrics summarise this situation: market data returns tend to be uncorrelated,
but dependent, they are heavy tailed, extremes appear in clusters and volatility is
random.
Our contribution aims at providing tools for going one step further: what would be the
stylised facts of dependence in financial data? Is there a way of understanding so-called
normal (i.e., Gaussian) dependence and how can we construct models which allow to
go beyond normal dependence? Other problems we would like to understand better are
spillover, the behaviour of correlations under extreme market movements, the pros and
contras of linear correlation as a measure of dependence, the construction of risk measures
for functions of dependent risks. One example concerning the latter is the following: suppose
we have two VaR numbers corresponding to two different lines of business. In order
to cover the joint position, can we just add the VaR? Under which conditions is this always
the upper bound? What can go wrong if these conditions are not fulfilled? A further type
of risk where dependence play a crucial role is credit risk: how to define, stress test and
model default correlation. The present chapter is not solving the above problem, it presents
however tools which are crucial towards the construction of solutions.
The notion we concentrate on is that of copula, well known for some time within the
statistics literature. The word copula first appeared in the statistics literature 1959 (Sklar,
1959), although similar ideas and results can be traced back to Hoeffding (1940). Copulas
allow us to construct models which go beyond the standard ones at the level of dependence.
They yield an ideal tool to stress test a wide variety of portfolios and products in insurance
and finance for extreme moves in correlation and more general measures of dependence.
As such, they gradually are becoming an extra, but crucial, element of best practice IRM.
332 P. Embrechts et al.
After Section 2 in which we define the concept of copula in full generality, we turn in Section
3 to an overview of the most important notions of dependence used in IRM. Sections 4,
5 and 6 introduces the most important families of copulas, their properties both methodological
as well as with respect to simulation. Throughout these sections, we stress the
importance of the techniques introduced within an IRM framework. Finally in Section 7
we discuss some specific examples.
We would like to stress that the present chapter only gives a first introduction aimed at
bringing together from the extensive copula world those results which are immediately usable
in IRM. Topics not included are statistical estimation of copulas and the modelling of
dependence, through copulas, in a dynamic environment. As such, the topics listed correspond
to a one-period point of view. Various extensions are possible; the interested reader
is referred to the bibliography for further reading.
2. Copulas
The standard "operational" definition of a copula is a multivariate distribution function defined
on the unit cube [0,1]n, with uniformly distributed marginals. This definition is very
natural if one considers how a copula is derived from a continuous multivariate distribution
function; indeed in this case the copula is simply the original multivariate distribution
function with transformed univariate marginals. This definition however masks some of
the problems one faces when constructing copulas using other techniques, i.e., it does not
say what is meant by a multivariate distribution function. For that reason, we start with a
slightly more abstract definition, returning to the "operational" one later. Below, we follow
Nelsen (1999) in concentrating on general multivariate distributions at first and then
studying the special properties of the copula subset. For further details we refer to Nelsen
(1999).
Throughout this chapter, for a function H , we denote by DomH and RanH the domain
and range respectively of H . Furthermore, a function f will be called increasing whenever
x y implies that f (x) f (y). We may also refer to this as f is nondecreasing. A statement
about points of a set S  Rn, where S is typically the real line or the unit cube [0,1]n,
is said to hold almost everywhere if the set of points of S where the statement fails to hold
has Lebesgue measure zero.
2.1. Mathematical introduction
Definition 2.1. Let S1,...,Sn be nonempty subsets of R, where R denotes the extended
real line [-,]. Let H be a real function of n variables such that DomH = S1 ××Sn
and for a b (ak bk for all k) let B = [a,b] (= [a1,b1] ×  × [an,bn]) be an n-box
whose vertices are in DomH . Then the H -volume of B is given by
VH (B) = sgn(c)H(c),
Ch. 8: Modelling Dependence with Copulas 333
where the sum is taken over all vertices c of B, and sgn(c) is given by
sgn(c) =
1, if ck = ak for an even number of k's,
-1, if ck = ak for an odd number of k's.
Equivalently, the H -volume of an n-box B = [a,b] is the n-th order difference of H
on B
VH (B) = b
aH(t) = bn
an
 b1
a1
H(t),
where the n first order differences are defined as
bk
ak
H(t) = H(t1,...,tk-1,bk,tk+1,...,tn) - H(t1,...,tk-1,ak,tk+1,...,tn).
Definition 2.2. A real function H of n variables is n-increasing if VH (B) 0 for all
n-boxes B whose vertices lie in DomH .
Suppose that the domain of a real function H of n variables is given by DomH = S1 ×
 × Sn where each Sk has a smallest element ak. We say that H is grounded if H(t) = 0
for all t in DomH such that tk = ak for at least one k. If each Sk is nonempty and has a
greatest element bk, then H has marginals, and the one-dimensional marginals of H are the
functions Hk with DomHk = Sk and with Hk(x) = H(b1,...,bk-1,x,bk+1,...,bn) for
all x in Sk. Higher-dimensional marginals are defined in an obvious way. One-dimensional
marginals are just called marginals.
Lemma 2.1. Let S1,...,Sn be nonempty subsets of R, and let H be a grounded
n-increasing function with domain S1 ×  × Sn. Then H is increasing in each argument.
Lemma 2.2. Let S1,...,Sn be nonempty subsets of R, and let H be a grounded
n-increasing function with marginals and domain S1 ×  × Sn. Then, if x = (x1,...,xn)
and y = (y1,...,yn) are any points in S1 ×  × Sn,
H(x) - H(y)
n
k=1
Hk(xk) - Hk(yk) .
For the proof, see Schweizer and Sklar (1983).
Definition 2.3. An n-dimensional distribution function is a function H with domain R
n
such that H is grounded, n-increasing and H(,...,) = 1.
It follows from Lemma 2.1 that the marginals of an n-dimensional distribution function
are distribution functions, which we denote F1,...,Fn.
334 P. Embrechts et al.
Definition 2.4. An n-dimensional copula is a function C with domain [0,1]n such that
(1) C is grounded and n-increasing.
(2) C has marginals Ck, k = 1,2,...,n, which satisfy Ck(u) = u for all u in [0,1].
Note that for any n-copula C, n 3, each k-dimensional marginal of C is a k-copula.
Equivalently, an n-copula is a function C from [0,1]n to [0,1] with the following proper-
ties:
(1) For every u in [0,1]n, C(u) = 0 if at least one coordinate of u is 0, and C(u) = uk if
all coordinates of u equal 1 except uk.
(2) For every a and b in [0,1]n such that ai bi for all i, VC([a,b]) 0.
Since copulas are joint distribution functions (on [0,1]n), a copula C induces a probability
measure on [0,1]n via
VC [0,u1] ×  × [0,un] = C(u1,...,un)
and a standard extension to arbitrary (not necessarily n-boxes) Borel subsets of [0,1]n.
A standard result from measure theory says that there is a unique probability measure on
the Borel subsets of [0,1]n which coincides with VC on the set of n-boxes of [0,1]n. This
probability measure will also be denoted VC.
From Definition 2.4 it follows that a copula C is a distribution function on [0,1]n with
uniformly distributed (on [0,1]) marginals. The following theorem follows directly from
Lemma 2.2.
Theorem 2.1. Let C be an n-copula. Then for every u and v in [0,1]n,
C(v) - C(u)
n
k=1
|vk - uk|.
Hence C is uniformly continuous on [0,1]n.
2.2. Sklar's Theorem
The following theorem is known as Sklar's Theorem. It is perhaps the most important result
regarding copulas, and is used in essentially all applications of copulas.
Theorem 2.2. Let H be an n-dimensional distribution function with marginals F1,...,Fn.
Then there exists an n-copula C such that for all x in Rn,
H(x1,...,xn) = C F1(x1),...,Fn(xn) . (2.1)
If F1,...,Fn are all continuous, then C is unique; otherwise C is uniquely determined
on RanF1 ×  × RanFn. Conversely, if C is an n-copula and F1,...,Fn are distribution
functions, then the function H defined above is an n-dimensional distribution function with
marginals F1,...,Fn.
For the proof, see Sklar (1996).
Ch. 8: Modelling Dependence with Copulas 335
From Sklar's Theorem we see that for continuous multivariate distribution functions, the
univariate marginals and the multivariate dependence structure can be separated, and the
dependence structure can be represented by a copula.
Let F be a univariate distribution function. We define the generalized inverse of F as
F-1(t) = inf{x  R | F(x) t} for all t in [0,1], using the convention inf = -.
Corollary 2.1. Let H be an n-dimensional distribution function with continuous marginals
F1,...,Fn and copula C (where C satisfies (2.1)). Then for any u in [0,1]n,
C(u1,...,un) = H F-1
1 (u1),...,F-1
n (un) .
Without the continuity assumption, care has to be taken; see Nelsen (1999) or Marshall
(1996).
Example 2.1. Let  denote the standard univariate normal distribution function and let
n
R denote the standard multivariate normal distribution function with linear correlation
matrix R. Then
C(u1,...,un) = n
R -1
(u1),...,-1
(un)
is the Gaussian or normal n-copula.
2.3. The Fréchet­Hoeffding bounds for joint distribution functions
Consider the functions Mn, n and Wn defined on [0,1]n as follows:
Mn
(u) = min(u1,...,un),
n
(u) = u1 un,
Wn
(u) = max(u1 +  + un - n + 1,0).
The functions Mn and n are n-copulas for all n 2 whereas the function Wn is not a
copula for any n 3 as shown in the following example.
Example 2.2. Consider the n-cube [1/2,1]n  [0,1]n.
VWn
1
2 ,1
n
= max(1 +  + 1 - n + 1,0) - n max 1
2 + 1 +  + 1 - n + 1,0
+ n
2 max 1
2 + 1
2 + 1 +  + 1 - n + 1,0 + 
+ max 1
2 +  + 1
2 - n + 1,0
= 1 - n
2 + 0 +  + 0.
Hence Wn is not a copula for n 3.
336 P. Embrechts et al.
The following theorem is called the Fréchet­Hoeffding bounds inequality (Fréchet, 1957).
Theorem 2.3. If C is any n-copula, then for every u in [0,1]n,
Wn
(u) C(u) Mn
(u).
For more details, including geometrical interpretations, see Mikusinski, Sherwood and
Taylor (1992). Although the Fréchet­Hoeffding lower bound Wn is never a copula for
n 3, it is the best possible lower bound in the following sense.
Theorem 2.4. For any n 3 and any u in [0,1]n, there is an n-copula C (which depends
on u) such that
C(u) = Wn
(u).
For the proof, see Nelsen (1999), p. 42.
We denote by C the joint survival function for n random variables with joint distribution
function C, i.e., if (U1,...,Un)T has distribution function C, then C(u1,...,un) =
P{U1 > u1,...,Un > un}.
Definition 2.5. If C1 and C2 are copulas, C1 is smaller than C2 (written C1  C2) if
C1(u) C2(u) and C1(u) C2(u),
for all u in [0,1]n.
Note that in the bivariate case, C(u1,u2) = 1 - u1 - u2 + C(u1,u2) and hence,
C1(u1,u2) C2(u1,u2) if and only if C1(u1,u2) C2(u1,u2).
The Fréchet­Hoeffding lower bound W2 is smaller than every 2-copula, and every
n-copula is smaller than the Fréchet­Hoeffding upper bound Mn. This partial ordering of
the set of copulas is called a concordance ordering. It is a partial ordering since not every
pair of copulas is comparable in this order. However many important parametric families
of copulas are totally ordered. We call a one-parameter family {C } positively ordered if
C1  C2 whenever 1 2. Examples of such one-parameter families will be given later.
2.4. Copulas and random variables
Let X1,...,Xn be random variables with continuous distribution functions F1,...,Fn,
and joint distribution function H . Then (X1,...,Xn)T has a unique copula C, where C is
given by (2.1). The standard copula representation of the distribution of the random vector
(X1,...,Xn)T then becomes:
H(x1,...,xn) = P{X1 x1,...,Xn xn} = C F1(x1),...,Fn(xn) .
Ch. 8: Modelling Dependence with Copulas 337
The transformations Xi  Fi(Xi) used in the above representation are usually referred
to as the probability-integral transformations (to uniformity) and form a standard tool in
simulation methodology. Since X1,...,Xn are independent if and only if H(x1,...,xn) =
F1(x1)Fn(xn) for all x1,...,xn in R, the following result follows from Theorem 2.2.
Theorem 2.5. Let (X1,...,Xn)T be a vector of continuous random variables with copula
C, then X1,...,Xn are independent if and only if C = n.
One nice property of copulas is that for strictly monotone transformations of the random
variables, copulas are either invariant, or change in certain simple ways. Note that if the
distribution function of a random variable X is continuous, and if  is a strictly monotone
function whose domain contains RanX, then the distribution function of the random variable
(X) is also continuous.
Theorem 2.6. Let (X1,...,Xn)T be a vector of continuous random variables with copula
C. If, for k = 1,...,n, k is strictly increasing on RanXk, then also (1(X1),...,n(Xn))T
has copula C.
Proof: Let F1,...,Fn denote the distribution functions of X1,...,Xn and let G1,...,Gn
denote the distribution functions of 1(X1),...,n(Xn), respectively. Let (X1,...,Xn)T
have copula C, and let (1(X1),...,n(Xn))T have copula C. Since k is strictly increas-
ing,
Gk(x) = P k(Xk) x = P Xk -1
k (x) = Fk -1
k (x)
for any x in R, hence
C G1(x1),...,Gn(xn) = P 1(X1) x1,...,n(Xn) xn
= P X1 -1
1 (x1),...,Xn -1
n (xn)
= C F1 -1
1 (x1) ,...,Fn -1
n (xn)
= C G1(x1),...,Gn(xn) .
Since X1,...,Xn are continuous, Ran G1 =  = RanGn = [0,1]. Hence it follows that
C = C on [0,1]n.
From Theorem 2.2 we know that the copula function C "separates" an n-dimensional
distribution function from its univariate marginals. The next theorem will show that there
is also a function, C, that separates an n-dimensional survival function from its univariate
survival marginals. Furthermore this function can be shown to be a copula, and this survival
copula can rather easily be expressed in terms of C and its k-dimensional marginals.
338 P. Embrechts et al.
Theorem 2.7. Let (X1,...,Xn)T be a vector of continuous random variables with copula
CX1,...,Xn . For i = 1,...,n, let i be strictly monotone on RanXi , and let (1(X1),...,
n(Xn))T have copula C1(X1),...,n(Xn). Furthermore let k be strictly decreasing for
some k. Without loss of generality let k = 1. Then
C1(X1),...,n(Xn)(u1,u2,...,un)
= C2(X2),...,n(Xn)(u2,...,un) - CX1,2(X2),...,n(Xn)(1 - u1,u2,...,un).
Proof: For i = 1,...,n, let Xi have distribution function Fi and let 1(Xi) have distribution
function Gi. Then
C1(X1),2(X2),...,n(Xn) G1(x1),...,Gn(xn)
= P 1(X1) x1,...,n(Xn) xn
= P X1 > -1
1 (x1),2(X2) x2,...,n(Xn) xn
= P 2(X2) x2,...,n(Xn) xn
- P X1 -1
1 (x1),2(X2) x2,...,n(Xn) xn
= C2(X2),...,n(Xn) G2(x2),...,Gn(xn)
- CX1,2(X2),...,n(Xn) F1 -1
1 (x1) ,G2(x2),...,Gn(xn)
= C2(X2),...,n(Xn) G2(x2),...,Gn(xn)
- CX1,2(X2),...,n(Xn) 1 - G1(x1),G2(x2),...,Gn(xn) ,
from which the conclusion follows directly.
By using the two theorems above recursively it is clear that the copula C1(X1),...,n(Xn)
can be expressed in terms of the copula CX1,...,Xn and its lower-dimensional marginals.
This is exemplified below.
Example 2.3. Consider the bivariate case. Let 1 be strictly decreasing and let 2 be
strictly increasing. Then
C1(X1),2(X2)(u1,u2) = u2 - CX1,2(X2)(1 - u1,u2)
= u2 - CX1,X2 (1 - u1,u2).
Let 1 and 2 be strictly decreasing. Then
C1(X1),2(X2)(u1,u2) = u2 - CX1,2(X2)(1 - u1,u2)
= u2 - 1 - u1 - CX1,X2 (1 - u1,1 - u2)
= u1 + u2 - 1 + CX1,X2 (1 - u1,1 - u2).
Ch. 8: Modelling Dependence with Copulas 339
Here C1(X1),2(X2) is the survival copula, C, of (X1,X2)T, i.e.,
H(x1,x2) = P{X1 > x1,X2 > x2} = C F1(x1),F2(x2) .
Note also that the joint survival function of n U(0,1) random variables whose joint
distribution function is the copula C is C(u1,...,un) = C(1 - u1,...,1 - un).
The mixed k-th order partial derivatives of a copula C, kC(u)/u1 uk, exist for
almost all u in [0,1]n. For such u, 0 kC(u)/u1 uk 1. For details, see Nelsen
(1999, p. 11). With this in mind, let
C(u1,...,un) = AC(u1,...,un) + SC(u1,...,un),
where
AC(u1,...,un) =
u1
0
...
un
0
n
s1 sn
C(s1,...,sn)ds1  dsn,
SC(u1,...,un) = C(u1,...,un) - AC(u1,...,un).
Unlike multivariate distributions in general, the marginals of a copula are continuous,
hence a copula has no individual points u in [0,1]n for which VC(u) > 0. If C =
AC on [0,1]n, then C is said to be absolutely continuous. In this case C has density
n
u1un
C(u1,...,un). If C = SC on [0,1]n, then C is said to be singular, and
n
u1un
C(u1,...,un) = 0 almost everywhere in [0,1]n. The support of a copula is the
complement of the union of all open subsets A of [0,1]n with VC(A) = 0. When C is singular
its support has Lebesgue measure zero and conversely. However a copula can have
full support without being absolutely continuous. Examples of such copulas are so-called
Marshall­Olkin copulas which are presented later.
Example 2.4. Consider the bivariate Fréchet­Hoeffding upper bound M given by
M(u,v) = min(u,v) on [0,1]2. It follows that 2
uv M(u,v) = 0 everywhere on [0,1]2 except
on the main diagonal (which has Lebesgue measure zero), and VM(B) = 0 for every
rectangle B in [0,1]2 entirely above or below the main diagonal. Hence M is singular.
One of the main aims of this chapter is to present effective algorithms for random variate
generation from the various copula families studied. The properties of the specific copula
family is often essential for the efficiency of the corresponding algorithm. We now present
a general algorithm for random variate generation from copulas. Note however that in most
cases it is not an efficient one to use.
Consider the general situation of random variate generation from the n-copula C. Let
Ck(u1,...,uk) = C(u1,...,uk,1,...,1), k = 2,...,n - 1,
340 P. Embrechts et al.
denote k-dimensional marginals of C, with
C1(u1) = u1 and Cn(u1,...,un) = C(u1,...,un).
Let U1,...,Un have joint distribution function C. Then the conditional distribution of Uk
given the values of U1,...,Uk-1, is given by
Ck(uk|u1,...,uk-1) = P{Uk uk|U1 = u1,...,Uk-1 = uk-1}
=
k-1Ck(u1,...,uk)
u1 uk-1
k-1Ck-1(u1,...,uk-1)
u1 uk-1
,
given that the numerator and denominator exist and that the denominator is not zero. The
following algorithm generates a random variate (u1,...,un)T from C. As usual, let U(0,1)
denote the uniform distribution on [0,1].
Algorithm 2.1.
* Simulate a random variate u1 from U(0,1).
* Simulate a random variate u2 from C2( | u1).
...
* Simulate a random variate un from Cn( | u1,...,un-1).
This algorithm is in fact a particular case of what is called "the standard construction". The
correctness of the algorithm can be seen from the fact that for independent U(0,1) random
variables Q1,...,Qn,
Q1,C-1
2 (Q2|Q1),...,C-1
n (Qn|Q1,C-1
2 (Q2|Q1),...)
T
has distribution function C. To simulate a value uk from Ck(|u1,...,uk-1) in general
means simulating q from U(0,1) from which uk can be obtained from the equation
q = Ck(uk|u1,...,uk-1) by numerical rootfinding. When C-1
k (q|u1,...,uk-1) has a
closed form (and hence there is no need for numerical rootfinding) this algorithm can be
recommended.
Example 2.5. Let the copula C be given by C(u,v) = (u- + v- - 1)-1/ , for  > 0.
Then
C2|1(v|u) =
C
u
(u,v) = -
1

u+
v-
1
-1/-1
-u--1
= u (-1-)/
u+
v-
1
-1/-1
= 1 + u
v-
1
(-1-)/
.
Ch. 8: Modelling Dependence with Copulas 341
Solving the equation q = C2|1(v|u) for v yields
C-1
2|1(q | u) = v = q-/(1+)
- 1 u+
1
-1/
.
The following algorithm generates a random variate (u,v)T from the above copula C.
* Simulate two independent random variates u and q from U(0,1).
* Set v = ((q-/(1+) - 1)u- + 1)-1/.
3. Dependence concepts
Copulas provide a natural way to study and measure dependence between random variables.
As a direct consequence of Theorem 2.6, copula properties are invariant under
strictly increasing transformations of the underlying random variables. Linear correlation
(or Pearson's correlation) is most frequently used in practice as a measure of dependence.
However, since linear correlation is not a copula-based measure of dependence, it can often
be quite misleading and should not be taken as the canonical dependence measure. Below
we recall the basic properties of linear correlation, and then continue with some copula
based measures of dependence.
3.1. Linear correlation
Definition 3.1. Let (X,Y)T be a vector of random variables with nonzero finite variances.
The linear correlation coefficient for (X,Y)T is
(X,Y) =
Cov(X,Y)

Var(X)

Var(Y)
, (3.1)
where Cov(X,Y) = E(XY) - E(X)E(Y) is the covariance of (X,Y)T, and Var(X) and
Var(Y) are the variances of X and Y.
Linear correlation is a measure of linear dependence. In the case of perfect linear dependence,
i.e., Y = aX + b almost surely for a  R \ {0}, b  R, we have |(X,Y)| = 1. More
important is that the converse also holds. Otherwise, -1 < (X,Y) < 1. Furthermore linear
correlation has the property that
(X + , Y + ) = sign( )(X,Y),
for ,  R \ {0}, ,  R. Hence linear correlation is invariant under strictly increasing
linear transformations. Linear correlation is easily manipulated under linear operations.
Let A, B be m × n matrices; a,b  Rm and let X, Y be random n-vectors. Then
Cov(AX + a,BY + b) = ACov(X,Y)BT
.
342 P. Embrechts et al.
From this it follows that for   Rn,
Var T
X = T
Cov(X),
where Cov(X) := Cov(X,X). Hence the variance of a linear combination is fully determined
by pairwise covariances between the components, a property which is crucial in
portfolio theory.
Linear correlation is a popular but also often misunderstood measure of dependence. The
popularity of linear correlation stems from the ease with which it can be calculated and it is
a natural scalar measure of dependence in elliptical distributions (with well known members
such as the multivariate normal and the multivariate t-distribution). However most
random variables are not jointly elliptically distributed, and using linear correlation as a
measure of dependence in such situations might prove very misleading. Even for jointly
elliptically distributed random variables there are situations where using linear correlation,
as defined by (3.1), does not make sense. We might choose to model some scenario using
heavy-tailed distributions such as t2-distributions. In such cases the linear correlation
coefficient is not even defined because of infinite second moments.
3.2. Perfect dependence
For every n-copula C we know from the Fréchet­Hoeffding inequality (Theorem 2.3) that
Wn
(u1,...,un) C(u1,...,un) Mn
(u1,...,un).
Furthermore, for n = 2 the upper and lower bounds are themselves copulas and we
have seen that W and M are the bivariate distributions functions of the random vectors
(U,1 - U)T and (U,U)T, respectively, where U  U(0,1) (i.e., U is uniformly distributed
on [0,1]). In this case we say that W describes perfect negative dependence and M
describes perfect positive dependence.
Theorem 3.1. Let (X,Y)T have one of the copulas W or M. Then there exist two
monotone functions , :R  R and a random variable Z so that
(X,Y) =d (Z),(Z) ,
with  increasing and  decreasing in the former case (W) and both  and  increasing
in the latter case (M). The converse of this result is also true.
For a proof, see Embrechts, McNeil and Straumann (2002). In a different form this result
was already in Fréchet (1951).
Definition 3.2. If (X,Y)T has the copula M then X and Y are said to be comonotonic; if
it has the copula W they are said to be countermonotonic.
Ch. 8: Modelling Dependence with Copulas 343
Note that if any of F and G (the distribution functions of X and Y, respectively) have
discontinuities, so that the copula is not unique, then W and M are possible copulas. In the
case of F and G being continuous, a stronger version of the result can be stated:
C = W  Y = T (X) a.s., T = G-1
 (1 - F) decreasing,
C = M  Y = T (X) a.s., T = G-1
 F increasing.
Other characterizations of comonotonicity can be found in Denneberg (1994).
3.3. Concordance
Let (x,y)T and (~x, ~y)T be two observations from a vector (X,Y)T of continuous random
variables. Then (x,y)T and (~x, ~y)T are said to be concordant if (x - ~x)(y - ~y) > 0, and
discordant if (x - ~x)(y - ~y) < 0.
The following theorem can be found in Nelsen (1999, p. 127). Many of the results in
this section are direct consequences of this theorem.
Theorem 3.2. Let (X,Y)T and (X,Y)T be independent vectors of continuous random
variables with joint distribution functions H and H, respectively, with common marginals
F (of X and X) and G (of Y and Y). Let C and C denote the copulas of (X,Y)T and
(X,Y)T, respectively, so that H(x,y) = C(F(x),G(y)) and H(x,y) = C(F(x),G(y)).
Let Q denote the difference between the probability of concordance and discordance of
(X,Y)T and (X,Y)T, i.e., let
Q = P X - X Y - Y > 0 - P X - X Y - Y < 0 .
Then
Q = Q C,C = 4
[0,1]2
C(u,v)dC(u,v) - 1.
Proof: Since the random variables are all continuous,
P X - X Y - Y < 0 = 1 - P X - X Y - Y > 0
and hence Q = 2P{(X - X)(Y - Y) > 0} - 1. But
P X - X Y - Y > 0 = P X > X,Y > Y + P X < X,Y < Y ,
and these probabilities can be evaluated by integrating over the distribution of one of the
vectors (X,Y)T or (X,Y)T. Hence
P X > X,Y > Y = P X < X,Y < Y
344 P. Embrechts et al.
=
R2
P X < x,Y < y dC F(x),G(y)
=
R2
C F(x),G(y) dC F(x),G(y) .
Employing the probability-integral transforms u = F(x) and v = G(y) then yields
P X > X,Y > Y =
[0,1]2
C(u,v)dC(u,v).
Similarly,
P X < X,Y < Y =
R2
P X > x,Y > y dC F(x),G(y)
=
R2
1 - F(x) - G(y) + C F(x),G(y) dC F(x),G(y)
=
[0,1]2
1 - u - v + C(u,v) dC(u,v).
But since C is the joint distribution function of a vector (U,V )T of U(0,1) random variables,
E(U) = E(V ) = 1/2, and hence
P X < X,Y < Y = 1 -
1
2
-
1
2
+
[0,1]2
C(u,v)dC(u,v)
=
[0,1]2
C(u,v)dC(u,v).
Thus
P X - X Y - Y > 0 = 2
[0,1]2
C(u,v)dC(u,v),
and the conclusion follows.
Corollary 3.1. Let C, C, and Q be as given in Theorem 3.2. Then
(1) Q is symmetric in its arguments: Q(C,C) = Q(C,C).
(2) Q is nondecreasing in each argument: if C  C , then Q(C,C) Q(C ,C).
(3) Copulas can be replaced by survival copulas in Q, i.e., Q(C,C) = Q(C,C).
Ch. 8: Modelling Dependence with Copulas 345
The following definition can be found in Scarsini (1984).
Definition 3.3. A real valued measure  of dependence between two continuous random
variables X and Y whose copula is C is a measure of concordance if it satisfies the following
properties:
(1)  is defined for every pair X,Y of continuous random variables.
(2) -1 X,Y 1, X,X = 1 and X,-X = -1.
(3) X,Y = Y,X.
(4) If X and Y are independent, then X,Y =  = 0.
(5) -X,Y = X,-Y = -X,Y .
(6) If C and C are copulas such that C  C, then C C.
(7) If {(Xn,Yn)} is a sequence of continuous random variables with copulas Cn, and if
{Cn} converges pointwise to C, then limn Cn = C.
Let  be a measure of concordance for continuous random variables X and Y. As a consequence
of Definition 3.3, if If Y is almost surely an increasing function of X, then X,Y =
M = 1, and if Y is almost surely a decreasing function of X, then X,Y = W = -1.
Moreover, if  and  are almost surely strictly increasing functions on RanX and Ran Y
respectively, then (X),(Y) = X,Y .
3.4. Kendalĺs tau and Spearman's rho
In this section we discuss two important measures of dependence (concordance) known as
Kendalĺs tau and Spearman's rho. They provide the perhaps best alternatives to the linear
correlation coefficient as a measure of dependence for nonelliptical distributions, for which
the linear correlation coefficient is inappropriate and often misleading. For more details
about Kendalĺs tau and Spearman's rho and their estimators (sample versions) we refer to
Kendall and Stuart (1979), Kruskal (1958), Lehmann (1975), Capéra and Genest (1993).
For other interesting scalar measures of dependence see Schweizer and Wolff (1981).
Definition 3.4. Kendalĺs tau for the random vector (X,Y)T is defined as
(X,Y) = P X - X Y - Y > 0 - P X - X Y - Y < 0 ,
where (X,Y)T is an independent copy of (X,Y)T.
Hence Kendalĺs tau for (X,Y)T is simply the probability of concordance minus the
probability of discordance.
Theorem 3.3. Let (X,Y)T be a vector of continuous random variables with copula C.
Then Kendalĺs tau for (X,Y)T is given by
(X,Y) = Q(C,C) = 4
[0,1]2
C(u,v)dC(u,v) - 1.
346 P. Embrechts et al.
Note that the integral above is the expected value of the random variable C(U,V ), where
U,V  U(0,1) with joint distribution function C, i.e., (X,Y) = 4E(C(U,V )) - 1.
Definition 3.5. Spearman's rho for the random vector (X,Y)T is defined as
S(X,Y) = 3 P X - X Y - Y > 0 - P X - X Y - Y < 0 ,
where (X,Y)T, (X,Y)T and (X ,Y )T are independent copies.
Note that X and Y are independent. Using Theorem 3.2 and the first part of Corollary 3.1
we obtain the following result.
Theorem 3.4. Let (X,Y)T be a vector of continuous random variables with copula C.
Then Spearman's rho for (X,Y)T is given by
S(X,Y) = 3Q(C,) = 12
[0,1]2
uv dC(u,v) - 3
= 12
[0,1]2
C(u,v)dudv - 3.
Hence, if X  F and Y  G, and we let U = F(X) and V = G(Y), then
S(X,Y) = 12
[0,1]2
uv dC(u,v) - 3 = 12E(UV ) - 3
=
E(UV ) - 1/4
1/12
=
Cov(U,V )

Var(U)

Var(V )
=  F(X),G(Y) .
In the next theorem we will see that Kendalĺs tau and Spearman's rho are concordance
measures according to Definition 3.3.
Theorem 3.5. If X and Y are continuous random variables whose copula is C, then
Kendalĺs tau and Spearman's rho satisfy the properties in Definition 3.3 for a measure
of concordance.
For a proof, see Nelsen (1999, p. 137).
Example 3.1. Kendalĺs tau and Spearman's rho for the random vector (X,Y)T are invariant
under strictly increasing componentwise transformations. This property does not hold
for linear correlation. It is not difficult to construct examples, the following construction
is instructive in its own right. Let X and Y be standard exponential random variables with
Ch. 8: Modelling Dependence with Copulas 347
copula C, where C is a member of the Farlie­Gumbel­Morgenstern family, i.e., C is given
by
C(u,v) = uv + uv(1 - u)(1 - v),
for some  in [-1,1]. The joint distribution function H of X and Y is given by
H(x,y) = C 1 - e-x
,1 - e-y
.
Let  denote the linear correlation coefficient. Then
(X,Y) =
E(XY) - E(X)E(Y)

Var(X)

Var(Y)
= E(XY) - 1,
where
E(XY) =

0

0
xy dH(x,y)
=

0

0
xy (1 + )e-x-y
- 2 e-2x-y
- 2 e-x-2y
+ 4 e-2x-2y
dx dy
= 1 +

4
.
Hence (X,Y) = /4. But
 1 - e-X
,1 - e-Y
= S(X,Y) = 12
[0,1]2
C(u,v)dudv - 3
= 12
[0,1]2
(uv + uv(1 - u)(1 - v))dudv - 3
= 12
1
4
+

36
- 3 =

3
.
Hence (X,Y) is not invariant under strictly increasing transformations of X and Y and
therefore linear correlation is not a measure of concordance.
Although the properties listed under Definition 3.3 are useful, there are some additional
properties that would make a measure of concordance even more useful. Recall that for a
random vector (X,Y)T with copula C,
C = M  C = C = 1,
C = W  C = C = -1.
The following theorem states that the converse is also true.
348 P. Embrechts et al.
Theorem 3.6. Let X and Y be continuous random variables with copula C, and let 
denote Kendalĺs tau or Spearman's rho. Then the following are true:
(1) (X,Y) = 1  C = M.
(2) (X,Y) = -1  C = W.
For a proof, see Embrechts, McNeil and Straumann (2002).
From the definitions of Kendalĺs tau and Spearman's rho it follows that both are increasing
functions of the value of the copula under consideration. Thus they are increasing
with respect to the concordance ordering given in Definition 2.5. Moreover, for continuous
random variables all values in the interval [-1,1] can be obtained for Kendalĺs tau or
Spearman's rho by a suitable choice of the underlying copula. This is however not the case
with linear correlation as is shown in the following example from Embrechts, McNeil and
Straumann (2002).
Example 3.2. Let X  LN(0,1) (Lognormal) and Y  LN(0,2), > 0. Then min =
(eZ,e-Z) and max = (eZ,eZ), where Z  N(0,1). min and max can be calculated,
yielding:
min =
e- - 1

e - 1 e2
- 1
, max =
e - 1

e - 1 e2
- 1
,
from which follows that lim min = lim max = 0. Hence the linear correlation
coefficient can be almost zero, even if X and Y are comonotonic or countermonotonic.
Kendalĺs tau and Spearman's rho are measures of dependence between two random
variables. However the extension to higher dimensions is obvious, we simply write pairwise
correlations in an n × n matrix in the same way as is done for linear correlation.
3.5. Tail dependence
The concept of tail dependence relates to the amount of dependence in the upper-rightquadrant
tail or lower-left-quadrant tail of a bivariate distribution. It is a concept that is
relevant for the study of dependence between extreme values. It turns out that tail dependence
between two continuous random variables X and Y is a copula property and hence
the amount of tail dependence is invariant under strictly increasing transformations of X
and Y.
Definition 3.6. Let (X,Y)T be a vector of continuous random variables with marginal
distribution functions F and G. The coefficient of upper tail dependence of (X,Y)T is
lim
u 1
P Y > G-1
(u)|X > F-1
(u) = U
Ch. 8: Modelling Dependence with Copulas 349
provided that the limit U  [0,1] exists. If U  (0,1], X and Y are said to be asymptotically
dependent in the upper tail; if U = 0, X and Y are said to be asymptotically
independent in the upper tail.
Since P{Y > G-1(u) | X > F-1(u)} can be written as
1 - P{X F-1(u)} - P{Y G-1(u)} + P{X F-1(u),Y G-1(u)}
1 - P{X F-1(u)}
,
an alternative and equivalent definition (for continuous random variables), from which it is
seen that the concept of tail dependence is indeed a copula property, is the following which
can be found in Joe (1997, p. 33).
Definition 3.7. If a bivariate copula C is such that
lim
u 1
1 - 2u + C(u,u)
1 - u
= U
exists, then C has upper tail dependence if U  (0,1], and upper tail independence if
U = 0.
Example 3.3. Consider the bivariate Gumbel family of copulas given by
C (u,v) = exp - (-lnu)
+ (-lnv) 1/
,
for  1. Then
1 - 2u + C(u,u)
1 - u
=
1 - 2u + exp(21/ lnu)
1 - u
=
1 - 2u + u21/
1 - u
,
and hence
lim
u 1
1 - 2u + C(u,u)
1 - u
= 2 - lim
u 1
21/
u21/ -1
= 2 - 21/
.
Thus for  > 1, C has upper tail dependence.
For copulas without a simple closed form an alternative formula for U is more useful.
An example is given in the case of the Gaussian copula
CR(u,v) =
-1(u)
-
-1(v)
-
1
2 1 - R2
12
exp s2
- 2R12st + t2
2(1 - R2
12)
ds dt,
350 P. Embrechts et al.
where -1 < R12 < 1 and  is the univariate standard normal distribution function.
Consider a pair of U(0,1) random variables (U,V ) with copula C. First note that
P{V v|U = u} = C(u,v)/u and P{V > v|U = u} = 1 - C(u,v)/u, and similarly
when conditioning on V . Then
U = lim
u 1
C(u,u)
1 - u
= - lim
u 1
dC(u,u)
du
= - lim
u 1
-2 +

s
C(s,t)
s=t=u
+

t
C(s,t)
s=t=u
= lim
u 1
P V > u|U = u + P{U > u|V = u} .
Furthermore, if C is an exchangeable copula, i.e., C(u,v) = C(v,u), then the expression
for U simplifies to
U = 2 lim
u 1
P{V > u|U = u}.
Example 3.4. Let (X,Y)T have the bivariate standard normal distribution function with
linear correlation coefficient . That is (X,Y)T  C((x),(y)), where C is a member of
the Gaussian family given above with R12 = . Since copulas in this family are exchange-
able,
U = 2 lim
u 1
P{V > u|U = u},
and because  is a distribution function with infinite right endpoint,
lim
u 1
P{V > u|U = u} = lim
x
P -1
(V ) > x|-1
(U) = x
= lim
x
P{X > x|Y = x}.
Using the well known fact that Y|X = x  N(x,1 - 2) we obtain
U = 2 lim
x

x - x
1 - 2
= 2 lim
x

x

1 - 

1 + 
,
from which it follows that U = 0 for R12 < 1. Hence the Gaussian copula C with  < 1
does not have upper tail dependence.
The concept of lower tail dependence can be defined in a similar way. If the limit
limu 0 C(u,u)/u = L exists, then C has lower tail dependence if L  (0,1], and lower
Ch. 8: Modelling Dependence with Copulas 351
tail independence if L = 0. For copulas without a simple closed form an alternative formula
for L is more useful. Consider a random vector (U,V )T with copula C. Then
L = lim
u 0
C(u,u)
u
= lim
u 0
dC(u,u)
du
= lim
u 0

s
C(s,t)
s=t=u
+

t
C(s,t)
s=t=u
= lim
u 0
P{V < u|U = u} + P{U < u|V = u} .
Furthermore if C is an exchangeable copula, i.e., C(u,v) = C(v,u), then the expression
for L simplifies to
L = 2 lim
u 0
P{V < u|U = u}.
Recall that the survival copula of two random variables with copula C is given by
C(u,v) = u + v - 1 + C(1 - u,1 - v),
and the joint survival function for two U(0,1) random variables whose joint distribution
function is C is given by
C(u,v) = 1 - u - v + C(u,v) = C(1 - u,1 - v).
Hence it follows that
lim
u 1
C(u,u)
1 - u
= lim
u 1
C(1 - u,1 - u)
1 - u
= lim
u 0
C(u,u)
u
,
so the coefficient of upper tail dependence of C is the coefficient of lower tail dependence
of C. Similarly the coefficient of lower tail dependence of C is the coefficient of upper tail
dependence of C.
4. Marshall­Olkin copulas
In this section we discuss a class of copulas called Marshall­Olkin copulas. To be able to
derive these copulas and present explicit expressions for rank correlation and tail dependence
coefficients without tedious calculations, we begin with bivariate Marshall­Olkin
copulas. We then continue with the general n-dimensional case and suggest applications
of Marshall­Olkin copulas to the modelling of dependent risks. For further details about
Marshall­Olkin distributions we refer to Marshall and Olkin (1967). Similar ideas are contained
in Muliere and Scarsini (1987).
352 P. Embrechts et al.
4.1. Bivariate Marshall­Olkin copulas
Consider a two-component system where the components are subject to shocks, which
are fatal to one or both components. Let X1 and X2 denote the lifetimes of the two components.
Furthermore assume that the shocks follow three independent Poisson processes
with parameters 1,2,12 0, where the index indicates whether the shocks effect only
component 1, only component 2 or both. Then the times Z1, Z2 and Z12 of occurrence of
these shocks are independent exponential random variables with parameters 1, 2 and 12
respectively. Hence
H(x1,x2) = P{X1 > x1,X2 > x2}
= P{Z1 > x1}P{Z2 > x2}P Z12 > max(x1,x2) .
The univariate survival functions for X1 and X2 are F1(x1) = exp(-(1 + 12)x1) and
F2(x2) = exp(-(2 + 12)x2). Furthermore, since max(x1,x2) = x1 + x2 - min(x1,x2),
H(x1,x2) = exp -(1 + 12)x1 - (2 + 12)x2 + 12 min(x1,x2)
= F 1(x1)F2(x2)min exp(12x1),exp(12x2) .
Let 1 = 12/(1 + 12) and 2 = 12/(2 + 12). Then exp(12x1) = F1(x1)-1 and
exp(12x2) = F2(x2)-2 , and hence the survival copula of (X1,X2)T is given by
C(u1,u2) = u1u2 min u-1
1 ,u-2
2 = min u1-1
1 u2,u1u1-2
2 .
This construction leads to a copula family given by
C1,2 (u1,u2) = min u1-1
1 u2,u1u1-2
2 =
u1-1
1 u2, u1
1 u2
2 ,
u1u1-2
2 , u1
1 u2
2 .
This family is known as the Marshall­Olkin family. Marshall­Olkin copulas have both an
absolutely continuous and a singular component. Since
2
u1u2
C1,2 (u1,u2) =
u
-1
1 , u
1
1 > u
2
2 ,
u-2
2 , u1
1 < u2
2 ,
the mass of the singular component is concentrated on the curve u1
1 = u2
2 in [0,1]2 as
seen in Figure 1.
Kendalĺs tau and Spearman's rho are quite easily evaluated for this copula family. For
Spearman's rho, applying Theorem 3.4 yields:
S(C1,2 ) = 12
[0,1]2
C1,2 (u,v)dudv - 3
Ch. 8: Modelling Dependence with Copulas 353
Fig. 1. A simulation from the Marshall­Olkin copula with 1 = 1.1, 2 = 0.2 and 12 = 0.6.
= 12
1
0
u1/2
0
u1-1 v dv +
1
u1/2
uv1-2 dv du - 3
=
312
21 + 22 - 12
.
To evaluate Kendalĺs tau we use the following theorem, a proof of which is found in Nelsen
(1999, p. 131).
Theorem 4.1. Let C be a copula such that the product (C/u)(C/v) is integrable on
[0,1]2. Then
[0,1]2
C(u,v)dC(u,v) =
1
2
-
[0,1]2

u
C(u,v)

u
C(u,v)dudv.
Using Theorems 3.3 and 4.1 we obtain
(C1,2 ) = 4
[0,1]2
C1,2 (u,v)dC1,2 (u,v) - 1
= 4
1
2
-
[0,1]2

u
C1,2 (u,v)

u
C1,2 (u,v)dudv - 1
=
12
1 + 2 - 12
.
354 P. Embrechts et al.
Thus, all values in the interval [0,1] can be obtained for S(C1,2 ) and (C1,2 ). The
Marshall­Olkin copulas have upper tail dependence. Without loss of generality assume
that 1 > 2, then
lim
u 1
C(u,u)
1 - u
= lim
u 1
1 - 2u + u2 min(u-1 ,u-2 )
1 - u
= lim
u 1
1 - 2u + u2u-2
1 - u
= lim
u 1
2 - 2u1-2 + 2u1-2 = 2,
and hence U = min(1,2) is the coefficient of upper tail dependence.
4.2. A multivariate extension
We now present the natural multivariate extension of the bivariate Marshall­Olkin family.
Consider an n-component system, where each nonempty subset of components is assigned
a shock which is fatal to all components of that subset. Let S denote the set of nonempty
subsets of {1,...,n}. Let X1,...,Xn denote the lifetimes of the components, and assume
that shocks assigned to different subsets s, s  S, follow independent Poisson processes
with intensities s. Let Zs, s  S, denote the time of first occurrence of a shock event
for the shock process assigned to subset s. Then the occurrence times Zs are independent
exponential random variables with parameters s, and Xj = mins:js Zs for j = 1,...,n.
There are in total 2n - 1 shock processes, each in one-to-one correspondence with a
nonempty subset of {1,...,n}.
Example 4.1. Let n = 4. Then
X1 = min(Z1,Z12,Z13,Z14,Z123,Z124,Z134,Z1234),
X2 = min(Z2,Z12,Z23,Z24,Z123,Z124,Z234,Z1234),
X3 = min(Z3,Z13,Z23,Z34,Z123,Z134,Z234,Z1234),
X4 = min(Z4,Z14,Z24,Z34,Z124,Z134,Z234,Z1234).
If for example 13 = 0, then Z13 =  almost surely.
We now turn to the question of random variate generation from Marshall­Olkin
n-copulas. Order the l := |S| = 2n - 1 nonempty subsets of {1,...,n} in some arbitrary
way, s1,...,sl, and set k := sk (the parameter of Zsk ) for k = 1,...,l. The following
algorithm generates random variates from the Marshall­Olkin n-copula.
Ch. 8: Modelling Dependence with Copulas 355
Algorithm 4.1.
* Simulate l independent random variates v1,...,vl from U(0,1).
* Set xi = min1 k l,isk,k=0(-lnvk/k), i = 1,...,n.
* Set i = l
k=1 1{i  sk}k, i = 1,...,n.
* Set ui = exp(-ixi), i = 1,...,n.
Then (x1,...,xn)T is an n-variate from the n-dimensional Marshall­Olkin distribution
and (u1,...,un)T is an n-variate from the corresponding Marshall­Olkin n-copula. Furthermore,
i is the shock intensity "felt" by component i.
Since the (i,j)-bivariate marginal of a Marshall­Olkin n-copula is a Marshall­Olkin
copula with parameters
i =
s: is,js
s
s: is
s and j =
s: is,js
s
s: js
s ,
the Kendalĺs tau and Spearman's rho rank correlation matrices are easily evaluated. The
(i,j) entries are given by
ij
i + j - ij
and
3ij
2i + 2j - ij
,
respectively. As seen above, evaluating the rank correlation matrix given the full parameterization
of the Marshall­Olkin n-copula is straightforward. However given a (Kendalĺs
tau or Spearman's rho) rank correlation matrix we cannot in general obtain a unique parameterization
of the copula. By setting the shock intensities for subgroups with more then
two elements to zero, we obtain the perhaps most natural parameterization of the copula in
this situation. However this also means that the copula only has bivariate dependence.
4.3. A useful modelling framework
In general the huge number of parameters for high-dimensional Marshall­Olkin copulas
make them unattractive for high-dimensional risk modelling. However, we now give an
example of how an intuitively appealing and easier parameterized model for modelling dependent
loss frequencies can be set up, for which the survival copula of times to first losses
is a Marshall­Olkin copula.
Suppose we are interested in insurance losses occurring in several different lines of business
or several different countries. In credit-risk modelling we might be interested in losses
related to the default of various different counterparties or types of counterparty. A natural
approach to modelling this dependence is to assume that all losses can be related to a series
of underlying and independent shock processes. In insurance these shocks might be natural
catastrophes; in credit-risk modelling they might be a variety of underlying economic
356 P. Embrechts et al.
events. When a shock occurs this may cause losses of several different types; the common
shock causes the numbers of losses of each type to be dependent. It is commonly assumed
that the different varieties of shocks arrive as independent Poisson processes, in which case
the counting processes of the losses are also Poisson and can be handled easily analytically.
In reliability such models are known as fatal shock models, when the shock always destroys
the component, and nonfatal shock models, when components have a chance of surviving
the shock. A good basic reference on such models is Barlow and Proschan (1975).
Suppose there are m different types of shocks and for e = 1,...,m, let {N(e)(t), t 0}
be a Poisson process with intensity (e) recording the number of events of type e occurring
in (0,t]. Assume further that these shock counting processes are independent. Consider
losses of n different types and for j = 1,...,n, let {Nj (t), t 0} be a counting process
that records the frequency of losses of the jth type occurring in (0,t]. At the rth occurrence
of an event of type e the Bernoulli variable I
(e)
j,r indicates whether a loss of type j
occurs. The vectors
I(e)
r = I
(e)
1,r ,...,I(e)
n,r
T
for r = 1,...,N(e)(t) are considered to be independent and identically distributed with a
multivariate Bernoulli distribution. In other words, each new event represents a new independent
opportunity to incur a loss but, for a fixed event, the loss trigger variables for
losses of different types may be dependent. The form of the dependence depends on the
specification of the multivariate Bernoulli distribution with independence as a special case.
We use the following notation for p-dimensional marginal probabilities of this distribution
(the subscript r is dropped for simplicity):
P I
(e)
j1
= ij1 ,...,I
(e)
jp
= ijp = p
(e)
j1,...,jp
(ij1 ,...,ijp ), ij1 ,...,ijp  {0,1}.
We also write p(e)
j (1) = p(e)
j for one-dimensional marginal probabilities, so that in the
special case of conditional independence we have p(e)
j1,...,jp
(1,...,1) =
p
k=1 p(e)
jk
. The
counting processes for events and losses are thus linked by
Nj (t) =
m
e=1
N(e)(t)
r=1
I
(e)
j,r .
Under the Poisson assumption for the event processes and the Bernoulli assumption for
the loss indicators, the loss processes {Nj (t), t 0} are clearly Poisson themselves, since
they are obtained by superpositioning m independent (possibly thinned) Poisson processes
generated by the m underlying event processes. The random vector (N1(t),...,Nn(t))T
can be thought of as having a multivariate Poisson distribution.
The presented nonfatal shock model has an equivalent fatal shock model representation,
i.e., of the type presented in Section 4.2. Hence the random vector (X1,...,Xn)T of times
Ch. 8: Modelling Dependence with Copulas 357
to first losses of different types, where Xj = inf{t 0 | Nj (t) > 0}, has an n-dimensional
Marshall­Olkin distribution whose survival copula is a Marshall­Olkin n-copula. From
this it follows that Kendalĺs tau, Spearman's rho and coefficients of tail dependence for
(Xi,Xj )T can be easily calculated. For more details on this model, see Lindskog and McNeil
(2001).
5. Elliptical copulas
The class of elliptical distributions provides a rich source of multivariate distributions
which share many of the tractable properties of the multivariate normal distribution and
enables modelling of multivariate extremes and other forms of nonnormal dependences.
Elliptical copulas are simply the copulas of elliptical distributions. Simulation from elliptical
distributions is easy, and as a consequence of Sklar's Theorem so is simulation from
elliptical copulas. Furthermore, we will show that rank correlation and tail dependence coefficients
can be easily calculated. For further details on elliptical distributions we refer to
Fang, Kotz and Ng (1987) and Cambanis, Huang and Simons (1981).
5.1. Elliptical distributions
Definition 5.1. If X is a n-dimensional random vector and, for some   Rn and some
n × n nonnegative definite, symmetric matrix , the characteristic function X-(t) of
X -  is a function of the quadratic form tTt, X-(t) = (tTt), we say that X has an
elliptical distribution with parameters ,  and , and we write X  En(,,).
When n = 1, the class of elliptical distributions coincides with the class of onedimensional
symmetric distributions. A function  as in Definition 5.1 is called a characteristic
generator.
Theorem 5.1. X  En(,,) with rank() = k if and only if there exist a random
variable R 0 independent of U, a k-dimensional random vector uniformly distributed on
the unit hypersphere {z  Rk | zTz = 1}, and an n × k matrix A with AAT = , such that
X =d  + RAU.
For the proof of Theorem 5.1 and the relation between R and  see Fang, Kotz and Ng
(1987) or Cambanis, Huang and Simons (1981).
Example 5.1. Let X  Nn(0,In). Since the components Xi  N(0,1), i = 1,...,n, are
independent and the characteristic function of Xi is exp(-t2
i /2), the characteristic function
of X is
exp -
1
2
t2
1 +  + t2
n = exp -
1
2
tT
t .
From Theorem 5.1 it then follows that X  En(0,In,), where (u) = exp(-u/2).
358 P. Embrechts et al.
If X  En(,,), where  is a diagonal matrix, then X has uncorrelated components
(if 0 < Var(Xi) < ). If X has independent components, then X  Nn(,). Note
that the multivariate normal distribution is the only one among the elliptical distributions
where uncorrelated components imply independent components. A random vector
X  En(,,) does not necessarily have a density. If X has a density it must be of
the form ||-1/2g((X - )T-1(X - )) for some nonnegative function g of one scalar
variable. Hence the contours of equal density form ellipsoids in Rn. Given the distribution
of X, the representation En(,,) is not unique. It uniquely determines  but  and
 are only determined up to a positive constant. More precisely, if X  En(,,) and
X  En(,,), then

= , 
= c, 
() = 

c
,
for some constant c > 0.
In order to find a representation such that Cov(X) = , we use Theorem 5.1 to obtain
Cov(X) = Cov( + RAU) = AE R2
Cov(U)AT
,
provided that E(R2) < . Let Y  Nn(0,In). Then Y =d Y U, where Y is independent
of U. Furthermore Y 2  2
n , so E( Y 2) = n. Since Cov(Y) = In we see that
if U is uniformly distributed on the unit hypersphere in Rn, then Cov(U) = In/n. Thus
Cov(X) = AATE(R2)/n. By choosing the characteristic generator (s) = (s/c), where
c = E(R2)/n, we get Cov(X) = . Hence an elliptical distribution is fully described by
,  and , where  can be chosen so that Cov(X) =  (if Cov(X) is defined). If Cov(X)
is obtained as above, then the distribution of X is uniquely determined by E(X), Cov(X)
and the type of its univariate marginals, e.g., normal or t4, say.
Theorem 5.2. Let X  En(,,), let B be a q × n matrix and b  Rq. Then
b + BX  Eq b + B,BBT
, .
Proof: By Theorem 5.1, b + BX has the stochastic representation
b + BX =d b + B + RBAU.
Partition X,  and  into
X =
X1
X2
,  =
1
2
,  =
11 12
21 22
,
where X1 and 1 are r × 1 vectors and 11 is a r × r matrix.
Ch. 8: Modelling Dependence with Copulas 359
Corollary 5.1. Let X  En(,,). Then
X1  Er(1,11,), X2  En-r (2,22,).
Hence marginal distributions of elliptical distributions are elliptical and of the same type
(with the same characteristic generator). The next result states that the conditional distribution
of X1 given the value of X2 is also elliptical, but in general not of the same type as X1.
Theorem 5.3. Let X  En(,,) with  strictly positive definite. Then
X1|X2 = x  Er ~,, ~ ,
where ~ = 1 + 12-1
22 (x - 2) and  = 11 - 12-1
22 21. Moreover, ~ =  if and
only if X  Nn(,).
For the proof and details about ~, see Fang, Kotz and Ng (1987). For the extension to the
case where rank() < n, see Cambanis, Huang and Simons (1981).
The following lemma states that linear combinations of independent, elliptically distributed
random vectors with the same dispersion matrix  (up to a positive constant) remain
elliptical.
Lemma 5.1. Let X  En(,,) and X  En( ~,c, ~) for c > 0 be independent. Then
for a,b  R, aX + bX  En(a + b ~,,) with (u) = (a2u) ~(b2cu).
Proof: By Definition 5.1, it is sufficient to show that for all t  Rn
aX+bX-a-b ~(t) = a(X-)(t)b(X- ~)(t)
=  (at)T
(at) ~ (bt)T
(c)(bt)
=  a2
tT
t ~ b2
ctT
t .
As usual, let X  En(,,). Whenever 0 < Var(Xi),Var(Xj ) < ,
(Xi,Xj ) :=
Cov(Xi,Xj )
Var(Xi)Var(Xj )
=
ij
iijj
.
This explains why linear correlation is a natural measure of dependence between random
variables with a joint nondegenerate (ii > 0 for all i) elliptical distribution. Throughout
this section we call the matrix R, with Rij = ij / iijj , the linear correlation matrix
of X. Note that this definition is more general than the usual one and in this situation
(elliptical distributions) makes more sense. Since an elliptical distribution is uniquely determined
by ,  and , the copula of a nondegenerate elliptically distributed random
vector is uniquely determined by R and .
360 P. Embrechts et al.
One practical problem with elliptical distributions in multivariate risk modelling is that
all marginals are of the same type. To construct a realistic multivariate distribution for some
given risks, it may be reasonable to choose a copula of an elliptical distribution but different
types of marginals (not necessarily elliptical). One big drawback with such a model seems
to be that the copula parameter R can no longer be estimated directly from data. Recall
that for nondegenerate elliptical distributions with finite variances, R is just the usual linear
correlation matrix. In such cases, R can be estimated using (robust) linear correlation
estimators. One such robust estimator is provided by the next theorem. For nondegenerate
nonelliptical distributions with finite variances and elliptical copulas, R does not correspond
to the linear correlation matrix. However, since the Kendalĺs tau rank correlation
matrix for a random vector is invariant under strictly increasing transformations of the vector
components, and the next theorem provides a relation between the Kendalĺs tau rank
correlation matrix and R for nondegenerate elliptical distributions, R can in fact easily be
estimated from data.
Theorem 5.4. Let X  En(,,) with P{Xi = i} < 1 and P{Xj = j } < 1. Then
(Xi,Xj ) = 1 -
xR
P{Xi = x}
2 2

arcsin(Rij ), (5.1)
where the sum extends over all atoms of the distribution of Xi. If rank() 2, then (5.1)
simplifies to
(Xi,Xj ) = 1 - P{Xi = i}
2 2

arcsin(Rij ).
For a proof, see Lindskog, McNeil and Schmock (2001). Note that if P{Xi = i} = 0 for
all i, which is true for, e.g., multivariate t-distribution or normal distributions with strictly
positive definite dispersion matrices , then
(Xi,Xj ) =
2

arcsin(Rij )
for all i and j.
The nonparametric estimator of R, sin(/2) (dropping the subscript for simplicity),
provided by the above theorem, inherits the robustness properties of the Kendalĺs tau estimator
and is an efficient (low variance) estimator of R for both elliptical distributions and
nonelliptical distributions with elliptical copulas.
5.2. Gaussian copulas
The copula of the n-variate normal distribution with linear correlation matrix R is
CGa
R (u) = n
R -1
(u1),...,-1
(un) ,
Ch. 8: Modelling Dependence with Copulas 361
where n
R denotes the joint distribution function of the n-variate standard normal distribution
function with linear correlation matrix R, and -1 denotes the inverse of the distribution
function of the univariate standard normal distribution. Copulas of the above form
are called Gaussian copulas. In the bivariate case the copula expression can be written as
CGa
R (u,v) =
-1(u)
-
-1(v)
-
1
2 1 - R2
12
exp s2
- 2R12st + t2
2(1 - R2
12)
ds dt.
Note that R12 is simply the usual linear correlation coefficient of the corresponding bivariate
normal distribution. Example 3.4 shows that Gaussian copulas do not have upper
tail dependence. Since elliptical distributions are radially symmetric, the coefficient of upper
and lower tail dependence are equal. Hence Gaussian copulas do not have lower tail
dependence.
We now address the question of random variate generation from the Gaussian copula
CGa
R . For our purpose, it is sufficient to consider only strictly positive definite matrices R.
Write R = AAT for some n × n matrix A, and if Z1,...,Zn  N(0,1) are independent,
then
 + AZ  Nn(,R).
One natural choice of A is the Cholesky decomposition of R. The Cholesky decomposition
of R is the unique lower-triangular matrix L with LLT = R. Furthermore Cholesky
decomposition routines are implemented in most mathematical software. This provides an
easy algorithm for random variate generation from the Gaussian n-copula CGa
R .
Algorithm 5.1.
* Find the Cholesky decomposition A of R.
* Simulate n independent random variates z1,...,zn from N(0,1).
* Set x = Az.
* Set ui = (xi), i = 1,...,n.
* (u1,...,un)T  CGa
R .
As usual  denotes the univariate standard normal distribution function.
5.3. t-copulas
If X has the stochastic representation
X =d  +



S
Z, (5.2)
where   Rn, S  2
 and Z  Nn(0,) are independent, then X has an n-variate tdistribution
with mean  (for  > 1) and covariance matrix 
-2  (for  > 2). If  2
362 P. Embrechts et al.
then Cov(X) is not defined. In this case we just interpret  as being the shape parameter
of the distribution of X.
The copula of X given by (5.2) can be written as
Ct
,R(u) = tn
,R t-1
 (u1),...,t-1
 (un) ,
where Rij = ij / iijj for i,j  {1,...,n} and where tn
,R denotes the distribution
function of

Y/

S, where S  2
 and Y  Nn(0,R) are independent. Here t denotes
the (equal) marginals of tn
,R, i.e., the distribution function of

Y1/

S. In the bivariate
case the copula expression can be written as
Ct
,R(u,v) =
t-1
 (u)
-
t-1
 (v)
-
1
2 1 - R2
12
1 +
s2 - 2R12st + t2
(1 - R2
12)
-(+2)/2
ds dt.
Note that R12 is simply the usual linear correlation coefficient of the corresponding bivariate
t-distribution if  > 2.
If (X1,X2)T has a standard bivariate t-distribution with  degrees of freedom and linear
correlation matrix R, then X2|X1 = x is t-distributed with v + 1 degrees of freedom and
E(X2|X1 = x) = R12x, Var(X2|X1 = x) =
 + x2
 + 1
1 - R2
12 .
This can be used to show that the t-copula has upper (and because of radial symmetry)
equal lower tail dependence:
U = 2 lim
x
P(X2 > x|X1 = x)
= 2 lim
x
t+1
 + 1
 + x2
1/2
x - R12x
1 - 2
l
= 2 lim
x
t+1
 + 1
/x2 + 1
1/2 
1 - R12

1 + R12
= 2t+1

 + 1

1 - R12

1 + R12
.
From this it is also seen that the coefficient of upper tail dependence is increasing in R12
and decreasing in , as one would expect. Furthermore, the coefficient of upper (lower) tail
dependence tends to zero as the number of degrees of freedom tends to infinity for R12 < 1.
Coefficients of upper tail dependence for the bivariate t-copula are given in Table 1. The
last row represents the Gaussian copula, i.e., no tail dependence.
Ch. 8: Modelling Dependence with Copulas 363
Table 1
\R12 -0.5 0 0.5 0.9 1
2 0.06 0.18 0.39 0.72 1
4 0.01 0.08 0.25 0.63 1
10 0.00 0.01 0.08 0.46 1
 0 0 0 0 1
It should be mentioned that the expression given above is just a special case of a general
formula for the coefficient(s) of tail dependence for elliptical distributions with tail
dependence. It turns out that if ii > 0 for all i and -1 < ij / iijj < 1 for all
i = j, then the bivariate marginal distributions of an elliptically distributed random vector
X =d  + RAU  En(,,) has tail dependence if and only if R is so-called regularly
varying (at ). For more details, see Hult and Lindskog (2002), and for details about
regular variation in general see Resnick (1987) or Embrechts, Mikosch and Klüppelberg
(1997).
Equation (5.2) provides an easy algorithm for random variate generation from the
t-copula, Ct
,R.
Algorithm 5.2.
* Find the Cholesky decomposition A of R.
* Simulate n independent random variates z1,...,zn from N(0,1).
* Simulate a random variate s from 2
 independent of z1,...,zn.
* Set y = Az.
* Set x =



s
y.
* Set ui = t(xi), i = 1,...,n.
* (u1,...,un)T  Ct
,R.
Figures 2 and 3 show samples from bivariate distributions with Gaussian and t-copulas.
In Figure 2, we have contrasted a real example (BMW-Siemens daily return data) with
simulated data using marginal t4 tails, corresponding Kendalĺs tau (0.5) and varying copulas.
Note that the Gaussian copula does not get the extreme joint tail observations clearly
present in the real data. The t2-copula seems to be able to do a much better job in that respect.
Indeed the t2-generated scatter plot shows most of the graphical features in the real
data. Note that these examples were only introduced to highlight the simulation procedures
and do not constitute a detailed statistical analysis. Figure 3 (a simulated example) further
highlights the difference between the Gaussian and t-copulas, this time with standard normal
marginals.
The algorithms presented for the Gaussian and t-copulas are fast and easy to implement.
We want to emphasize the potential usefulness of t-copulas as an alternative to Gaussian
copulas. Both Gaussian and t-copulas are easily parameterized by the linear correlation
matrix, but only t-copulas yield dependence structures with tail dependence.
364 P. Embrechts et al.
Fig. 2. The upper left plot shows BMW-Siemens daily log returns from 1989 to 1996. The other plots show
samples from bivariate distributions with t4-marginals and Kendalĺs tau 0.5.
Fig. 3. Samples from two distributions with standard normal marginals, R12 = 0.8 but different dependence
structures. (X1,Y1)T has a Gaussian copula and (X2,Y2)T has a t2-copula.
Ch. 8: Modelling Dependence with Copulas 365
6. Archimedean copulas
The copula families we have discussed so far have been derived from certain families
of multivariate distribution functions using Sklar's Theorem. We have seen that elliptical
copulas are simply the distribution functions of componentwise transformed elliptically
distributed random vectors. Since simulation from elliptical distributions is easy, so is simulation
from elliptical copulas. There are however drawbacks: elliptical copulas do not
have closed form expressions and are restricted to have radial symmetry (C = C). In many
finance and insurance applications it seems reasonable that there is a stronger dependence
between big losses (e.g., a stock market crash) than between big gains. Such asymmetries
cannot be modelled with elliptical copulas.
In this section we discuss an important class of copulas called Archimedean copulas.
This class of copulas is worth studying for a number of reasons. Many interesting parametric
families of copulas are Archimedean and the class of Archimedean copulas allow
for a great variety of different dependence structures. Furthermore, in contrast to elliptical
copulas, all commonly encountered Archimedean copulas have closed form expressions.
Unlike the copulas discussed so far these copulas are not derived from multivariate distribution
functions using Sklar's Theorem. A consequence of this is that we need somewhat
technical conditions to assert that multivariate extensions of Archimedean 2-copulas are
proper n-copulas. A further disadvantage is that multivariate extensions of Archimedean
copulas in general suffer from lack of free parameter choice in the sense that some of the
entries in the resulting rank correlation matrix are forced to be equal. At the end of this
section we present one possible multivariate extension of Archimedean copulas. For other
multivariate extensions we refer to Joe (1997).
There is much written about Archimedean copulas. For some background on bivariate
Archimedean copulas see Genest and MacKay (1986b). For parameter estimation and a
discussion on other statistical questions we refer to Genest and Rivest (1993). Good references
on Archimedean copulas in general are Genest and MacKay (1986a), Nelsen (1999),
Joe (1997). See also the webpage http://www.mat.ulaval.ca/pages/genest/ for further related
work.
6.1. Definitions
We begin with a general definition of Archimedean copulas, which can be found in Nelsen
(1999, p. 90). As our aim is the construction of multivariate extensions of Archimedean
2-copulas, this general definition will later prove to be a bit more general than needed.
Definition 6.1. Let  be a continuous, strictly decreasing function from [0,1] to [0,]
such that (1) = 0. The pseudo-inverse of  is the function [-1] : [0,]  [0,1] given
by
[-1]
(t) =
-1(t), 0 t (0),
0, (0) t .
366 P. Embrechts et al.
Note that [-1] is continuous and decreasing on [0,], and strictly decreasing on
[0,(0)]. Furthermore, [-1]((u)) = u on [0,1], and
 [-1]
(t) =
t, 0 t (0),
(0), (0) t .
Finally, if (0) = , then [-1] = -1.
Theorem 6.1. Let  be a continuous, strictly decreasing function from [0,1] to [0,]
such that (1) = 0, and let [-1] be the pseudo-inverse of . Let C be the function from
[0,1]2 to [0,1] given by
C(u,v) = [-1]
(u) + (v) . (6.1)
Then C is a copula if and only if  is convex.
For a proof, see Nelsen (1999, p. 91).
Copulas of the form (6.1) are called Archimedean copulas. The function  is called
a generator of the copula. If (0) = , we say that  is a strict generator. In this case,
[-1] = -1 and C(u,v) = -1((u) + (v)) is said to be a strict Archimedean copula.
Example 6.1. Let (t) = (-lnt) , where  1. Clearly (t) is continuous and (1) =
0.  (t) = -(-lnt)-1 1
t , so  is a strictly decreasing function from [0,1] to [0,].
 (t) 0 on [0,1], so  is convex. Moreover (0) = , so  is a strict generator. From
(6.1) we get
C (u,v) = -1
(u) + (v) = exp - (-lnu)
+ (-lnv) 1/
.
Furthermore C1 =  and lim C = M (recall that (u,v) = uv and M(u,v) =
min(u,v)). This copula family is called the Gumbel family. As shown in Example 3.3
this copula family has upper tail dependence.
Example 6.2. Let (t) = (t- - 1)/, where   [-1,) \ {0}. This gives the Clayton
family
C (u,v) = max u+
v-
1
-1/
,0 .
For  > 0 the copulas are strict and the copula expression simplifies to
C (u,v) = u+
v-
1
-1/
. (6.2)
The Clayton family has lower tail dependence for  > 0, and C-1 = W, lim0 C = 
and lim C = M. Since most of the following results are results for strict Archimedean
copulas we will refer to (6.2) as the Clayton family.
Ch. 8: Modelling Dependence with Copulas 367
Example 6.3. Let (t) = -ln((e-t - 1)/(e- - 1)), where   R \ {0}. This gives the
Frank family
C (u,v) = -
1

ln 1 +
(e-u - 1)(e-v - 1)
e- - 1
.
Frank copulas are strict Archimedean copulas. Furthermore
lim
C
= W, lim
0
C =  and lim

C = M.
Members of the Frank family are the only Archimedean copulas which satisfy the equation
C(u,v) = C(u,v) for so-called radial symmetry, see Frank (1979) for details.
Example 6.4. Let (t) = 1 - t for t in [0,1]. Then [-1](t) = 1 - t for t in [0,1], and 0
for t > 1; i.e., [-1](t) = max(1 - t,0). Since C(u,v) = max(u + v - 1,0) =: W(u,v),
we see that the bivariate Fréchet­Hoeffding lower bound W is Archimedean.
6.2. Properties
The results in the following theorem will enable us to formulate multivariate extensions of
Archimedean copulas.
Theorem 6.2. Let C be an Archimedean copula with generator . Then
(1) C is symmetric, i.e., C(u,v) = C(v,u) for all u,v in [0,1].
(2) C is associative, i.e., C(C(u,v),w) = C(u,C(v,w)) for all u,v,w in [0,1].
Proof: The first part follows directly from (6.1). For (2),
C C(u,v),w = [-1]
 [-1]
(u) + (v) + (w)
= [-1]
(u) + (v) + (w)
= [-1]
(u) +  [-1]
((v) + (w)) = C u,C(v,w) .
The associativity property of Archimedean copulas is not shared by copulas in general
as shown by the following example.
Example 6.5. Let C be a member of the bivariate Farlie­Gumbel­Morgenstern family of
copulas, i.e., C (u,v) = uv + uv(1 - u)(1 - v), for   [-1,1]. Then
C
1
4
,C
1
2
,
1
3
= C C
1
4
,
1
2
,
1
3
for all   [-1,1] \ {0}. Hence the only member of the bivariate Farlie­GumbelMorgenstern
family of copulas that is Archimedean is .
368 P. Embrechts et al.
Theorem 6.3. Let C be an Archimedean copula generated by  and let
KC(t) = VC (u,v)  [0,1]2
| C(u,v) t .
Then for any t in [0,1],
KC(t) = t -
(t)
 (t+)
. (6.3)
For a proof, see Nelsen (1999, p. 102).
Corollary 6.1. If (U,V )T has distribution function C, where C is an Archimedean copula
generated by , then the function KC given by (6.3) is the distribution function of the
random variable C(U,V ).
The next theorem will provide the basis for a general algorithm for random variate generation
from Archimedean copulas. Before the theorem can be stated we need an expression
for the density of an absolutely continuous Archimedean copula. From (6.1) it follows that
 C(u,v)

u
C(u,v) =  (u),
 C(u,v)

v
C(u,v) =  (v),
 C(u,v)

u
C(u,v)

v
C(u,v) +  C(u,v)
2
uv
C(u,v) = 0,
and hence
2
uv
C(u,v) = -
 (C(u,v)) 
u C(u,v) 
v C(u,v)
 (C(u,v))
= -
 (C(u,v)) (u) (v)
[ (C(u,v))]3
.
Thus, when C is absolutely continuous, its density is given by
2
uv
C(u,v) = -
 (C(u,v)) (u) (v)
[ (C(u,v))]3
. (6.4)
Theorem 6.4. Under the hypotheses of Corollary 6.1, the joint distribution function
H(s,t) of the random variables S = (U)/[(U) + (V )] and T = C(U,V ) is given
by H(s,t) = sKC(t) for all (s,t) in [0,1]2. Hence S and T are independent, and S is
uniformly distributed on [0,1].
Ch. 8: Modelling Dependence with Copulas 369
Proof: [This proof, for the case when C is absolutely continuous, can be found in Nelsen
(1999, p. 104). For the general case, see Genest and Rivest (1993).] The joint density h(s,t)
of S and T is given by
h(s,t) =
2
uv
C(u,v)
(u,v)
(s,t)
,
where 2C(u,v)/uv is given by (6.4) and (u,v)/(s,t) denotes the Jacobian of the
transformation (u) = s(t), (v) = (1 - s)(t). But
(u,v)
(s,t)
=
(t) (t)
 (u) (v)
,
and hence
h(s,t) = -
 (t) (u) (v)
[ (t)]3
(t)
(t)
 (u) (v)
=
 (t)(t)
[ (t)]2
.
Therefore
H(s,t) =
s
0
t
0
 (y)(y)
[ (y)]2
dy dx = s y -
(y)
 (y)
t
0
= sKC(t),
from which the conclusion follows.
An application of Theorem 6.4 is the following algorithm for generating random variates
(u,v)T whose joint distribution is an Archimedean copula C with generator .
Algorithm 6.1.
* Simulate two independent U(0,1) random variates s and q.
* Set t = K-1
C (q), where KC is the distribution function of C(U,V ).
* Set u = [-1](s(t)) and v = [-1]((1 - s)(t)).
Note that the variates s and t correspond to the random variables S and T in Theorem 6.4
and from the proof it follows that this algorithm yields the desired result.
Example 6.6. Consider the Archimedean copula family given by
C (u,v) = 1 + u-1
- 1

+ v-1
- 1
 1/ -1
generated by  (t) = (t-1 -1) for  1. To generate a random variate from C we simply
apply Algorithm 6.1 with
370 P. Embrechts et al.
 (t) = t-1
- 1

,
-1
 (t) = t1/
+ 1
-1
,
K-1
C
(t) =
i + 1
2
i
+ 1
2
2
- is.
6.3. Kendalĺs tau revisited
Recall that Kendalĺs tau for a copula C can be expressed as a double integral of C.
This double integral is in most cases not straightforward to evaluate. However for an
Archimedean copula, Kendalĺs tau can be expressed as an (one-dimensional) integral
of the generator and its derivative, as shown in the following theorem from Genest and
MacKay (1986a).
Theorem 6.5. Let X and Y be random variables with an Archimedean copula C generated
by . Kendalĺs tau of X and Y is given by
C = 1 + 4
1
0
(t)
 (t)
dt. (6.5)
Proof: Let U and V be U(0,1) random variables with joint distribution function C, and
let KC denote the distribution function of C(U,V ). Then from Theorem 3.3 we have
C = 4E C(U,V ) - 1 = 4
1
0
t dKC(t) - 1
= 4 tKC(t)
1
0
-
1
0
KC(t)dt - 1 = 3 - 4
1
0
KC(t)dt.
From Theorem 6.3 and Corollary 6.1 it follows that
KC(t) = t -
(t)
 (t+)
.
Since  is convex,  (t+) and  (t-) exist for all t in (0,1) and the set {t  (0,1) |  (t+) =
 (t-)} is at most countable (i.e., it has Lebesgue measure zero). Hence
C = 3 - 4
1
0
t -
(t)
 (t+)
dt = 1 + 4
1
0
(t)
 (t)
dt.
Ch. 8: Modelling Dependence with Copulas 371
Example 6.7. Consider the Gumbel family with generator (t) = (-lnt) , for  1.
Then (t)/ (t) = (t lnt)/. Using Theorem 6.5 we can calculate Kendalĺs tau for the
Gumbel family.
 = 1 + 4
1
0
t lnt

dt = 1 +
4

t2
2
lnt
1
0
-
1
0
t
2
dt = 1 -
1

.
As a consequence, in order to have Kendalĺs tau equal to 0.5 in Figure 2 (the Gumbel
case), we put  = 2.
Example 6.8. Consider the Clayton family with generator (t) = (t- - 1)/, for  
[-1,) \ {0}. Then (t)/ (t) = (t+1 - t)/. Using Theorem 6.5 we can calculate
Kendalĺs tau for the Clayton family.
 = 1 + 4
1
0
t+1 - t

dt = 1 +
4

1
 + 2
-
1
2
=

 + 2
.
Example 6.9. Consider the Frank family presented in Example 6.3. It can be shown that
[see, e.g., Genest (1987)] Kendalĺs tau is  = 1 - 4(1 - D1())/, where Dk(x) is the
Debye function, given by
Dk(x) =
k
xk
x
0
tk
et - 1
dt
for any positive integer k.
6.4. Tail dependence revisited
For Archimedean copulas, tail dependence can be expressed in terms of the generators.
Theorem 6.6. Let  be a strict generator such that -1 belongs to the class of Laplace
transforms of strictly positive random variables. If -1 (0) is finite, then
C(u,v) = -1
(u) + (v)
does not have upper tail dependence. If C has upper tail dependence, then -1 (0) = and
the coefficient of upper tail dependence is given by
U = 2 - 2 lim
s 0
-1 (2s)
-1 (s)
.
372 P. Embrechts et al.
Proof: [This proof can be found in Joe (1997, p. 103).] Note that
lim
u 1
C(u,u)
1 - u
= lim
u 1
1 - 2u + -1(2(u))
1 - u
= 2 - 2 lim
u 1
-1 (2(u))
-1 ((u))
= 2 - 2 lim
s 0
-1 (2s)
-1 (s)
.
If -1 (0)  (-,0), then the limit is zero and C does not have upper tail dependence.
Since -1 (0) is the negative of the expectation of a strictly positive random variable,
-1 (0) < 0 from which the conclusion follows.
The additional condition on the generator  might seem somewhat strange. It will however
prove quite natural when we turn to the construction of multivariate Archimedean copulas.
Furthermore, the condition is satisfied by the majority of the commonly encountered
Archimedean copulas.
Example 6.10. The Gumbel copulas are strict Archimedean with generator (t) =
(-lnt) . Hence -1(s) = exp(-s1/ ) and its derivative -1 (s) = -s1/-1 exp(-s1/)/.
Using Theorem 6.6 we get
U = 2 - 2 lim
s 0
-1 (2s)
-1 (s)
= 2 - 21/
lim
s 0
exp(-(2s)1/)
exp(-s1/ )
= 2 - 21/
,
see also Example 3.3.
Theorem 6.7. Let  be as in Theorem 6.6. The coefficient of lower tail dependence for the
copula C(u,v) = -1((u) + (v)) is equal to
L = 2 lim
s
-1 (2s)
-1 (s)
.
The proof is similar to that of Theorem 6.6.
Example 6.11. Consider the Clayton family given by C (u,v) = (u- +v- -1)-1/, for
 > 0. This strict copula family has generator (t) = (t- -1)/. It follows that -1(s) =
(1 + s)-1/ . Using Theorems 6.6 and 6.7 shows that U = 0 and that the coefficient of
lower tail dependence given by
L = 2 lim
s
-1 (2s)
-1 (s)
= 2 lim
s
(1 + 2s)-1/-1
(1 + s)-1/-1
= 2-1/
.
Ch. 8: Modelling Dependence with Copulas 373
Example 6.12. Consider the Frank family given by
C (u,v) = -
1

ln 1 +
(e-u - 1)(e-v - 1)
e- - 1
for   R \ {0}. This strict copula family has generator (t) = -ln((e-t - 1)/(e- - 1)).
It follows that
-1
(s) = -
1

ln 1 - 1 - e-
e-s
and -1
(s) = -
1

(1 - e- e-s
1 - (1 - e- )e-s
.
Since
-1
(0) = e
- 1

is finite, the Frank family does not have upper tail dependence according to Theorem 6.6.
Furthermore, members of the Frank family are radially symmetric, i.e. C = C, and hence
the Frank family does not have lower tail dependence.
6.5. Multivariate Archimedean copulas
In this section we look at the construction of one particular multivariate extension of
Archimedean 2-copulas. For other multivariate extensions see Joe (1997). It should be
noted that in order to show that other multivariate extensions are proper copulas, we essentially
have to go through the same arguments as those given below.
The expression for the n-dimensional product copula n, with u = (u1,...,un)T, can
be written as n(u) = u1 ...un = exp(-[(-lnu1)++(-lnun)]). This naturally leads
to the following generalization of (6.1):
Cn
(u) = [-1]
(u1) +  + (un) . (6.6)
In the 3-dimensional case,
C3
(u1,u2,u3) = [-1]
  [-1]
(u1) + (u2) + (u3) = C C(u1,u2),u3 ,
and in the 4-dimensional case,
C4
(u1,...,u4)
= [-1]
  [-1]
  [-1]
(u1) + (u2) + (u3) + (u4)
= C C3
(u1,u2,u3),u4 = C C C(u1,u2),u3 ,u4 .
374 P. Embrechts et al.
Whence in general, n 3, Cn(u1,...,un) = C(Cn-1(u1,u2,...,un-1),un). This technique
of constructing higher-dimensional copulas generally fails. But since Archimedean
copulas are symmetric and associative it seems more likely that Cn as defined above, given
certain additional properties of  (and [-1]), is indeed a copula for n 3.
Definition 6.2. A function g(t) is completely monotone on the interval I if it has derivatives
of all orders which alternate in sign, i.e., if it satisfies
(-1)k dk
dtk
g(t) 0
for all t in the interior of I and k = 0,1,2,....
If g :[0,)  [0,) is completely monotone on [0,) and there is a t  [0,) such
that g(t) = 0, then g(t) = 0 for all t  [0,). Hence if the pseudo-inverse [-1] of an
Archimedean generator  is completely monotone, then [-1](t) > 0 for all t  [0,)
and hence [-1] = -1.
The following theorem from Kimberling (1974) gives necessary and sufficient conditions
for the function (6.6) to be an n-copula.
Theorem 6.8. Let  be a continuous strictly decreasing function from [0,1] to [0,] such
that (0) =  and (1) = 0, and let -1 denote the inverse of . If Cn is the function from
[0,1]n to [0,1] given by (6.6), then Cn is an n-copula for all n 2 if and only if -1 is
completely monotone on [0,).
This theorem can be partially extended to the case where  is nonstrict and [-1] is mmonotone
on [0,) for some m 2, that is, the derivatives of [-1] alter sign up to and
including the mth order on [0,). Then the function Cn given by (6.6) is an n-copula for
2 n m. However, for most practical purposes, the class of strict generators  such that
-1 is completely monotone is a rich enough class.
The following corollary shows that the generators suitable for extensions to arbitrary dimensions
of Archimedean 2-copulas correspond to copulas which can model only positive
dependence.
Corollary 6.2. If the inverse -1 of a strict generator  of an Archimedean copula C is
completely monotone, then C , i.e., C(u,v) uv for all u, v in [0,1].
For a proof, see Nelsen (1999, p. 122).
While it is simple to generate n-copulas of the form given by (6.6), they suffer from a
very limited dependence structure since all k-marginals are identical, they are distribution
functions of n exchangeable U(0,1) random variables. One would like to have a multivariate
extension of the Archimedean 2-copula given by (6.1) which allows for nonexchangeability.
Such multivariate extensions are discussed in Joe (1997). We will now discuss one
Ch. 8: Modelling Dependence with Copulas 375
such extension in detail. Since any multivariate extension should contain (6.6) as a special
case, clearly the necessary conditions for (6.6) to be a copula has to be satisfied. In the light
of Theorem 6.8, we restrict ourselves to strict generators.
The expression for the general multivariate extension of (6.1) we will now discuss is
notationally complex. For that reason we will discuss sufficient conditions for the 3- and
4-dimensional extensions to be proper 3- and 4-copulas respectively. The pattern and conditions
indicated generalize in an obvious way to higher dimensions. The 3-dimensional
generalization of (6.1) is
-1
1 1  -1
2 2(u1) + 2(u2) + 1(u3) , (6.7)
where 1 and 2 are generators of strict Archimedean copulas. The 4-dimensional generalization
of (6.1) is
-1
1 1  -1
2 2  -1
3 3(u1) + 3(u2) + 2(u3) + 1(u4) , (6.8)
where 1, 2 and 3 are generators of strict Archimedean copulas. The expressions (6.7)
and (6.8) can be written as
C1 C2(u1,u2),u3 and C1 C2 C3(u1,u2),u3 ,u4 ,
respectively, where Ci denotes an Archimedean copula generated by i.
If generators i are chosen so that certain conditions are satisfied, then multivariate
copulas can be obtained such that each bivariate marginal has the form (6.1) for some i.
However, the number of distinct generators i among the n(n - 1)/2 bivariate marginals
is only n - 1, so that the resulting dependence structure is one of partial exchangeability.
Clearly the generators have to satisfy the necessary conditions for the n-copula given by
(6.6) in order to make (6.7) and (6.8) valid copula expressions. What other conditions are
needed to make these proper copulas? To answer that question we now introduce function
classes Ln and L
n. Let
Ln = : [0,)  [0,1] | (0) = 1, () = 0, (-1)j
(j)
0,
j = 1,...,n ,
n = 1,2,...,, with L being the class of Laplace transforms of strictly positive random
variables.
Also introduce
L
n = : [0,)  [0,) | (0) = 0, () = , (-1)j-1
(j)
0,
j = 1,...,n ,
n = 1,2,...,. Note that -1  L1 if  is the generator of a strict Archimedean copula.
The functions in L
n are usually compositions of the form -1   with ,   L1.
376 P. Embrechts et al.
Note also that with this notation, the necessary and sufficient conditions for (6.6) to be
a proper copula is that -1  Ln and that, if (6.6) is a copula for all n, then -1 must
be completely monotone and hence be a Laplace transform of a strictly positive random
variable.
It turns out that if -1
1 and -1
2 are completely monotone (Laplace transforms of strictly
positive random variables) and 1  -1
2  L
, then (6.7) is a proper copula. Note that
(6.7) has (1,2) bivariate marginal of the form (6.1) with generator 2 and (1,3) and (2,3)
bivariate marginals of the form (6.1) with generator 1. Also (6.6) is the special case of
(6.7) with 1 = 2. The 3-dimensional copula in (6.7) has a (1,2) bivariate marginal copula
which is larger than the (1,3) and (2,3) bivariate marginal copulas (which are identical).
As one would expect, there are similar conditions for the 4-dimensional case. If -1
1 ,
-1
2 and -1
3 are completely monotone (Laplace transforms of strictly positive random
variables) and 1  -1
2 and 2  -1
3 are in L
, then (6.8) is a proper copula. Note that all
3-dimensional marginals of (6.8) have the form (6.7) and all bivariate marginals have the
form (6.1). Clearly the idea underlying (6.7) and (6.8) generalize to higher dimensions.
Example 6.13. Let i(t) = (-lnt)i with i 1 for i = 1,...,n, i.e., the generators of
Gumbel copulas. What conditions do we have to impose on 1,...,n in order to obtain an
n-dimensional extension of the Gumbel family of the form indicated above (expressions
(6.7) and (6.8)). It should first be noted that -1
i  L for all i, so (6.6) with the above
generators gives an n-copula for all n 2. Secondly, i -1
i+1(t) = ti /i+1 . If i/i+1 / N,
then the nth derivative of i  -1
i+1(t) is given by
i
i+1
...
i
i+1
- (n - 1) ti /i+1-n
.
Hence if i/i+1 / N, then i  -1
i+1  L
 if and only if i/i+1 < 1. If i/i+1  N,
then i  -1
i+1  L
 if and only if i/i+1 = 1. Hence an n-dimensional extension of the
Gumbel family of the form indicated above, given by
exp - (-lnu1)2 + (-lnu2)2 1/2
+ (-lnu3)1 1/1
in the 3-dimensional case, is a proper n-copula if 1  n.
Example 6.14. Consider the Archimedean copula family given by
C (u,v) = 1 + u-1
- 1

+ v-1
- 1
 1/ -1
generated by  (t) = (t-1 - 1) for  1. Set i(t) = i (t) for i = 1,...,n. Can the
above copulas be extended to n-copulas of the form indicated by (6.7) and (6.8), and if
so under what conditions on 1,...,n? By calculating derivatives of -1
i and i  -1
i+1 it
Ch. 8: Modelling Dependence with Copulas 377
follows that -1
i  L and i  -1
i+1  L
 if and only if i/i+1 1. Hence the n-dimensional
extension of the above copulas are n-copulas if 1  n.
Copulas of the above form have upper and lower tail dependence, with coefficients of
upper and lower tail dependence given by 2 - 21/ and 2-1/ respectively. One limiting
factor for the usefulness of this copula family might be that they only allow for a limited
range of positive dependence, as seen from the expression for Kendalĺs tau given by  =
1 - 2/(3), for  1.
Note that the results presented in this section hold for strict Archimedean copulas. With
some additional constraints most of the results can be generalized to hold also for nonstrict
Archimedean copulas. However for practical purposes it is sufficient to only consider strict
Archimedean copulas. This basically means (there are exceptions such as the Frank family)
that we consider copula families with only positive dependence. Furthermore, risk models
are often designed to model positive dependence, since in some sense it is the "dangerous"
dependence: assets (or risks) move in the same direction in periods of extreme events.
7. Modelling extremal events in practice
7.1. Insurance risk
Consider a portfolio consisting of n risks X1,...,Xn, representing potential losses in different
lines of business for an insurance company. Suppose that the insurance company, in
order to reduce the risk in its portfolio, seeks protection against simultaneous big losses
in different lines of business. One suitable reinsurance contract might be the one which
pays the excess losses Xi - ki for i  B  {1,...,n} (where B is some prespecified set of
business lines), given that Xi > ki for all i  B. Hence the payout function f is given by
f (Xi,ki); i  B =
iB
1{Xi>ki }
iB
(Xi - ki) . (7.1)
In order to price this contract the seller (reinsurer) would typically need to estimate
E(f ((Xi,ki); i  B)). Without loss of generality let B = {1,...,l} for l n. If the joint
distribution H of X1,...,Xl could be accurately estimated, calculating the expected value
of (7.1) (possibly by using numerical methods) would not be difficult. Unfortunately, accurate
estimation of H is seldom possible due to lack of reliable data. It is more realistic, and
we will assume this, that the data available allow for estimation of the marginals F1,...,Fn
of H and pairwise rank correlations. The probability of payout is given by
H(k1,...,kl) = C F1(k1),...,Fl(kl) , (7.2)
where H and C denotes the joint survival function and survival copula of X1,...,Xl. If
the thresholds are chosen to be quantiles of the Xi s, i.e., if ki = F-1
i (i) for all i, then the
378 P. Embrechts et al.
right-hand side of (7.2) simplifies to C(1 - 1,...,1 - l). In a reinsurance context, these
quantile levels are often given as return periods and are known to the underwriter.
For a specific copula family, Kendalĺs tau estimates can typically be transformed into
an estimate of the copula parameters. For Gaussian (elliptical) n-copulas this is due to
the relation Rij = sin((Xi,Xj )/2), where Rij = ij / iijj with  being the dispersion
matrix of the corresponding normal (elliptical) distribution. For the multivariate
extension of the Gumbel family presented in Section 6.5 this is due to the relation
 = 1/(1 - (Xi,Xj )), where  denotes the copula parameter for the bivariate Gumbel
copula of (Xi,Xj )T. Hence, once a copula family is decided upon, calculating the probability
of payout or the expected value of the contract is easy. However there is much
uncertainty in choosing a suitable copula family representing the dependence between potential
losses for the l lines of business. The data may give indications of properties such
as tail dependence but it should be combined with careful consideration of the nature of
the underlying loss causing mechanisms. To show the relevance of good dependence modelling,
we will consider marginal distributions and pairwise rank correlations to be given
and compare the effect of the Gaussian and Gumbel copula on the probability of payout
and expected value of the contract. To be able to interpret the results more easily, we make
some further simplifications: let Xi  F for all i, where F is the distribution function of
the standard Lognormal distribution LN(0,1), let ki = k for all i and let (Xi,Xj ) = 0.5
for all i = j. Then,
H(k,...,k) = 1 + (-1)
l
1
C1 F(k) +  + (-1)l l
l
Cl F(k),...,F(k) ,
where Cm, for m = 1,...,l - 1, are m-dimensional marginals of C = Cl (the copula of
(X1,...,Xl)). In the Gaussian case,
Cm F(k),...,F(k) = m
Rm
-1
F(k) ,...,-1
F(k) ,
where m
Rm
denotes the distribution function of m multivariate normally distributed random
variables with linear correlation matrix Rm with off-diagonal entries sin(0.5/2) = 1/

2.
m
l
(-1(F(k)),...,-1(F(k))) can be calculated by numerical integration using the fact
that [see Johnson and Kotz (1972, p. 48)]
m
l
(a,...,a) =

(x)

a -

lx

1 - l
m
dx,
where  denotes the univariate standard normal density function. In the Gumbel case,
Cm F(k),...,F(k) = exp - -lnF(k)

+  + - lnF(k)
 1/
= F(k)m1/
,
where  = 1/(1 - 0.5) = 2.
Ch. 8: Modelling Dependence with Copulas 379
For illustration, let l = 5, i.e., we consider 5 different lines of business. Figure 4 shows
payout probabilities (probabilities of joint exceedances) for thresholds k  [0,15], when
the dependence structure among the potential losses are given by a Gaussian copula (lower
curve) and a Gumbel copula (upper curve). If we let k = F-1(0.99)  10.25, i.e., payout
occurs when all 5 losses exceed their respective 99% quantile, then Figure 5 shows
that if one would choose a Gaussian copula when the true dependence structure between
Fig. 4. Probability of payout for l = 5 when the dependence structure is given by a Gaussian copula (lower curve)
and Gumbel copula (upper curve).
Fig. 5. Ratios of payout probabilities (Gumbel/Gaussian) for l = 3 (lower curve) and l = 5 (upper curve).
380 P. Embrechts et al.
Fig. 6. Estimates of E(f (X1,X2,k)) for Gaussian (lower curve) and Gumbel (upper curve) copulas.
the potential losses X1,...,X5 is given by a Gumbel copula, the probability of payout is
underestimated almost by a factor 8.
Figure 6 shows estimates of E(f (X1,X2,k)) for k = 1,...,18. The lower curve shows
estimates for the expectation when (X1,X2)T has a Gaussian copula and the upper curve
when (X1,X2)T has a Gumbel copula. The estimates are sample means from samples
of size 150000. Since F-1(0.99)  10.25, Figure 6 shows that if one would choose a
Gaussian copula when the true dependence between the potential losses X1 and X2 is given
by a Gumbel copula, the expected loss to the reinsurer is underestimated by a factor 2.
7.2. Market risk
We now consider the problem of measuring the risk of holding an equity portfolio over a
short time horizon (one day, say) without the possibility of rebalancing. More precisely,
consider a portfolio of n equities with current value given by
Vt =
n
i=1
iSi,t ,
where i is the number of units of equity i and Si,t is the current price of equity i. Let
t+1 = -(Vt+1 - Vt)/Vt, the (negative) relative loss over time period (t,t + 1], be our
aggregate risk. Then
t+1 =
n
i=1
i,t i,t+1
Ch. 8: Modelling Dependence with Copulas 381
where i,t = iSi,t /Vt is the portion of the current portfolio value allocated to equity i,
and i,t+1 = -(Si,t+1 - Si,t )/Si,t is the (negative) relative loss over time period (t,t + 1]
of equity i.
We will highlight the techniques introduced by studying the effect of different distributional
assumptions for  := (1,t+1,...,n,t+1)T on the aggregate risk := t+1. The
classical distributional assumption on , widely used within market risk management, is
that of multivariate normality. However, in general the empirical distribution of  has (onedimensional)
marginal distributions which are heavier tailed than the normal distribution.
Furthermore, there is an even more critical problem with multivariate normal distributions
in this context. Extreme falls in equity prices are often joint extremes, in the sense that a
big fall in one equity price is accompanied by simultaneous big falls in other equity prices.
This is for instance seen in Figure 7, an example already encountered in Figure 2. Loosely
speaking, a problem with the multivariate normal distributions (or models based on them)
is that they do not assign a high enough probability of occurrence to the event in which
many thing go wrong a the same time ­ the "perfect storm" scenario. More precisely, daily
equity return data often indicate that the underlying dependence structure has the property
of tail dependence, a property which we know Gaussian copulas lack.
Suppose  is modelled by a multivariate normal distribution Nn(,), where  and
 are estimated from historical prices of the equities in the portfolio. There seems to
be much agreement on the fact that the quantiles of =  T  N( T, T ) do not
Fig. 7. Daily log returns from 1989 to 1996.
382 P. Embrechts et al.
capture the portfolio risk due to extreme market movements; see for instance Embrechts,
Mikosch and Klüppelberg (1997), Embrechts (2000) and the references therein. Therefore,
different stress test solutions have been proposed. One such "solution" is to choose s and
s in such a way that s  Nn(s,s) represents the distribution of the relative losses of
the different equities under more adverse market conditions. The aim is that the quantiles
of s =  Ts  N( Ts, Ts ) should be more realistic risk estimates. To judge this
approach we note that
Fig. 8. Quantile curves: VaR (), VaR s () and VaR  () from lower to upper.
Fig. 9. Quantile curves: VaR () and VaR  () from lower to upper.
Ch. 8: Modelling Dependence with Copulas 383
F-1
s () -  Ts
F-1() -  T
=
 Ts
 T
,
where F and Fs denotes the distribution functions of and s respectively. Hence the
effect of this is simply a translation and scaling of the quantile curve F-1(). As a comparison,
let  have a t4-distribution with mean  and covariance matrix  and let  be
the corresponding portfolio return. Furthermore let n = 10, i = s,i = 
i = 0, i = 1/10
for all i and let (i,j ) = (
i ,
j ) = 0.4, (s,i,s,j ) = 0.6, ij = sin((i,j )/2),
s,ij = 1.5 sin((s,i,s,j )/2) for all i,j. Then Figure 8 shows from lower to upper the
quantile curves of , s and  respectively. If  were the true portfolio return, Figure
8 shows that the approach described above would eventually underestimate the quantiles
of the portfolio return. It should be noted that this is not mainly due to the heavier
tailed t4-marginals. This can be seen in Figure 9 which shows quantile curves of  and
=  T , where  is a random vector with t4-marginals, a Gaussian copula, E( ) = E()
and Cov( ) = Cov().
There are of course numerous alternative applications of copula techniques to integrated
risk management. Besides the references already quoted, also see Embrechts, Hoeing and
Juri (2001) where the calculation of Value-at-Risk bounds for functions of dependent risks
is discussed. The latter paper also contains many more relevant references to this important
topic.
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Sklar, A., 1959. Fonctions de répartition  n dimensions et leurs marges. Publications de ĺInstitut de Statistique
de ĺUniversité de Paris 8, 229­231.
Sklar, A., 1996. Random variables, distribution functions, and copulas ­ a personal look backward and forward.
In: Rüschendorff, L., Schweizer, B., Taylor, M.D. (Eds.), Distributions with Fixed Marginals and Related
Topics. Institute of Mathematical Statistics, Hayward, CA, pp. 1­14.
Chapter 9
PREDICTION OF FINANCIAL DOWNSIDE-RISK
WITH HEAVY-TAILED CONDITIONAL DISTRIBUTIONS
STEFAN MITTNIK
Institute of Statistics, University of Munich, Akademiestr. 1, D-80799 Munich, Germany
Ifo Institute for Economic Research, Munich, Germany
Center for Financial Studies, Frankfurt, Germany
MARC S. PAOLELLA
Institute of Statistics and Econometrics, University of Kiel, Olshausenstr. 40, D-24098 Kiel, Germany
Contents
Abstract 386
1. Introduction 387
2. GARCH-stable processes 388
3. Modeling exchange-rate returns 389
3.1. Approximate maximum likelihood estimation 390
3.2. Estimation results and volatility persistence 392
3.3. Goodness of fit 395
4. Prediction of densities and downside risk 398
5. Conclusions 402
References 403
 The research of S. Mittnik was supported by the Deutsche Forschungsgemeinschaft.
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
386 S. Mittnik and M.S. Paolella
Abstract
The use of GARCH models with stable Paretian innovations in financial modeling has
been recently suggested in the literature. This class of processes is attractive because it
allows for conditional skewness and leptokurtosis of financial returns without ruling out
normality. This contribution illustrates their usefulness in predicting the downside risk of
financial assets in the context of modeling foreign exchange-rates and demonstrates their
superiority over use of normal or Studenťs t GARCH models.
Ch. 9: Prediction of Financial Downside-Risk 387
1. Introduction
Risk managers of financial institutions are particularly interested in the left ­ i.e., downside
­ tail of the return distribution of financial assets. To assess the short-term exposure
to market risks, they are required to evaluate future shortfall probabilities or value-at-risk
levels of financial investments. Such estimates can be based on the distribution of the returns
themselves. For example, ever since the pioneering works of Mandelbrot (1963) and
Fama (1965) there have been numerous studies investigating the appropriateness of the
stable Paretian distribution for modeling the unconditional distribution of asset returns [for
an overview, see, for example, Mittnik and Rachev (1993), McCulloch (1997)].
However, short-term prediction often benefits substantially when taking conditional
volatility into account. The GARCH class of conditional models has been widely and both
from an academic and applied perspective ­ successfully used to model returns on
financial assets [see Palm (1997), Gouriéroux (1997), for surveys]. Although a stationary
GARCH model with normally distributed innovations gives rise to an unconditional
distribution with higher (possibly nonexistent) kurtosis than the normal, it is often found
that residuals from estimated GARCH models of financial return data still tend to exhibit
nonnegligible kurtosis. To allow for this, other fatter tailed distributions for GARCH innovations
have been considered in the literature, most notably the Studenťs t. Only very
recently has the stable Paretian distribution been considered in the context of modeling the
conditional heteroscedastic distribution of asset returns. Special cases of the model considered
herein were developed by McCulloch (1985), Nelson (1990), Panorska, Mittnik and
Rachev (1995), and Mittnik, Rachev and Paolella (1998), while a more general case was
examined in Liu and Brorsen (1995), Paolella (1999) and Mittnik, Paolella and Rachev
(2000, 2002).
Like the Studenťs t, the stable Paretian distribution includes the normal distribution as
a special, limiting case and permits heavy-tailed distributions for GARCH innovations.
However, the stable Paretian distribution allows for skewness, an attractive property in financial
applications not shared by the Studenťs t. In addition to this practical aspect, the
stable Paretian distribution also has the appealing theoretical property that it is the only
valid distribution that arises as a limiting distribution of sums of independently, identically
distributed (iid) random variables. This is highly desirable, given that error terms in econometric
models are usually interpreted as random variables that represent the sum of the
external effects not being captured by the model.
This contribution investigates the use of asymmetric stable Paretian power GARCH
models for modeling downside risk and demonstrates that this model class is more suitable
than the class of Studenťs t GARCH models, particularly when one uses a goodness-of-fit
criterion that focuses on the tails of the conditional distribution.
The remainder is organized as follows. Section 2 discusses GARCH processes with stable
Paretian innovations and stationarity conditions. Section 3 reconsiders the empirical
analysis of the five exchange-rate series in Liu and Brorsen (1995) using the appropriate
measure for persistence of volatility and compares the goodness of fit of the estimated stable
Paretian and Studenťs t GARCH models. The problem of out-of-sample conditional
388 S. Mittnik and M.S. Paolella
density prediction with particular focus on predicting downside market risk is considered
in Section 4. Section 5 concludes.
2. GARCH-stable processes
Sequence yt is said to be a stable Paretian power GARCH process or, in short, an
S
,GARCH(r,s) process [see Panorska, Mittnik and Rachev (1995), Paolella (1999),
Rachev and Mittnik (2000)], if
yt =  + ctt, t
iid
 S,(0,1) (1)
and
c
t = 0 +
r
i=1
i|yt-i - |
+
s
j=1
j c
t-j , (2)
where S, (0,1) denotes the standard asymmetric stable Paretian distribution with stable
index , skewness parameter   [-1,1], zero location parameter, and unit scale parameter.
There exist several notational varieties of the stable Paretian distribution; we use the
same as in Samorodnitsky and Taqqu (1994) and Rachev and Mittnik (2000), whereby

-
eitx
dH(x) =



exp -c
|t|
1 - i sign(t)tan

2
+ it , if  = 1,
exp -c|t| 1 + i
2

sign(t)ln|t| + it , if  = 1,
(3)
is the characteristic function and H denotes the distribution function corresponding to
S,(,c). The density is symmetric for  = 0 and skewed to the right (left) for  > 0
( < 0). Stable index , which, in general, assumes values in interval (0,2], determines the
tail-thickness of the distribution. As  approaches 2, tails become thinner; and for  = 2
the standard stable Paretian distribution coincides with normal distribution N(0,2). For
 < 2, t does not possess moments of order  or higher.
Mittnik, Paolella and Rachev (2002) derived sufficient conditions under which the
S
,GARCH(r,s) process has a unique strictly stationary solution. These are given by
1 <  2, 0 <  < , c0 > 0, ci 0, i = 1,...,r, r 1, dj 0, j = 1,...,s, s 0, and
that the volatility persistence, VS, defined by
VS := E|Z|
r
i=1
i +
s
j=1
j (4)
Ch. 9: Prediction of Financial Downside-Risk 389
for Z  S,(0,1), satisfies
VS 1. (5)
If 1 <  2 and 0 <  < , they also showed that
,, := E|Z|
=
1

1 -


1 + 2
,
/(2)
cos


arctan, , (6)
where , :=  tan(/2) and
 =



(1 - )cos

2
, if  = 1,

2
, if  = 1.
(7)
Restrictions 1 <  2 and 0 <  <  not only appear to be satisfied for the data sets used
below, but also for other, even more volatile series, such as stock price indices and East
Asian currencies [see Mittnik, Rachev and Paolella (1998), Mittnik, Paolella and Rachev
(2000), respectively].
Analogous to the ordinary normal GARCH model (Engle and Bollerslev, 1986), we say
that yt is an integrated S
,GARCH(r,s) process, denoted S
,IGARCH(r,s), if, in (5),
VS = 1. In practice, the estimated volatility persistence, VS, tends to be quite close to one
for highly volatile series, so that an integrated model might offer a reasonable data description.
Because both finite sample and even asymptotic properties of VS and the associated
likelihood ratio test statistics are not known [see, however, Mittnik, Paolella and Rachev
(2000)], it is not immediately clear how one can test for an integrated process. Instead
of formally testing, we suggest fitting both models and examining the change in various
goodness-of-fit statistics, most notably the Anderson­Darling statistic, which is particularly
relevant for assessing the models' ability to successfully model the value-at-risk (see
Section 3.3 below).
3. Modeling exchange-rate returns
To examine the appropriateness of the stable GARCH hypothesis, we model returns1 on
five daily spot foreign exchange rates against the U.S. dollar, namely the British pound,
Canadian dollar, German mark, Japanese yen, and the Swiss franc. The choice of exchange
rate allows us to compare our more general GARCH specification to that used by Liu and
Brorsen (1995), who set  =  in (2). However, our sample is somewhat larger than theirs,
1 We define the return rt in period t by rt = 100 × (lnPt - lnPt-1), where Pt is the exchange rate at time t.
390 S. Mittnik and M.S. Paolella
covering the period January 2, 1980 to July 28, 1994, yielding series of lengths 3681, 3682,
3661, 3621, and 3678, respectively. Serial correlation was found to be negligible, and, as is
common in practice, a GARCH(r,s) specification with r = s = 1 was sufficient to capture
serial correlation in the absolute returns. Therefore, we specify a model of the form
rt =  + ctt, (8)
c
t = 0 + 1|rt-1 - |
+ 1c
t-1 (9)
for each of the five currencies.
3.1. Approximate maximum likelihood estimation
Evaluation of the probability density function (pdf) and, thus, the likelihood function
of the S, distribution is nontrivial, because it lacks an analytic expression. The maximum
likelihood (ML) estimate of parameter vector  = (,c0,0,1,1,,,) for the
S
,GARCH(1,1) models (8), (9) is obtained by maximizing the logarithm of the likelihood
function
L(;r1,...,rT ) =
T
t=1
c-1
t S,
rt - 
ct
, (10)
where c0 denotes the unknown initial value of ct .
The ML estimation we conduct is approximate in the sense that the stable Paretian density
function S,((rt -)/ct) needs to be approximated. To do so, we follow the algorithm
of Mittnik et al. (1999), which approximates the stable Paretian density via fast Fourier
transform of the characteristic function. DuMouchel (1973) shows that the ML estimator
of the parameters of the stable density is consistent and asymptotically normal with the
asymptotic covariance matrix being given by the inverse of the Fisher information matrix.
Approximate standard errors of the estimates can be obtained via numerical approximation
of the Hessian matrix.
Below, we will demonstrate that ­ for the five series under consideration ­ the
S
,GARCH(r,s) model outperforms its Studenťs t counterpart. However, it is of practical
interest to know at least three things before adopting a new and more complex method:
first, how easy the stable ML estimation routine is to implement; second, whether it is
numerically well-behaved; and third, how fast it performs. When implemented in highlevel
software which provide both FFT and linear interpolation routines (such as Matlab
and Splus), the algorithm becomes a straightforward programming exercise. Our experience
has shown that the method is extremely well behaved, giving rise to numerical problems
only for grossly misspecified and/or overspecified models (for which the Studenťs t
GARCH model also has difficulties) or, in the case of the more general class of ARMAGARCH
models, when there is near zero-pole cancellation in the ARMA structure ­ a
well-known difficulty in ARMA estimation.
Ch. 9: Prediction of Financial Downside-Risk 391
The satisfactory behavior of the algorithm is actually not surprising for at least two
reasons. First, there is no explicit numerical integration involved [as in the approach of
Liu and Brorsen (1995)] and, second, the method can be made arbitrarily accurate by the
choice of several tuning constants [recommendations for which are given in Mittnik et al.
(1999)].Nevertheless, it is clear that the method will take longer than the (essentially closed
form) evaluation of the Studenťs t density. For the series considered in this paper, use
of a quasi-Newton minimization algorithm (BFGS, as implemented in Matlab 5.2) with
convergence tolerance of 10-4 resulted in convergence after about 150 to 350 function
evaluations (including gradient calculations). Rather contrary to our initial expectations
­ and fears ­, the choice of initial values is of surprisingly little importance. Given any
"reasonable" set of values, say  > 1.4, || < 0.7, || < 0.2, 0 > 0, 1 > 0 and 1 > 0.2,
convergenceto the same respective maxima occurred for all five exchange-rate series under
consideration, and also for the vast majority of trials from simulation experiments. From a
purely numerical standpoint then, the method appears both highly reliable and "stable".
Evaluation of the GARCH recursion requires presample values 0 and c0. Following
Nelson and Cao (1992), one could set those to their unconditional expected values, i.e.,
^c0 =
^0
1 - ^, ^,^
r
i=1
^i - s
j=1
^j
and ^0 = ^^c0. (11)
In the IGARCH case, (11) will be invalid, so we instead estimate c0 as an additional parameter.
In fact, we chose to do this for all models considered here, as (11) will clearly be
problematic for nearly integrated GARCH models.
For the integrated model S
,IGARCH(1,1), the restriction 1 = 1 - ,,1 needs
to be imposed. Notice that this entails evaluation of (4) at each iteration, as 1 is also
dependent on values ^, ^ and ^.
We compare the S
,GARCH model to the most commonly used heavy-tailed variant of
the GARCH model, the Studenťs t-GARCH models in power form, say t
 -GARCH(r,s),
given by
rt =  + ctt, t
iid
 t(), (12)
c
t = 0 +
r
i=1
i|rt-i - |
+
s
j=1
j c
t-j , (13)
where t() refers to the Studenťs t distribution with  degrees of freedom, i.e.,
f (x;) = K 1 +
x2

-(+1)/2
(14)
392 S. Mittnik and M.S. Paolella
and
K =
(( + 1)/2)-1/2

 (/2)
. (15)
Assuming 0 <  <  and  > 1,2 taking unconditional expectations of c
t in (13) shows
that Ec
t exists if E|T | r
i=1 i + s
j=1 j < 1, where T  t() and
, := E|T |
=


 + 1
2
 - 
2
-1 
2
. (16)
Analogous to (4), the measure of volatility persistence for t
 -GARCH(r,s) models is defined
to be
Vt := ,
r
i=1
i +
s
j=1
j . (17)
Similar remarks regarding treatment of presample values and the imposing of the
IGARCH constraint apply to the Studenťs t model as well.
3.2. Estimation results and volatility persistence
The parameter estimates of the models are presented in Table 1. Noteworthy are the estimates
of the skewness parameter : all ^ values are (statistically) significantly different
from zero, although those for the British pound and German mark series are quite close to
zero. In addition, when || < 0.3 and  is over 1.8, the amount of skewness is, for practical
purposes, slight. Skewness is most pronounced for the Japanese yen, for which ^ = 1.81
and ^ = -0.418.
The persistence-of-volatility measure given in the last column of Table 1 reflects the
speed with which volatility shocks die out. A V -value near one is indicative of an integrated
GARCH process, in which volatility shocks have persistent effects. Under the
S, assumption, the models for the Canadian dollar (VS = ^, ^,^
^1 + ^1 = 1.001) and
Japanese yen (VS = 1.002) series would suggest that they are very close to being integrated.
Under the Studenťs t assumption, Vt = ^,^
^1 + ^1 = 0.992 for the Canadian dollar,
which is also rather close to being integrated, while Vt is only 0.972 for the Japanese
yen. Thus, for these two currencies, the indications regarding persistence of volatility differ
under the two distributional assumptions. For the other currencies, the measures are
strikingly close, most notably for the German mark (VS = Vt = 0.969) and the Swiss franc
2 The condition  > 1 is analogous to requiring  > 1 in the stable Paretian case and implies existence of a finite
first moment of the innovations.
Ch. 9: Prediction of Financial Downside-Risk 393
Table 1
GARCH parameter estimatesa
Intercept GARCH Distribution Persistence
parameters parameters measureb
 0 1 1  Shape Skew V
British
S, -9.773e-3 8.085e-3 0.04132 0.9171 1.359 1.850 -0.1368 0.984
(0.012) (2.39e-3) (6.42e-3) (0.0118) (0.0892) (0.0245) (0.0211)
t -2.312e-3 0.01190 0.06373 0.9071 1.457 6.218 ­ 0.976
(0.010) (3.56e-3) (0.0115) (0.0200) (0.167) (0.615)
Canadian
S, 5.167e-3 1.034e-3 0.04710 0.9164 1.404 1.823 0.3577 1.001
(0.0614) (3.12e-4) (6.63e-3) (0.0118) (0.0143) (0.0104) (0.0209)
t -2.240e-3 7.774e-4 0.06112 0.9118 1.793 5.900 ­ 0.992
(3.83e-3) (6.90e-4) (5.98e-3) (7.27e-3) (0.0150) (0.0801)
German
S, 2.580e-3 0.01525 0.05684 0.8971 1.101 1.892 -0.06779 0.969
(0.016) (1.61e-3) (3.44e-3) (7.42e-3) (9.78e-3) (0.0216) (0.0184)
t 6.643e-3 0.01812 0.07803 0.8938 1.261 7.297 ­ 0.969
(9.21e-4) (2.25e-3) (6.45e-3) (4.43e-3) (0.147) (0.186)
Japanese
S, -0.01938 4.518e-3 0.06827 0.8865 1.337 1.814 -0.4175 1.002
(0.0166) (1.12e-3) (7.91e-3) (0.0124) (0.0132) (0.0107) (8.80e-3)
t 5.318e-3 9.949e-3 0.07016 0.8756 1.816 5.509 ­ 0.972
(8.87e-3) (3.03e-3) (0.0119) (0.0205) (0.162) (0.461)
Swiss
S, -2.677e-3 0.01595 0.04873 0.9115 1.041 1.902 -0.2836 0.971
(0.0124) (3.30e-3) (6.84e-3) (0.0132) (0.144) (0.0206) (0.0722)
t 8.275e-3 0.02099 0.06825 0.9061 1.159 8.294 ­ 0.968
(0.0118) (3.91e-3) (6.85e-3) (7.25e-3) (0.179) (0.933)
a Estimated models: rt =  + ct t , c
t = 0 + 1|rt-1 - | + 1c
t-1. "Shape" denotes the degrees of freedom
parameter  for the Studenťs t distribution and stable index  for the stable Paretian distribution; "Skew" refers
to the stable Paretian skewness parameter . Standard deviations resulting from ML estimation are given in
parentheses.
b V corresponds to VS in the stable Paretian and Vt in the Studenťs t case. V = 1 implies an IGARCH model.
(VS = 0.971, Vt = 0.968). It is interesting to note that, for each of these two currencies,
the log-likelihood values Lt and LS are also extremely close. These are discussed further
in the next section.
For all five series, we also estimated the models with the IGARCH condition imposed.
Table 2 shows the resulting parameter estimates. Not surprisingly, for those models for
which the persistence measure was close to unity, the IGARCH-restricted parameter estimates
differ very little. For the remaining models, the greatest changes occur with the
394 S. Mittnik and M.S. Paolella
Table 2
IGARCH parameter estimatesa
Intercept IGARCH Distribution
parameters parameters
 0 1 1  Shape Skew
British
S, -0.01023 7.050e-3 0.03781 0.9114 1.598 1.846 -0.1340
(0.0103) (1.79e-3) (5.64e-3) ­ (0.0677) (0.0224) (0.0147)
t -3.033e-3 4.237e-3 0.05774 0.9130 1.949 5.543 (0.0101)
(1.68e-3) (9.83e-3) ­ (0.264) (0.484)
Canadian
S, 5.148e-3 1.115e-3 0.04689 0.9154 1.404 1.823 0.3578
(3.65e-3) (2.14e-4) (5.71e-3) ­ (0.0143) (0.0105) (0.0209)
t -2.098e-3 4.998e-4 0.06468 0.9146 1.796 5.890 (3.48e-3)
(1.37e-4) (7.54e-3) ­ (0.0226) (0.0838)
German
S, 8.959e-3 9.666e-3 0.04518 0.8896 1.676 1.881 0.03944
(0.0113) (1.85e-3) (6.10e-3) ­ (0.0662) (0.0217) (0.0930)
t 8.851e-3 5.505e-3 0.08124 0.9003 1.741 6.560 (0.0106)
(1.60e-3) (0.0106) ­ (0.231) (0.676)
Japanese
S, -0.01932 4.814e-3 0.06768 0.8858 1.336 1.814 -0.4175
(8.44e-3) (9.75e-4) (7.68e-3) ­ (0.0751) (0.0226) (0.0151)
t 6.136e-3 5.611e-3 0.06036 0.8689 2.314 5.066 (8.57e-3)
(1.31e-3) (0.0112) ­ (0.224) (0.410)
Swiss
S, 3.823e-3 0.01111 0.03700 0.9009 1.724 1.889 -0.1703
(0.0127) (2.65e-3) (5.40e-3) ­ (0.0419) (0.0169) (0.137)
t 9.130e-3 2.047e-3 0.07125 0.9347 1.166 8.194 (0.0119)
(8.34e-4) (9.13e-3) ­ (9.79e-3) (0.0996)
a Estimated models: rt =  + ct t , c
t = 0 + 1|rt-1 - | + (1 - 1)c
t-1 with IGARCH condition ^1 =
1 - ^ ^1 imposed. See footnote to Table 1 for further details.
power parameter  and, to a lesser extent, the shape parameters  and . The former increase,
while the latter decrease under IGARCH restrictions.
It should also be noted that the restriction  = , imposed by Liu and Brorsen (1995)
when estimating GARCH-stable models for the same five currencies, is not supported by
our results. This is important because, if  , the unconditional first moments of ct is
infinite for any  < 2. The knife-edge specification  =  does not only induce conceptual
difficulties, but also leads to a highly volatile evolution of the ct series in practical work.
For our estimates, we obtain ^ < ^, which suggest that conditional volatility c
t is a welldefined
quantity in the sense that E(c
t | rt-1,rt-2,...) <  for VS < 1.
Ch. 9: Prediction of Financial Downside-Risk 395
3.3. Goodness of fit
We employ three likelihood-based and one empirical CDF-based criteria for comparing
the goodness of fit of the candidate models. The first is the maximum log-likelihood value
obtained from ML estimation. This value may be viewed as an overall measure of goodness
of fit and allows us to judge which candidate is more likely to have generated the data.
The second is the AICC [Hurvich and Tsai (1989), see also Brockwell and Davis (1991),
Equation (9.3.7)] given by
AICC = -2L +
2T (k + 1)
T - k - 2
, (18)
where k denotes the number of estimated parameters and T the number of observations.
This is the bias-corrected information criterion of Akaike (1973), which corrects the latter's
tendency to overfit. Similarly, the SBC (Schwartz, 1978), defined as
SBC = -2L +
k log(T )
T
, (19)
is a similar penalizing strategy which is commonly used.
The fourth criterion is the Anderson­Darling statistic [Anderson and Darling (1952), see
also Press et al. (1991), and Tanaka (1996)], given by
AD = sup
xR
|Fs(x) - F (x)|
F(x)(1 - F(x))
, (20)
where F(x) denotes the cdf of the estimated parametric density, and Fs(x) is the empirical
sample distribution, i.e.,
Fs(x) =
1
T
T
t=1
I(-,x]
rt - ^
^ct
,
where I() is the usual indicator function. The AD statistic weights discrepancies appropriately
across the whole support of the distribution. This is especially important if one
is interested in determining conditional shortfall probabilities, i.e., the probability of large
investment losses, or so-called value-at-risk measures, where one focuses on the left tail of
the conditional return distribution.
Table 3 displays the aforementioned goodness-of-fit measures for the estimated models.
In both the unrestricted and IGARCH restricted cases, the inference suggested from the
maximum log-likelihood value L, and the AICC and SBC are identical. This is not too
surprising, given the large ratio of observations to parameters, and the fact that there is only
one parameter difference between the Studenťs t and stable Paretian GARCH models.
396 S. Mittnik and M.S. Paolella
Table 3
Goodness-of-fit measures of estimated modelsa
L AICC SBC AD
S, t S, t S, t S, t
Britain:
GARCH -3842.0 -3828.6 7700.0 7671.2 7684.0 7657.2 0.0375 0.0244
IGARCH -3842.3 -3837.1 7698.6 7686.2 7684.6 7674.2 0.0417 0.0420
Canada:
GARCH -159.92 -152.25 0335.9 0318.5 0319.9 0304.5 0.0532 0.0571
IGARCH -159.97 -153.71 0334.0 0319.4 0320.0 0307.4 0.0529 0.0633
Germany:
GARCH -3986.5 -3986.2 7989.0 7986.4 7973.0 7972.4 0.0368 0.345
IGARCH -3989.9 -3999.4 7993.8 8010.8 7979.8 7998.8 0.0506 0.200
Japan:
GARCH -3178.7 -3333.7 6373.4 6681.4 6357.4 6667.4 0.0401 0.0986
IGARCH -3178.8 -3334.6 6371.6 6681.2 6357.6 6669.2 0.0394 0.0793
Switzerland:
GARCH -4308.6 -4308.1 8633.2 8630.2 8617.2 8616.2 0.0457 0.287
IGARCH -4314.2 -4325.0 8642.4 8662.0 8628.4 8650.0 0.0460 0.278
a L refers to the maximum log-likelihood value; AICC is the corrected AIC criteria (18); SBC is the Schwarz
Bayesian criteria (19); and AD is the Anderson­Darling statistic (20).
It appears that L significantly favors the Studenťs t distribution for the British pound
(with values, in obvious notation, Lt = -3828.6 and LS = -3842.0) and the Canadian
dollar (Lt = -152.25, LS = -159.92). For the German mark (Lt = -3896.2,
LS = -3896.5) and the Swiss franc (Lt = -4308.1, LS = -4308.6), the log-likelihood
values, AICC and SBC are very close, albeit larger for the Studenťs t. On the other hand,
the S, assumption is favored quite strongly for the Japanese yen with LS = -3178.7 as
compared to Lt = -3331.7.
For the British pound, the AD statistic (ADt = 0.0244, ADS = 0.0375) slightly favors
the Studenťs t model, in agreement with L, although the difference is relatively small. The
AD statistics for the remaining countries all favor the stable Paretian model, particularly
for the German mark (ADt = 0.345, ADS = 0.0368), the Japanese yen (ADt = 0.0986,
ADS = 0.0401) and the Swiss franc (ADt = 0.287, ADS = 0.0457). The usual caveat applies,
in that, statistically speaking, it is not clear to what extent these differences are significant.
However, given virtually identical log-likelihood values, but AD statistics which
are several times smaller for the S, distribution, one might safely conclude that, particularly
in the tails of the conditional distribution, the S, model offers a distinct advantage,
irrespective of its desirable theoretical properties which are not shared by the Studenťs t
distribution.
Ch. 9: Prediction of Financial Downside-Risk 397
Fig. 1. Comparison of the variance adjusted differences between the sample and fitted distribution functions.
398 S. Mittnik and M.S. Paolella
For each currency and both distributional assumptions, Figure 1 plots the values
ADt =
|Fs(^t:T ) - F(^t:T )|
F(^t:T )(1 - F (^t:T ))
,
t = 1,...,T , where T is the sample size and ^t:T denotes the sorted GARCH-filtered
residuals. In most cases, most notably for the Studenťs t GARCH model of the German,
Japanese and Swiss currency returns, the maximum absolute value of the ADt occurs in
the (left) tail of the distribution.
Turning now to the IGARCH-restricted fits, it is clear that the log-likelihood values must
necessarily decrease, since none of the unrestricted GARCH models precisely satisfied the
IGARCH restrictions. However, for the S, model of the Canadian dollar (L = 159.97)
and Japanese yen (L = 3178.8), the log-likelihoods are very close to their unrestricted
counterparts. This was expected, as the IGARCH condition for the unrestricted models of
these two currencies were nearly met. Somewhat surprising, however, is the small decrease
in AD values for the S, model of the Canadian dollar (ADS = 0.0529) and Japanese yen
(ADS = 0.0394). Particularly for the latter two currencies, stable IGARCH models appear
to describe the daily returns quite plausibly.
4. Prediction of densities and downside risk
Decisions on financial investments are typically based on the expected return and the expected
risk of the assets under consideration. Rather than adhering to the conventional
mean-variance criterion, recent risk management concepts for financial institutions focus
on the downside risk or the value-at-risk of a financial position due to market movements.
In this context, a typical question would be: what is the probability that the value of a financial
position will drop by 50% or more over the next period, i.e., Pr(rt+1 < -0.50)?
Alternatively, one may ask what is the threshold or value-at-risk, -z( ), under which
a position will not fall with a probability of 100(1 -  )%; i.e., find -z( ) such that
Pr(rt+1 < -z( )) =  .
Under unconditional normality, it would be sufficient to simply predict the conditional
mean and variance to answer such questions. However, for GARCH processes driven by
nonnormal, asymmetric and, possibly, infinite-variance innovations, the predictive conditional
density
^ft+1|t(rt+1) = f
rt+1 - (^t)
ct+1(^t)
rt ,rt-1,... , (21)
needs to be computed. In (21), ^t refers to the estimated parameter vector using the sample
information up to and including period t; and ct+1() is obtained from the conditional-scale
Ch. 9: Prediction of Financial Downside-Risk 399
recursion (2) using ^t .3 Multistep density predictions,
^ft+n|t(rt+n) = f
rt+n - (^t)
ct+n(^t)
rt ,rt-1,... , (22)
are obtained by recursive application of (2) with unobserved quantities being replaced by
their conditional expectations.
For each of the five currencies under consideration, we evaluate ^ft+1|t(rt+1), t =
2000,...,T - 1, for the S
,GARCH(1,1) and t
 GARCH(1,1) models, as well as the
conventional GARCH(1,1) model with normal innovations.4 We re-estimate (via ML estimation)
the model parameters at each step, as would typically be done in actual applica-
tions.
The overall density forecasting performance of competing models can be compared by
evaluating their conditional densities at the future observed value rt+1, i.e., ^ft+1|t(rt+1).
A model will fare well in such a comparison if realization rt+1 is near the mode of ^ft+1|t()
and if the mode of the conditional density is more peaked. The conditional densities are
determined not only by the specification of the mean and GARCH equations, but also by
the distributional choice for the innovations.
Table 4 presents the means, standard deviations and medians of the density values
^ft+1|t(rt+1), t = 2000,...,T - 1, for each currency. Based on the means, values corTable
4
Comparison of overall predictive performancea
British Canadian German Japanese Swiss
Mean
Normal 0.4198 1.1248 0.4064 0.4796 0.3713
t 0.4429 1.1871 0.4258 0.5207 0.3851
S, 0.4380 1.1798 0.4213 0.5173 0.3820
Standard deviation
Normal 0.1934 0.5697 0.1888 0.1988 0.1620
t 0.2325 0.6802 0.2151 0.2782 0.1840
S, 0.2189 0.6482 0.2016 0.2662 0.1771
Median
Normal 0.4291 1.0824 0.4178 0.5172 0.3942
t 0.4483 1.1500 0.4452 0.5261 0.4069
S, 0.4493 1.1730 0.4477 0.5242 0.4041
a The entries represent average predictive likelihood values, T -1
t=2000
^ft+1|t (rt+1).
3 A conditionally varying location parameter, t , would be handled analogously.
4 Since the sample sizes, T , of the five currencies vary, the number of forecasts ranges from 1,621 to 1,682.
400 S. Mittnik and M.S. Paolella
responding to the S, and Studenťs t assumptions are extremely close, with the Studenťs
t values nevertheless larger in each case. Based on the medians, however, the stable
Paretian model is (slightly) favored by the British, Canadian and German currencies. Notice
that this is contrary to the model selection based on the goodness of fit measures; both
AICC and AD statistics favored use of stable Paretian innovations for the Japanese yen and
Studenťs t innovations for the British pound.
Next, we examine how well the models predict the downside risk. Consider the valueat-risk
implied by a particular model, M, namely
Pr rt+1 -zM
t+1( ) = , t = 2000,...,T - 1. (23)
For a correctly specified model we expect 100% of the observed rt+1-values to be less
than or equal to the implied threshold-values -zt+1( ). If the observed frequency
^ M
:=
1
T - 2000
T -1
t=2000
I(-,-zM
t+1( )](rt+1)
is less (higher) than  , then model M tends to overestimate (underestimate) the risk of the
currency position; i.e., the implied absolute zM
t+1( )-values tend to be too large (small).
The predictive performance for assessing the downside risk achieved by the normal,
Studenťs t and stable Paretian models are compared in Table 5 for the shortfall probabilities
 = 0.01, 0.025, 0.05, 0.10. A comparison of the stable Paretian and Studenťs t
Table 5
Comparison of predictive performance for downside riska
100 Model British German Canadian Japanese Swiss
Normal 1.9036 1.5051 1.3674 1.9124 1.4899
1.0 t 1.3682 0.9031 0.7134 1.4189 1.3707
S, 1.3682 0.9031 1.3080 1.3572 1.2515
Normal 3.0339 2.6490 2.3187 2.8994 3.2777
2.5 t 2.8554 2.9500 2.1403 3.2079 3.3969
S, 2.9149 2.9500 2.4970 2.5910 3.1585
Normal 4.7591 4.5756 3.6266 4.9969 4.7676
5.0 t 5.1160 5.2378 3.9834 5.7372 5.0656
S, 5.1160 5.2378 5.0535 5.2437 5.0656
Normal 8.3879 9.2113 8.5612 8.0814 8.9392
10.0 t 9.8751 10.6562 9.9287 10.3023 10.8462
S, 9.6966 10.4154 10.2259 9.8088 10.2503
a The entries show the observed frequencies ^ M = (T - 2000)-1 T -1
t=2000 I(-,-zM
t+1( )]
(rt+1) multiplied
by 100. For a correctly specified model, we expect ^ M   .
Ch. 9: Prediction of Financial Downside-Risk 401
GARCH models over the five currencies and four cutoff values,  , shows that, in 4 out the
20 cases, the Studenťs t GARCH model outperforms that of the stable Paretian, while the
latter is more accurate in 11 cases, sometimes considerably so (as for the Canadian dollar
with  = 0.025 and 0.05). The remaining 5 cases are tied.
Table 6 presents summary measures5 for the predictive performance of the three models
across all five currencies in the form of the mean error
MEM( ) =
1
5
5
i=1
100 ^ M
i -  ,
mean absolute error
MAEM ( ) =
1
5
5
i=1
100 ^ M
i - 
Table 6
Summary measures for the predictive performancea
100 Model ME( ) MAE( ) MSE( )
Normal 0.6357 0.6357 0.4558
1.0 t 0.1549 0.3083 0.1080
S, 0.2376 0.2764 0.0861
Normal 0.3357 0.4083 0.2209
2.5 t 0.4101 0.5540 0.3527
S, 0.3223 0.3235 0.1633
Normal -0.4548 0.4548 0.4357
5.0 t 0.0280 0.4346 0.3302
S, 0.1433 0.1433 0.0273
Normal -1.3638 1.3638 2.0195
10.0 t 0.3217 0.4002 0.2517
S, 0.0794 0.2772 0.0830
Normal -0.2118 0.7156 0.7830
Aggregate
t 0.2287 0.4243 0.2607
S, 0.1956 0.2551 0.0899
a Shown are the mean error (ME), mean absolute error (MAE) and mean squared error (MSE) of the observed
extreme-tail frequencies from Table 5 across the five currencies. The bottom panel is the aggregate over all
 -values considered.
5 The measures are evaluated for 100 rather than  because the resulting scales of the reported values enhance
readability.
402 S. Mittnik and M.S. Paolella
and the mean squared error
MSEM ( ) =
1
5
5
i=1
1002
^ M
i - 
2
.
The ME's for the normal show that it underestimates the probability of extreme downturns
(MENormal( ) > 0 for  = 0.01, 0.025) and overestimates the probability of moderate
downturns (MENormal( ) < 0 for  = 0.05, 0.10). With one exception, the ME's of the stable
Paretian and Studenťs t GARCH models are smaller (in absolute terms) than those for
the normal. However, they are always positive, indicating, on average, slight underprediction
of the downturn probabilities. For  = 0.01 and  = 0.05, the Studenťs t model has
smaller ME than the stable Paretian model. This is due to the Studenťs t modeĺs offsetting
prediction error for the Canadian dollar for these  -values.
While the ME's indicate possible systematic prediction bias, the MAEs and MSEs reflect
the size of the prediction error. With respect to both measures, the stable Paretian
model dominates those of both the normal and the Studenťs t for all  -values considered.
This is also evident from the bottom panel of Table 6, which aggregates the summary measures
over all  -values considered. In the aggregate, the model using the stable Paretian
innovation assumption outperforms those using the normal and Studenťs t in terms of all
three summary measures.
5. Conclusions
Power GARCH processes driven by either stable Paretian or Studenťs t innovations have
been evaluated and compared in the context of predicting downside market risk, an activity
which is particularly important for risk managers of financial institutions. For all five
exchange-rate series considered, the asymmetric stable Paretian distributional assumption
was found to be superior.
While there exist several popular model classes designed to parsimoniously and effectively
fit financial return data, the GARCH class of models is arguably the most common.
Furthermore, the usual assumption, and that which is implemented in popular software
packages, is that the driving innovations are either normally or Studenťs t distributed.
The former is the "standard" assumption in financial and even most econometric or statistical
models, but fails demonstrably in empirical applications [see, e.g., Palm (1997),
Gouriéroux (1997), and the references therein]. Indeed, normality is a special, limiting case
of the stable Paretian distribution, which, otherwise, allows for fatter-than-normal tails and
skewness, these being precisely two of the typical "stylized facts" associated with financial
returns data. The Studenťs t assumption does allow for fatter tails, but is restricted to being
symmetric. The latter restraint can actually be overcome if more general Studenťs t-like
distributions are used (Paolella, 1999; Mittnik and Paolella, 2000), but these suggestions,
while often providing admirable in- and out-of-sample fits, do not possess the theoretical
Ch. 9: Prediction of Financial Downside-Risk 403
property of summability, common only to the stable Paretian (and, thus, normal) class of
distributions.
With respect to the summability property, one might argue that the value of stable
Paretian models is, as shown here, their improved forecasting ability as compared to competing
models, with such "theoretical niceties" as summability being largely irrelevant. In
a larger context, however, the summability property can often be judiciously used when
building more complex financial models such as those used in portfolio analysis. In such
models, the ad hoc nature of, say, the Studenťs t distribution can become quite problematic.
Further discussion along these lines and a test for the summability property in the
context of GARCH models has been proposed in Paolella (2001) and further applied in
Mittnik, Paolella and Rachev (2000).
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Nelson, D.B., Cao, C.Q., 1992. Inequality constraints in the univariate GARCH model. Journal of Business and
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Chapter 10
STABLE NON-GAUSSIAN MODELS FOR CREDIT RISK
MANAGEMENT
BERNHARD MARTIN
Institute of Statistics and Mathematical Economics, University of Karlsruhe, Germany
SVETLOZAR T. RACHEV
Department of Statistics and Applied Probability, University of California, Santa Barbara, USA
Institute of Statistics and Mathematical Economics, University of Karlsruhe, Germany
e-mail: rachev@lsoe-4.wiwi.uni-karlsruhe.de
EDUARDO S. SCHWARTZ
Anderson School of Management, University of California, Los Angeles, USA
Contents
Abstract 406
1. Stable modeling in credit risk ­ recent advances 407
2. A one-factor model for stable credit returns 408
2.1. Credit risk evaluation for single assets 410
2.2. A stable portfolio model with independent credit returns 411
2.3. A stable portfolio model with dependent credit returns 413
3. Comparison of empirical results 415
3.1. The observed portfolio data 415
3.2. Generating comparable risk-free bonds from the yield curve 415
3.3. Fitting the empirical time series for Ri, Yi , and Ui 416
3.4. CVaR-results for the independence assumption 417
3.5. CVaR-results for the dependence assumption 418
4. The detection and measurement of long-range dependence 422
4.1. Fractal processes and the Hurst-Exponent 423
4.2. The Aggregated Variance Method 425
4.3. Absolute Values of the Aggregated Series 426
4.4. Classical R/S analysis 426
4.5. The modified approach by Lo 427
4.6. The statistic of Mansfield, Rachev and Samorodnitsky (MRS) 430
4.7. Empirical results for long-range dependence in credit data 431
5. Conclusion 439
References 440
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
406 B. Martin et al.
Abstract
Unlike the credit risk models based on the normal assumption, the model in this chapter
assumes credit returns to follow a stable distribution. As empirical studies show, the
daily returns of a bond and its credit spread obey a stable law, exhibiting peaked, heavy
tailed, and skewed distributions. This implies the application of stable Credit Value-at-Risk
(CVaR) in order to obtain a more precise measure for a bonďs risk compared to normal
Credit Value-at-Risk.
Describing the returns of a financial instrument subject to credit risk, our model is based
on the one-factor model proposed by Rachev, Schwartz and Khindanova (2000). It separates
the risky asset into a default free component subject to interest risk and a component
that represents the default risk. For this model, we change the definition of the bonds' returns
and derive the risk of individual corporate bonds directly from their historical market
prices. Thus, we avoid to construct yield curves mapping the individual credit risk of the
observed risky bonds.
In an empirical example consisting of a portfolio with two corporate bonds, we compare
the stable and normal Credit Value-at-Risk. Furthermore, we analyze the effects of
considering the dependence among different instruments compared to the independence
assumption.
The second part of the chapter analyzes the presence of long-range dependence (LRD)
in credit returns using time series of corporate bond indices. For the detection of LRD, we
apply the classical R/S analysis by Mandelbrot and Wallis (1968), the statistic of Lo (1991),
and the Mansfield­Rachev­Samorodnitsky (MRS) statistic (1999). Our results show that
the Hurst-Exponent is greater than 0.5 for all four time series. Under the Gaussian assumption,
the LRD hypothesis is significant for two of the four time series. Allowing a tail-index
 of less than 2, 1 <  2, the applied MRS-statistic also indicates significant LRD for
these two series.
Ch. 10: Stable Non-Gaussian Models 407
1. Stable modeling in credit risk ­ recent advances
Academics and practitioners1 have examined the application of stable distributions for
modeling asset returns. As it is well documented in the literature on empirical finance,2
changes in the value of a financial asset are heavy-tailed and peaked, whereas the mass of
the commonly used normal distribution is located around its center. Therefore, the normal
distribution3 fails to model crashes and strong upturns in financial markets.
Recent research has also examined the returns of instruments subject to credit risk. Those
studies4 found that credit returns are also peaked and heavy-tailed. Moreover, they turned
out to be skewed.
Rachev, Schwartz and Khindanova (2000) suggested the application of stable distributions
for credit instruments to meet those properties. As explained above, for stable distributions,
the peakedness and the heavy tails are determined by the stability index , whereas,
the parameter  is responsible for skewness or asymmetry.
In the following, we propose a model that describes the returns of individual corporate
bonds assuming those to follow a stable law. We are especially interested in determining
the Value-at-Risk (VaR) of such financial instruments subject to credit risk for a given
time horizon. VaR is a measure for the riskiness of an asset and determines the economic
capital required for holding the asset.5 VaR models seek to measure the maximum loss of
value on a given asset or liability over a given time period at a given confidence level (e.g.,
95%). The VaR is defined as a threshold regarding the price change of the instrument over
the observed time horizon. The return over time horizon  is expected to fall below that
threshold with a probability of 1 - c. It says, with a probability of 1 - c the returns are
expected to be less than -VaRc.6 The VaR is expressed as
P p() -VaRc = 1 - c, (1)
with
* p(): price change over time horizon ;
* c: confidence level of VaR, e.g., 95%;
* The probability that losses exceed VaRc is 1 - c.
1 See the work of Mandelbrot (1963), Fama (1965a, b), Fama and Roll (1971).
2 For example, see Rachev and Mittnik (2000).
3 The application of the normal distribution for financial returns goes back to the work of Bachelier (1900).
4 Federal Reserve System Task Force on Internal Credit Risk Models (1998), Basel Committee on Banking
Supervision (1999).
5 Saunders (1999).
6 The VaR is defined as a positive number.
408 B. Martin et al.
2. A one-factor model for stable credit returns
In their model for credit returns, Rachev, Schwartz and Khindanova (2000) assumed a
linear relationship between the returns of a risky credit instrument and the returns of a
comparable risk-free credit instrument.
For such a credit instrument i, the returns are described by
Ri = ai + biYi + Ui, (2)
where
* Ri are the log-returns of an asset i that is subject to credit risk;
* Yi are the log-returns of a risk-free asset;
* Ui is the disturbance. It represents the spread or the premium for the credit risk;
* ai and bi are constants which are obtained by ordinary least squares (OLS) estimation.
In this linear model, the returns of both the risky (Ri) and the risk-free (Yi) credit instrument
are assumed to follow a strictly stable law. Moreover, the disturbance term (Ui) is a
strictly stable random variable:
* Ui  S(,,), 1 <  < 2;
* Yi  S ( , , ), 1 <  < 2.
For credit instruments, the log-return Ri,t at time t is defined as
Ri,t = log
Pi,t,T
Pi,t-1,T -1
, (3)
where Pi,t,T is the price of an instrument i subject to credit risk with maturity date T
evaluated at time t. The log-returns of the riskless asset Yi,t are determined by
Yi,t = log
Bi,t,T
Bi,t-1,T -1
, (4)
with Bi,t,T as the price of the risk-free asset with maturity date T evaluated at time t. This
means that all prices used for the calculation of the returns are determined on the basis of
constant time-to-maturity. Therefore, the time series of log-returns (both Yi,t and Ri,t ) is
calculated such that the time-to-maturity is the same for all t.
It must be noted that Yi,t and Ri,t are not directly observable for individual bonds
whose market price movements are recorded on a daily basis. The prices Bi,t,T ,Bi,t-1,T -1,
Bi,t-2,T -2,... are calculated from the yield curve of riskless treasury bonds and
Pi,t,T ,Pi,t-1,T -1,Pi,t-2,T -2,... are derived from a yield curve generated from risky
bonds representing a similar level of credit risk (e.g., having equal credit ratings). Such
an approach enables us to deal with constant time-to-maturity. This is crucial, because for
the prices of individual bonds, time-to-maturity decreases with increasing time t. However,
a decreasing time to maturity does have an effect on the credit returns. Thus, the advantage
of the approach in (3) and (4) is that we do not have to pay attention on the influence of a
changing time-to-maturity.
Ch. 10: Stable Non-Gaussian Models 409
The effect of changing time-to-maturity on credit returns can be demonstrated by a small
example with two riskless zero-bonds: the first one has a time-to-maturity of one year, the
other one has a time to maturity of two years. Furthermore, the term structure is assumed
to be flat, and therefore, both securities have equal yields. If the yield of both increases by
the same percentage, then the price of the two-year bond reacts more sensitively, compared
to the one-year bond.
However, the approach of modelling the returns as in (3) and (4) is very difficult to implement
in practice. Historical data of daily yield curves is available for treasury bonds, but it
is practically impossible to observe a time series Pt,T ,Pt-1,T -1,Pt-2,T -2,... for an individual
bond. We would have to define a number of different credit risk categories and assign
individual bonds with different maturities to those.7 We would use the prices of the bonds
assigned to the same risk category in order to generate the corresponding yield curve.8
In order to avoid such difficulties, we look for a more practical way to define the credit
returns. Obviously, a risk manager would prefer to deal with the observed real prices of
a bond to fit a model, rather than deriving prices from yield curves that have to be generated
before. Moreover, each yield curve only represents an average credit risk level. Our
approach proposed in the following paragraph determines the individual credit risk of the
analyzed bond.
A new approach to define the returns Ri and Yi . From the historical yield curve data
of treasury bonds, we can construct daily prices for any riskless bond with given coupon,
coupon dates, and maturity. Thus, we can generate a corresponding riskless bond i with
identical specifications for each risky corporate bond i. We define the return Ri,t of a risky
corporate bond as its actual (observable) daily price movement:
Ri,t = log
Pi,t,T
Pi,t-1,T
. (5)
Here, time-to-maturity is no longer kept fixed. The return Ri,t is that of an individual
bond i with fixed maturity date T . The riskless returns Yi,t are defined the same way:
Yi,t = log
Bi,t,T
Bi,t-1,T
. (6)
This riskless bond i has the same specifications (maturity, coupon, coupon dates), as the
risky bond i.
With this new approach, the original linear risk-return relation Ri = ai + biYi + Ui
remains, but its components Ri , Yi , and Ui now have a different meaning. Ri and Yi are
individual bond returns, and the disturbance Ui incorporates both credit spread and the risk
of time-to-maturity.
7 For example, one can use the rating grades assigned by Standard & Poor's or Moody's to define a risk category.
8 For example, see McCulloch (1971, 1975).
410 B. Martin et al.
For all empirical examinations in this chapter, we used the model with the returns defined
in (5) and (6). In the following, we present a brief summary of advantages and disadvantages
of both approaches:
The model whose returns are defined by Equations (3) and (4), abandons the problem
of changing time-to-maturity. This is its main advantage. The disadvantage of such an approach
is that yield curves have to be modelled for a number of different risk levels (e.g.,
corporate credit ratings), and for the risk free (treasury) bonds. After fitting the parameters
a and b of Equation (2), we can simulate future scenarios for each yield curve integrating a
model for the riskless returns. Such a framework would enable us to simulate future daily
returns for each time-to-maturity. With the simulated yield curves, we would then be able
to calculate the future returns of individual bonds.
With our model defined by the returns in (5) and (6) we neither need to construct yield
curves for a number of risk levels (of risky bonds), nor do we have to simulate future
representations of those yield curves by applying a complex term structure model. Thus,
we can directly simulate future returns of individual bonds by generating representations
of Yi and Ui according to their fitted distributions.
The advantage of the chosen approach is that we can work with the actual historical price
data and spread information of the individual bonds, instead of generating yield curves,
each for a certain risk grade. Such yield curves only represent the average of a risk grade.
Studies found that in some cases a higher rated bond can even have a larger credit spread
than bonds with a lower rating grade. This is due to the fact that the range of credit spreads
within a given rating grade can be relatively wide and that the spread ranges of neighboring
grades are usually overlapping. A reason for this effect could be that the market values the
creditworthiness of an issuer differently than the rating agencies do. Sometimes the market
can anticipate a change in the credit quality of an issuer before the rating agencies react.
The construction of a yield curve for a given credit grade usually requires data from a
large number of bonds with various issuers. The yield curve of a single issuer is calculable
even only for large corporations with many issued bonds.
2.1. Credit risk evaluation for single assets
In order to obtain the Credit Value-at-Risk (CVaR) for a bond i over a time horizon of one
period, we perform the following steps:
* We create a corresponding risk-free treasury bond with equal maturity, coupon, and
coupon dates.
* The estimates for ai and bi are calculated with OLSE.
As in Rachev, Khindanova and Schwartz, the estimates are given by
^ai =
T
t=1 Y2
it
T
t=1 Rit - T
t=1 Yit
T
t=1 RitYit
T T
t=1 Y2
it - ( T
t=1 Yit )2
, (7)
^bi =
T T
t=1 Rit Yit - T
t=1 Yit
T
t=1 Rit
T T
t=1 Y2
it - ( T
t=1 Yit )2
, (8)
Ch. 10: Stable Non-Gaussian Models 411
where i = 1,...,N; t = 1,...,T .
With the estimates ^ai and ^bi, we obtain the residuals Ui,
Ui = Ri - ^ai - ^biYi. (9)
* Finally, we perform a stable fit for Ui and Yi.
* In order to calculate the CVaR of asset i for one period, we simulate 1000 representations
of Ri = ai + biYi + Ui.
2.2. A stable portfolio model with independent credit returns
Suppose there are n different credit instruments i (bonds) in a portfolio, and let vi be the
weight of security i within the portfolio.9 The return of the portfolio is given by
Rp =
n
i=1
viRi, (10)
with
Rp =
n
i=1
vi(ai + biYi + Ui) =
n
i=1
viai +
n
i=1
vibiYi +
n
i=1
viUi, (11)
and
n
i=1
vi = 1. (12)
Rp can be expressed by
Rp =
n
i=1
viai + Yp + Up, (13)
with YP and Up given by
Yp =
n
i=1
vibiYi (14)
9 vi can also be negative in case short-selling is allowed.
412 B. Martin et al.
and
Up =
n
i=1
viUi. (15)
The constant ap of the total portfolio is
ap =
n
i=1
viai. (16)
As we assume the Ri to be driven by independent -stable distributions, this also means
that both the Ui and the Yi , i = 1,...,n, are independent of each other. We further assume
that both the Ui and the Yi, i = 1,...,n, are characterized by a common index of stability
( for the Ui,  for the Yi). A common stability index allows an easy analytical solution for
the parameters of the distributions for Up and Yp. For the properties of stable distributions,
see Samorodnitsky and Taqqu (1994).
The common index of stability is calculated as an average from the stability indices of
the distributions of the individual Ui and Yi , weighted according to formula (10):
 =
n
i=1 |vi|i
n
i=1 |vi|
(17)
and
 =
n
i=1 |vi|i
n
i=1 |vi|
. (18)
With the common stability index, the parameters , ,  have to be re-estimated for the
individual Ui and Yi first.
The assumption of independent returns gives us an analytical solution for the portfolio's
Up and Yp.
The parameters of Up and Yp are then determined by the following expressions:
Up =
n
i=1
|vi|Ui

1/
, (19)
Up =
n
i=1[sign(vi)Ui (|vi|Ui )]
n
i=1(|vi|Ui )
, (20)
Up =
n
i=1
viUi , (21)
Ch. 10: Stable Non-Gaussian Models 413
Yp =
n
i=1
|vi ^bi|Yi

1/
, (22)
Yp =
n
i=1[sign(vi ^bi)Yi (|vi ^bi|Yi ) ]
n
i=1(|vi ^bi|Yi )
, (23)
Yp =
n
i=1
viYi . (24)
The portfolio's returns Rp are given by (13).
2.3. A stable portfolio model with dependent credit returns
This section introduces a solution for modelling the dependence between credit returns
on the one hand, and integrating the skewness-property of their distributions on the other
hand.
Each variable Ui and Yi is split into a dependent symmetric and into an independent
skewed component. Both components are independent of each other;
Ui = U
(1)
i + U
(2)
i , (25)
Yi = Y
(1)
i + Y
(2)
i . (26)
By the example of Ui, we show the derivation of the parameters for the two independent
components. Both components are defined to have identical stability indices:
U
(1)
i  S(1,1,0), (27)
U
(2)
i  S(2,2,0). (28)
Because of the independence of U
(1)
i and U
(2)
i , the parameters' values of Ui are calculated
as follows:
 = 
1 + 
2
1/
, (29)
 =
1
1 + 2
2

1 + 
2
. (30)
U(1)
i is symmetric, therefore 1 = 0. We also set equal values for the scale parameters,
1 = 2 = .
Thus, the parameters of Ui are:
 = 21/

, (31)
414 B. Martin et al.
 =
1
2
2. (32)
Summing up the results for the parameters, we have: 1 = 2 =  = 2-1/, 2 = 2
(2 is for the skewed component U
(2)
i ), and 1 = 0 (1 is for the symmetrical component
U
(1)
i ),
U
(1)
i  S 2-1/
,0,0 , (33)
U
(2)
i  S 2-1/
,2,0 . (34)
Analogously, Yi is split into Y
(1)
i +Y
(2)
i , and their parameters are obtained the same way.
The return of the credit instrument i is then given by
Ri,t = a + b Y
(1)
i,t + Y
(2)
i,t + U
(1)
i,t + U
(2)
i,t . (35)
The symmetric components Y
(1)
i,t and U
(1)
i,t are used to incorporate the dependence among
the n assets. The dependence structure of the SS10 vectors (U
(1)
1 ,U
(1)
2 ,...,U
(1)
n ) and
(Y
(1)
1 ,Y
(1)
2 ,...,Y
(1)
n ) is modelled by representing them as sub-Gaussian vectors. Thus,
(U(1)
1 ,U(1)
2 ,...,U(1)
n ) is represented as
U(1)
1 ,U(1)
2 ,...,U(1)
n  A1/2
G1,A1/2
G2,...,A1/2
Gn , (36)
where A is a totally skewed /2-stable random variable with
A  S/2 cos

4
2/
,1,0
and G = (G1,G2,...,Gn) is an n-dimensional Gaussian zero mean random vector.
Let Rij = EGiGj , i,j = 1,...,n, be the covariances within the vector G = (G1,G2,
...,Gn). Then (U(1)
1 ,U(1)
2 ,...,U(1)
n ) is generated by simulating a representation of the
Gaussian vector G with correlated elements G1,G2,...,Gn and an independent representation
of the /2-stable random variable A.11
The generation of vector (Y
(1)
1 ,Y
(1)
2 ,...,Y
(1)
n ) is performed analogously.
10 A SS vector is a symmetrically stable random vector.
11 There are various ways to model the dependence. For example, see Rachev, Khindanova and Schwartz (2000).
Ch. 10: Stable Non-Gaussian Models 415
3. Comparison of empirical results
In order to illustrate the effects of the different assumptions on our stable credit modelling,
we perform an empirical example. We chose a portfolio consisting of two corporate bonds
and calculated its daily Credit Value-at-Risk (CVaR) for the independent and the dependent
case.
3.1. The observed portfolio data
As our sample-portfolio, two corporate bonds (country: US market; currency: US-Dollars)
were selected. Both bonds pay coupons twice a year. Historical prices were obtained from
Bloomberg12 for the past four years (March 14th 1996 up to March 13th 2000). According
to their credit ratings, the bonds exhibit considerable credit risk. For our portfolio, we
assume to have one unit of each security. Both have a nominal value of 100 US $ (see
Table 1).
3.2. Generating comparable risk-free bonds from the yield curve
First, we calculate the daily returns of the above listed bonds using market prices. Then,
for each bond a corresponding riskless bond was generated in order to derive the values
for the Yi. The corresponding riskless bond has the same specifications (maturity, coupon,
coupon date) as the risky corporate bond. The history of daily prices of these artificial treasury
bonds were calculated from the daily yield curves. The treasury-yield curve for each
day was approximated by prices of 9 risk-free zero bonds with maturities: 0.25, 0.5, 1, 2,
3, 4, 5, 7, 10 years. These 9 points were interpolated by a natural cubic spline algorithm.13
Table 1
Bonds selected for sample-portfolio
Corporation Coupon Rating (S&P/Moodys) Maturity
Pennzoil (Bond 1) 10.25 BBB+/Baa2 11/05
United Airlines (Bond 2) 9.0 BB+/Baa2 12/03
Table 2
Estimates for a and b
Corporation ^ai ^bi
Pennzoil (Bond 1) 0.0000 0.9262
United Airlines (Bond 2) 0.0000 0.9878
12 Bloomberg Information System, Corporate Bonds Section.
13 Burden and Faires (1997).
416 B. Martin et al.
With the obtained daily yield curves, we can generate historical prices for our artificial
treasury bonds and calculate their daily returns according to (6).
Next, we perform the linear regressions to estimate the parameters a and b of the equations
Ri = ai + biYi + Ui. The resulting ^ai and ^bi are OLS-estimates (see Equations (7)
and (8), and Table 2).
With the values ^b and ^a the estimates for the disturbances Ui can be calculated: Ui =
Ri - ^a - ^bYi. We now have the empirical distributions for Ri , Yi, and Ui . Next, we apply
both a stable and a normal fit to those.
During the available sample period from 1996 to 2000, time-to-maturity for the observed
bonds reduces by 41% and 52%. The question rises if this has a systematic effect on the
fitted parameters of the Yi over time. However, in our case empirical analysis found no
evidence for this.
3.3. Fitting the empirical time series for Ri, Yi, and Ui
For the stable fit, we applied the Maximum Likelihood Estimation (MLE) to obtain the
four parameters of the distribution. The stable densities were approximated via Fast Fourier
Transformation.14 The procedure was implemented with Matlab 5.3.
Table 3
Parameters for R fitted with stable and normal distribution
Corporation Stable Normal
alpha beta sigma mu mean std-dev
Pennzoil (Bond 1) 1.5451 -0.0690 0.0019 0.0000 -0.0001 0.0041
United Airlines (Bond 2) 1.5199 -0.0744 0.00164 0.0000 -0.0001 0.0035
Table 4
Parameters for Y fitted with stable and normal distribution
Corporation Stable Normal
alpha beta sigma mu mean std-dev
Pennzoil (Bond 1) 1.3639 -0.0297 0.0012 0.0000 -0.0001 0.0027
United Airlines (Bond 2) 1.2811 0.0062 0.0009 0.0000 -0.0001 0.0022
Table 5
Parameters for the disturbance U fitted with stable and normal distribution
Corporation Stable Normal
alpha beta sigma mu mean std-dev
Pennzoil (Bond 1) 1.0348 -0.0247 0.0006 0.0000 0.0001 0.0027
United Airlines (Bond 2) 1.1663 0.0117 0.0008 0.0000 0.0000 0.0032
14 For example, see Rachev and Mittnik (2000).
Ch. 10: Stable Non-Gaussian Models 417
The parameters of the stable and Gaussian distributions fitted for the Ri, Yi , and Ui are
shown in Tables 3, 4, and 5.
3.4. CVaR-results for the independence assumption
The assumption of independence between the bonds in our portfolio leads to the application
of the equations in Section 2.2. We perform the stable fit for both the Yi and the Ui based
on a common stability index, and select  = 1.10 for the Ui, and  = 1.32 for the Yi . Reestimating
the parameters by a stable fit applying common stability indices, we obtain the
results presented in Tables 6 and 7.
The parameters of the portfolio's Up and Yp, given by
Up = v1U1 + v2U2 and Yp = v1 ^b1Y1 + v2 ^b1Y2, (37)
are determined by the relationships presented in Section 2.2. With v1 = v2 = 0.5, we have
Up = 0.5U1 + 0.5U2 and Yp = 0.5 ^b1Y1 + 0.5 ^b1Y2. (38)
Table 6
Parameters for the disturbance U fitted with stable and normal distribution assuming  =
1.10
Corporation Stable
alpha beta sigma mu
Pennzoil (Bond 1) 1.1000 -0.0047 0.0007 0.0000
United Airlines (Bond 2) 1.1000 0.0828 0.0011 0.0000
Table 7
Parameters for Y fitted with stable and normal distribution assuming  = 1.32
Corporation Stable
alpha beta sigma mu
Pennzoil (Bond 1) 1.3200 0.0089 0.0013 0.0001
United Airlines (Bond 2) 1.3200 -0.0430 0.0010 0.0000
Table 8
Stable parameters for portfolio Up and Yp
Stable
alpha beta sigma mu
Yp 1.3200 -0.0137 0.0009 0.0000
Up 1.1000 0.0497 0.0008 0.0000
418 B. Martin et al.
The results for the parameters of Up and Yp are printed in Table 8. Their calculation is
performed according to Equations (19)­(24).
The resulting equation describing the portfolio's returns is
Rp = 0.5(^a1 + ^a2) + Yp + Up = 0.0000 + Yp + Up. (39)
Based on this, we can simulate 1000 daily returns. This provides us the daily Credit
Value-at-Risk of the portfolio. For the stable model with independence assumption, we
obtain a one-day CVaR of 0.67% at the 95% level and a one-day CVaR of 2.24% at the
99% level.
So far, we have assumed both bonds to be independent of each other. However, empirical
examinations exhibit strong dependence among the Yi and among the Ui. Therefore, the
following section presents the results of the model in Section 2.3 incorporating dependence
among the Ui and dependence among the Yi.
3.5. CVaR-results for the dependence assumption
Calculating the Gaussian covariances and correlations between the Ui and Yi of our example
portfolio, the results are presented in Tables 9­12.
The modelling of the dependent case ­ as demonstrated in the former section ­ is performed
by splitting both the Yi and the Ui into two components. The first component
includes the dependence which is modelled by a sub-Gaussian random vector. The second
component exhibits the skewness (see Section 2.3).
Table 13 provides the Credit-Value-at-Risk (CVaR) for the 95% and 99% level with
horizon one day, comparing both stable models (independent and dependent case) with the
empirical data.
Table 9
cov(Yi,Yj ), i,j = 1,2
cov(Yi,Yj )  10-4
Y1 Y2
Y1 0.7699 0.5821
Y2 0.5821 0.4785
Table 10
cov(Ui,Uj ), i,j = 1,2
cov(Ui,Uj )  10-4
U1 U2
U1 0.1038 0.0850
U2 0.0850 0.0748
Table 11
cor(Yi,Yj ), i,j = 1,2
cov(Yi,Yj )
Y1 Y2
Y1 1.0000 0.9591
Y2 0.9591 1.0000
Table 12
cor(Ui,Uj ), i,j = 1,2
cov(Ui,Uj )
U1 U2
U1 1.0000 0.9653
U2 0.9653 1.0000
Ch. 10: Stable Non-Gaussian Models 419
Table 13
Stable portfolio Credit Value-at-Risk (one-day) as log-price and percental price changes
95% 99%
log-price change perc. change (%) log-price change perc. change (%)
Empirical 0.0054 0.54 0.0108 1.08
Dependent 0.0060 0.60 0.0242 2.40
Independent 0.0068 0.67 0.0226 2.24
Table 14
Gaussian portfolio Credit Value-at-Risk (one-day) as log-price and percentile price changes
95% 99%
log-price change perc. change (%) log-price change perc. change (%)
Dependent 0.0061 0.61 0.0087 0.87
Independent 0.0044 0.44 0.0063 0.63
For comparison, Table 14 presents the CVaR assuming the returns to follow a Gaussian
law.
The results for CVaR confirm the earlier findings15 that for credit returns the Gaussian
VaR is only acceptable for the 95% level, but does underestimate the 99% level. The stable
VaR is also appropriate for the 95% level, but it is a more conservative measure for the
99% level. This is actually good because the empirical VaR tends to underestimate the true
VaR due to the low number of observations in the tails.16
Calculating the CVaR also for longer horizons, e.g., 10 days, we would have to build the
10-day returns for both the corporate bonds and their corresponding treasury bonds from
the empirical data, and fit the above models with those data. It has to be pointed out that
longer horizons cannot be calculated by taking the one-day return model and extend it to
the desired horizon by simply applying a Lévy process with independent increments. Subsequent
observations of the returns are not i.i.d. as volatility clustering can be observed and
long-memory effects might occur. Thus, the volatility for a multiple-day horizon cannot be
obtained by a simple scaling approach.17 Longer forecast horizons require new types of
models while sample data should be available for longer periods.
So far we have dealt with the phenomenon of heavy tails in credit returns. Two others,
volatility clustering and long-range dependence, have already been mentioned. The following
part of the chapter explains the phenomenon of long-range dependence. It introduces
the theory and possible ways of detection. Finally, we examine such behavior for credit
return data.
15 Rachev, Schwartz and Khindanova (2000).
16 See Rachev and Mittnik (2000).
17 See Christoffersen, Diebold and Schuermann (1998).
420 B. Martin et al.
Fig.1.Stablemodelsforindependentanddependentcase:densityofdailylog-returns(portfolio).
Ch. 10: Stable Non-Gaussian Models 421
Fig.2.Stablemodelsforindependentanddependentcase:densityofdailylog-returns(portfolio)­tailsonleft-handside.
422 B. Martin et al.
4. The detection and measurement of long-range dependence
Time series can have a long memory. Those systems are not independently identically
distributed. This phenomenon is often referred as burstiness in the literature.18 The underlying
stochastic processes for such burstiness are called fractal. Fractal processes with a
long memory are called persistent. A common characteristic of those fractal processes is
that their space time is governed parsimoniously by power law distributions. This effect
is called the "Noah-Effect", explaining the occurrence of heavy tails and infinite variance.
It can be observed as the tendency of time series for abrupt and discontinuous changes.
Another property of fractal processes is hyperbolically decaying autocorrelations, which is
known as the "Joseph-Effect". It is the tendency of a persistent time series to have trends
and cycles.
The examination of fractal processes in finance has become a popular topic over the
years.19 For a long-memory process, we observe that larger-than-average representations
are more likely followed by larger-than-average representations instead of lower-thanaverage
representations. Hurst developed a statistic to examine the long memory of a stochastic
process. As significant autocorrelations are often not visible, he came up with a
new methodology to provide a measure (the Hurst-Exponent) for long-range dependence
within a time series.
Due to the failures of traditional capital market theory which is largely based on the
theory of martingales, researchers experienced that markets do not follow a purely random
walk. The fractal market hypothesis was developed. The existence of self-similar structures
is a major component of it. For self-similar processes, small increments of time are
statistically similar to larger increments of time.
Self-Similarity is defined as follows:20 Let Xt be a stochastic process with a continuous
time t. Xt is self-similar with self-similarity parameter H (H -ss), if the re-scaled process
with time scale ct, c-H Xct , is equal in distribution to the original process Xt ,
Xt
d
= c-H
Xct . (40)
This means, for a sequence of time points t1,...,tk and a positive stretch factor c,
the distribution of c-H (Xct1 ,...,Xctk ) is identical with the one of Xt1,...,Xtk . In other
words, the path covered by a self-similar process always looks the same, regardless of the
scale it is observed with. In terms of financial data this means: no matter if we have intraday,
daily, weekly, or monthly data, the plots of the resulting processes have similar looks.
For further information on self-similarity we refer to Samorodnitsky and Taqqu (1994), or
Beran (1994).
18 Willinger, Taqqu and Erramilli (1996).
19 For example, we refer to Mandelbrot (1997a, b, 1999), and Peters (1994).
20 Beran (1994).
Ch. 10: Stable Non-Gaussian Models 423
4.1. Fractal processes and the Hurst-Exponent
First, we consider a process without a long memory. A perfect example is Standard Brownian
Motion, which is characterized as a standard random walk. 21 Commonly known is
Einstein's "to the one-half" ­ rule, describing the distance covered by a particle driven by
Standard Brownian Motion. It states that the distance between consecutive values of the
observed time series of this particle is proportional to the square root of time:22
R  T 0.5
. (41)
The power of 0.5 refers to the Hurst-Exponent which is already known as the selfsimilarity
parameter. For Standard Brownian Motion, the Hurst-Exponent H is equal to
0.5 which means that we have an unbiased random walk. A process with a Gaussian limiting
distribution but a Hurst-Exponent H different from 0.5 is called Fractional Brownian
Motion. Fractional Brownian Motion differs from Standard Brownian Motion by the fact
that it is a biased random walk. The odds are biased in one direction or the other.
Definition of Fractional Brownian Motion. 23 Let us assume a self-similar Gaussian
process with Xt , t  R, having mean zero and the autocovariance function
Cov(Xt1 ,Xt2) =
1
2
|t1|2H
+ |t2|2H
- |t1 - t2|2H
VarX(1), (42)
where H is the self-similarity parameter and H  (0,1).
Such a process is called a Fractional Brownian Motion. For H = 1/2 it becomes a Standard
Brownian Motion.
The increments of Fractional Brownian Motion, Yj = BH (j + 1) - BH (j), j  Z, form
a stationary sequence Yj which is called Fractional Gaussian Noise.
Fractional Gaussian Noise. A sequence of Fractional Gaussian Noise has the following
properties:
(i) its mean is zero,
(ii) its variance EY2
j = EB2
H (1) = 2
0 , and
(iii) its autocovariance function is
r(j) =
2
0
2
(j + 1)2H
- 2j2H
+ (j - 1)2H
,
where j  Z, j 0, and r(j) = r(-j) for j < 0.
21 See Campbell, Lo and McKinlay (1997).
22 Peters (1994).
23 Samorodnitsky and Taqqu (1994).
424 B. Martin et al.
For j  , r(j) behaves like a power function.
lim
j
r(j)  0. (43)
The autocorrelations are given by
(j) =
1
2
(j + 1)2H
- 2j2H
+ (j - 1)2H
, (44)
where j 0 and (j) = (-j) for j < 0. As j tends to infinity, (j) is equivalent to
H(2H - 1)j2H-2.24
In the presence of long memory, 1/2 < H < 1, the correlations decay to zero so slowly
that they are no more summable:

j=(j)
= . (45)
For H = 1/2, i.e., a Gaussian i.i.d. process, all correlations at non-zero lags are zero.
For 0 < H < 1/2, the correlations are summable, and it holds:

j=(j)
= 0. (46)
H = 1 implies (j) = 1. For H > 1, the condition -1 (j) 1 is violated.
For 0 < H < 1, a Gaussian process with mean zero and the given autocovariance function
is self-similar and has stationary increments (H -sssi). The above autocovariance function
is shared by all Gaussian H -sssi processes.
Fractional processes with stable innovations. There are many different extensions of the
Fractional Brownian motion to the -stable case with  < 2. Most common is the so called
Linear Fractional Stable Motion or, Linear Fractional Lévy Motion.
In an analogy to the Gaussian case with  = 2, the increments of Linear Fractional
Stable Motion25 show long-range dependence for H > 1/. LRD for  < 1 does not exist,
as H must lie in (0,1). Processes with H = 1/ are called -stable Lévy Motion whose
increments X(tj+1) - X(tj ) are all mutually independent.
For -stable Lévy processes with infinite variance, we carefully have to interpret the
value obtained for H and how it is related to the parameter d measuring the degree of
long-range dependence.
24 Beran (1994).
25 Samorodnitsky and Taqqu (1994).
Ch. 10: Stable Non-Gaussian Models 425
H , the Hurst-Exponent, is the scaling parameter and describes asymptotical self-simi-
larity:
For finite variance processes, the relation between H and d is
H = d +
1
2
. (47)
For processes with infinite variance ( < 2), the relation is
H = d +
1

. (48)
If d > 0, the time series is governed by a long-memory process.
There is a number of methods to distinguish a purely random time series from a fractal
one. For example, the classical R/S analysis26 determines the parameter H of a time series.
The resulting graph is called pox-plot of R/S or rescaled-adjusted-range plot.
Before the classical R/S method will be described, we briefly explain two other methods
to derive the Hurst-Exponent H , the Aggregated Variance Method and the similar method
Absolute Values of Aggregated Series.27
4.2. The Aggregated Variance Method
The original time series X = (Xi, i = 1,...,N) is divided into blocks. Each block has the
size m elements. The index k labels the block. The aggregated series is calculated as the
mean of each block:
X(m)
(k) =
1
m
km
i=(k-1)m+1
Xi with k = 1,2,...,
N
m
. (49)
After building the aggregated series, we get the sample variance of X(m)(k) as
VarX(m)
=
1
N/m
N/m
k=1
X(m)
(k)
2
-
1
N/m
N/m
k=1
X(m)
(k)
2
. (50)
The procedure is repeated for different values of m {mi, i 1}. The chosen values for
m should be equidistant on a log-scale, i.e., mi+1/mi = C.
As X(m) scales like m(H-1), the sample variance VarX(m) behaves like m(2H-2). Thus,
plotting a log-log representation of m and VarX(m), the plots form a straight line with slope
2H - 2.
26 Mandelbrot and Wallis (1968).
27 Teverovsky, Taqqu and Willinger (1995) as well as Teverovsky, Taqqu and Willinger (1998).
426 B. Martin et al.
4.3. Absolute Values of the Aggregated Series
This method is similar to the Method of Aggregated Variance explained above. Starting
again with the aggregated series, we calculate the sum of the absolute values of the aggregated
series.
1
(N/m)
N/m
k=1
X(m)
(k) . (51)
If the original series has a long-range dependence parameter H , the log­log-plot of m
versus the corresponding values of the statistic provides us with a line of slope H - 1.
4.4. Classical R/S analysis
Let us assume we have a time series of N consecutive values. Y(n) = n
i=1 Xi, n 1, is
the partial sum and S2(n) = 1
n
n
i=1[Xi - n-1Y(n)]2, n 1, is the corresponding sample
variance.
We define Z(t) = Y(t) - t
n Y(n). The rescaled-adjusted-range statistic or R/S statistic
is defined by
R
S
(n) =
1
S(n)
max
0 t n
Z(t) - min
0 t n
Z(t) . (52)
R/S is called the rescaled adjusted range as its mean is zero, and it is expressed in terms
of the local standard deviation. For large n, the expected value of the statistic approaches
c1nH :
E R/S(n)  c1nH
, (53)
where c1 is a positive, finite constant and does not depend on n. In case of long-rangedependence
in a Gaussian process, the values for H range in the interval (0.5,1.0). For
an i.i.d. Gaussian process (i.e., pure random walk) or a short-range dependent process, the
value of R/S(n) approaches c2n0.5. c2 is independent of n, finite, and positive.
E R/S(n)  c2n0.5
. (54)
The practical application of the R/S analysis is performed graphically. It is described in
Mandelbrot and Wallis (1968).
With this procedure K different estimates of R/S(n) are obtained. It starts with dividing
the total sample of N consecutive values into K blocks, each of size N/K. We define
k(m) =
(m - 1)N
K
+ 1 (55)
Ch. 10: Stable Non-Gaussian Models 427
as the starting points of each block, where K is the total number of blocks and m =
1,...,K is the current block number. Now we compute the R(n,k(m))/S(n,k(m)) for each
lag n such that k(m) + n < N. All data points before k(m) are ignored in order to avoid the
influence of particular short-range dependence in the data.
Plotting the log(R(n,k(m))/S(n,k(m))) for each block versus log(n), we can estimate the
slope of the fitted straight line. The classical R/S analysis is quite robust against variations
in the marginal distribution of the data. This is also true for data with infinite variance.
Calculating the Hurst-Exponent H and the stability index  of the process innovations,
the long-range dependence parameter d is obtained by
d = H -
1
2
, (56)
for finite variance ( = 2), and b
d = H -
1

, (57)
for infinite variance ( < 2).
Long-range dependence occurs if d is greater than 0.
The R/S analysis is a nonparametric tool for examining long-memory effects. There is no
requirement for the time series' underlying limiting distribution. In case of an underlying
Gaussian process ( = 2), a Hurst-Exponent of H = 0.5 implies that there is no long-range
dependence among the elements of the time series.
For 0.5 < H < 1, a Gaussian time series is called persistent.28 A persistent time series is
characterized by long-memory effects. If long memory is present, the effects occur regardless
of the scale of the time series. All daily changes are correlated with all future daily
changes, and all weekly changes are correlated with all future weekly changes. The fact
that there is no characteristic time scale is an important property of fractal time series.
0 < H < 0.5 signals an antipersistent system for finite variance. Such a system reverses
itself more frequently than a purely random one. At the first glance, it looks like a meanreverting
process. But this would actually require a stable mean, which is not the case in
such systems.
4.5. The modified approach by Lo
Hursťs R/S statistic turned out to react sensitively towards short-memory processes. Thus,
Lo (1991) modified the classical R/S statistic, now showing robustness towards short-range
dependence.29 Lo's statistic only focuses on lag n = N, the length of the series.30 Multiple
lags are not analyzed, the statistic does not vary n over several lags < N.
28 Peters (1994).
29 Lo (1991).
30 Teverovsky, Taqqu and Willinger (1998).
428 B. Martin et al.
Compared to the graphical R/S method, which delivers an estimate of the parameter H ,
Lo's modified statistic just indicates the presence of long-range dependence, but does not
deliver an estimate of the Hurst-Exponent. The statistic performs a test of the hypotheses
* H0: no long-range dependence.
Instead of the ordinary sample standard deviation S for normalization, there is an adjusted
standard deviation Sq in the denominator. Sq considers the elimination of short term
memory to the statistic. As it is known that the R/S statistic responds very sensitively towards
short range dependence, the influence of short range dependence can be offset by
normalizing R with a weighted sum of short-lag autocovariances. To the variance S2 Lo
added weighted autocovariances up to order q.31 His modified statistic Vq(N) is defined
by
Vq(N) = N-1/2 R(N)
Sq(N)
, (58)
with
Sq(N) = S2 + 2
q
j=1
wj (q) ^j , (59)
where ^j is the autocovariance of order j for the observed time series. wj (q) is defined as
wj (q) = 1 -
j
q + 1
with q < N. (60)
The statistic Vq(N) is applied for a hypothesis test. It checks if the null hypothesis of
the test can be rejected or not, given a certain confidence level. The two hypotheses are:
* H0: no long-range dependence present in the observed data, 0 < H 0.5.
* H1: long-range dependence is present in the data, 0.5 < H < 1.
The statistic assumes a Gaussian process ( = 2). In cases where the value of Vq(N) lies
inside the interval [0.809,1.862], H0 is accepted as the statistic is in the 95% acceptance
region. For Vq(N) outside the interval [0.809,1.862], H0 is rejected.
Lo's results are asymptotic assuming N   and q = q(N)  .32 However, in practice
the sample size is finite and the value of the statistic depends on the chosen q. Thus,
the question arises, what would be the proper value for q in order to perform the hypothesis
test? Andrews (1991) has developed a data driven method for choosing q:33
qopt =
3N
2
1/3
2 ^
1 - ^2
2/3
, (61)
31 Peters (1994).
32 Teverowsky, Taqqu and Willinger (1998).
33 See Lo (1991).
Ch. 10: Stable Non-Gaussian Models 429
here [] stands for the greatest integer smaller than the value between. ^ is the first order autocorrelation
coefficient. Therefore, choosing Andrews' q assumes that the true underlying
process is AR(1).
Critique of Lo's statistic. Lo's statistic is applied by calculating Vq for a number of lags
q, plotting those values against q. The confidence interval for accepting H0 at the 95%
confidence level is plotted as well.
Simulations have shown that the acceptance of H0 (and therefore the value of Vq(N))
varies significantly with q. Taqqu, Willinger and Teverowsky (1998) found that the larger
the time series and the larger the value for q, the less likely H0 is rejected.
Whereas, Lo's statistic just checks for the significance of long-range dependence, the
graphical method of the classical R/S provides relatively good estimates of H .
For small q the results of Vq usually vary strongly. Then a range of stability follows
after the so called "extra" short-range dependence has been eliminated, and the only effect
measurable for the statistic would be long-range dependence.
Applying the statistic to Fractional Brownian Motion with H > 0.5, which is a purely
long-range dependent process without short memory effects, Vq is expected to stabilize at
very low values of q. Unfortunately this could not be confirmed by the testing of Taqqu,
Willinger and Teverowsky (1998). Moreover, they demonstrate that ­ if q is large enough
­ the following holds for Vq(N) and q1/2-H :
Vq(N) q1/2-H
. (62)
For H > 0.5, Vq decreases with increasing q. Even for strongly fractional processes
with time series containing 10000 samples, Taqqu, Willinger and Teverowsky found
that, with increasing values for q, the probability of Vq lying inside the H0 95% confidence
interval and accepting the null-hypothesis grows. To mention three cases only: for
q = 500 and H = 0.9 the null-hypothesis (no long-range dependence) is accepted with
90% for Fractional Brownian Motion, with 92% for FARIMA(0.5,d,0), and with 94% for
FARIMA(0.9,d,0).34
Lo's test is very conservative in rejecting the null-hypothesis. It works for short-range
dependence, but in cases of long-range dependence it mostly accepts the null-hypothesis.
The statistic of Lo is certainly an improvement compared to the short-range sensitive classical
R/S, but should not be used isolated without comparing its results with other tests for
LRD.
In practical applications, the question for a proper choice of q remains. The value of
Andrews' data driven qopt depends on the econometric model underlying the observed
time series, but, the appropriate model is not known in advance. Andrews' choice bears the
assumption that the time series obeys an AR(1) process.
34 FARIMA(0.5,d,0) means a fractional ARIMA process with an AR(1) coefficient of 0.5 and an MA(1) coefficient
of 0.
430 B. Martin et al.
It used to be a common way to asses long-range dependence by looking at the rate at
which the autocorrelations decay. With a Hurst-Exponent H different from 0.5 the correlations
are no longer summable. Such non-summability of autocorrelations used to be seen
as a comfortable way of assuming long-range dependence. But there are pitfalls: if the underlying
process is considered to follow a stable law with  < 2, a second moment does
not exist and therefore autocorrelations do not exist either.
It can be concluded that ­ if testing for long-range dependence ­ the application of a
single technique is insufficient.
4.6. The statistic of Mansfield, Rachev and Samorodnitsky (MRS)
Long-range dependence means that a time series exhibits a certain kind of order over a
long comprehensive period. Instead of pure chaos with no rule in the price movements of
an asset, we can find periods of time with its sample mean significantly different from the
theoretical mean. The stronger the long-memory effects in the time series, the longer an
interval of the series whose mean deviates from the expected value.
Mansfield, Rachev and Samorodnitsky (1999) concentrate on this property of LRDexhibiting
time series. This property of LRD is valid regardless of the assumed underlying
stochastic model.
The authors define a statistic that delivers the length of the longest interval within the
time series, where the sample mean lies beyond a certain threshold. The threshold is set
greater than the finite mean EXi of the whole time series. Furthermore, the time series is
assumed to follow a stationary ergodic process.
Expressed in mathematical terms, the statistic is defined as
Rn(A) = sup j - i: 0 i < j n,
Xi+1 +  + Xj
j - i
 A , (63)
which is defined for every n = 1,2,.... If the supremum is taken over the empty set, the
statistic is defined to be equal to zero.
The set A is defined either as
A = (,) with  > , (64)
or as
A = (-,) with  < , (65)
where  is the theoretical mean of the time series.
Rn(-,) and Rn(,) are interpreted as "greatest lengths of time intervals when
the system runs under effective load that is different from the nominal load".35 In the
following, the examination is restricted to Rn(,).
35 Mansfield, Rachev and Samorodnitsky (1999).
Ch. 10: Stable Non-Gaussian Models 431
A theoretical way to examine a time series for long-range dependence would be the
log­log plot of Rn(,) versus n. In the case of long-range dependence, the slope of the
plot would be expected to be greater than 1/ with  as the tail index. However,  is not
known in advance. Therefore, Mansfield, Rachev and Samorodnitsky developed a statistic
that does not rely on an a-priori tail index. They defined
Wn() =
Rn(,)
Mn
, (66)
where Mn = max(X1,...,Xn) is the largest of the first n observations, n 1. This statistic
has a self-normalizing nature, and because of the denominator it has the ability to
compensate for the effects of the tail-index .
In case of short-range dependence, the ratio Wn() approaches a weak limit as n  .
In case of long-range dependence, Rn grows faster than Mn and the statistic diverges.
For visualization, the statistic Wn() is plotted against . Its limiting distribution is
independent of . A difficult task is the selection of the proper range of . It has to be
determined empirically by looking where the values for Wn() stabilize.
Once the value of the statistic is at least 19 for a certain  then long-range dependence
is present at a significance level of 0.05.
4.7. Empirical results for long-range dependence in credit data
For our empirical examination of long-memory effects in daily credit return data, we use
the returns of bond indices provided by Merill Lynch.36 We have selected four indices with
time series of daily index prices from January 1988 to April 2000. Each index represents a
number of bonds with similar properties (see explanation in Table 15). As the analysis of
long-memory effects requires large data samples, an important criterion for the selection
of an index was the available sample size. The sample sizes are listed in Table 16.
We apply three different methods for estimating the self-similarity parameter H and two
methods performing a hypothesis test regarding the presence of LRD. As explained before,
we have chosen
(i) the "Aggregated Variance Method",
(ii) the method "Absolute Values of Aggregated Series",
(iii) the classical R/S analysis developed by Mandelbrot and Wallis,
(iv) Lo's modified R/S statistic, and
(v) the statistic of Mansfield, Rachev and Samorodnitsky (MRS).
All these methods have been implemented with Matlab 5.3. Methods (i)­(iii) provide an
estimate of the Hurst-Exponent H . Method (iv) tests if the null hypothesis "no long-range
dependence" has to be accepted or rejected at a given confidence level. Method (v) is also
a hypothesis test, however, contrary to Lo's test it works independently of the tail index.
36 The time series were obtained via Bloomberg's Index Section.
432 B. Martin et al.
Table 15
Explanation of the selected indices
Index Explanation
X0H0 High Yield 175
C8B0 Corporates C rated, cash pay
J0A3 AAA-AA rated corporates, time to maturity 15 yrs
C0A0 US Corporate master
Table 16
Data sets used for testing LRD
Index No. of observations Starting date Ending date
X0H0 3083 10­31­86 04­30­00
C8B0 3470 10­31­86 04­30­00
J0A3 2920 08­04­88 04­30­00
C0A0 3472 01­04­88 04­30­00
Table 17
Results for Aggregated Variance and Absolute Values of the Aggregated Series
Index H for Aggreg. Variance H for Abs. Values of Aggreg. Ser.
X0H0 0.7632 0.7596
C8B0 0.5527 0.5511
J0A3 0.8070 0.8022
C0A0 0.5856 0.5838
Testing the index-returns for long-range dependence, we computed the daily changes of
the index log-prices
rt = log(pt ) - log(pt-1). (67)
The results of the methods "Aggregated Variance" and "Absolute Values of the Aggregated
Series". For methods (i) and (ii), we plotted the values of the statistic over m (number
of elements in each block), with m ranging from 10 up to 40. Finally, we determined the
slope of the data points in order to obtain H . The values for H are printed in Table 17. Both
methods calculate Hurst-Exponents greater than 0.5 for all observed indices. Thus, under
the Gaussian assumption, the underlying processes are long-memory processes. X0H0 and
J0A3 show strong LRD, whereas C8B0 and C0A0 have a weaker long memory.
The results of classical R/S and Lo's statistic. As we only have about 3000 observations
for each time series, we do not divide the data set into several blocks for the classical R/S
statistic. Thus, we choose K = 1.
The results of classical R/S and the values of Lo's statistic Vq (for q we chose a range
of 1,...,50) are presented in Table 18. We plotted both the log(R/S)­log(n), and the
Ch. 10: Stable Non-Gaussian Models 433
Table 18
Results for the classical R/S statistic and Lo's test
Classical R/S Lo's statistic
Index Fitted H Range of Vq (q = 1,... ,50) Optimal q (Andrews)
XOHO 0.7579 [1.74, 3.44] 11
C8B0 0.4874 [1.33, 1.40] 6
J0A3 0.9213 [2.04, 4.29] 10
C0A0 0.4493 [1.23, 1.40] Vq­q
graphs for our observed indices X0H0, C8B0, J0A3, and C0A0 (see Figures 3­6
for classical R/S, and Figure 7 for Lo's test). The second column of Table 18 presents
the Hurst-Exponent estimated with the R/S statistic. In the third column the table presents
the intervals in which the values of Lo's Vq are located for q = 1,...,50. The fourth
column provides the optimal lag q, determined by Andrews' data driven method.37 The
results of the R/S-statistic are similar to the ones obtained by the Aggregated Variance
Method and the Absolute Values of the Aggregated Series. The time series of X0H0 and
J0A3 exhibit strong LRD according to their Hurst-Exponent H . This is supported by the
result of Lo's test that rejects the null-hypothesis "no LRD" at the 95% level. However,
for C8B0 and C0A0, the Hurst-Exponent appears already in the area of antipersistence.
Another interesting finding is that for C0A0 ­ which has the lowest value for H ­ the
sample autocorrelation of order 1 is negative. Therefore, we cannot calculate the optimal
q for C0A0.
4.7.0.1. The results of the statistic by Mansfield, Rachev and Samorodnitsky (MRS). Figures
8­11 show the plots of Wn() over the range of . For the time series of the index
X0H0, we found that the statistic Wn() is linearly increasing with  in the range
of [0.5 e-4,3.5 e-4] (the empirical mean of the whole series is 0.497 e-4). The value of
Wn() reaches levels of about 19 and then declines until it stabilizes at a level of about 1
(see Figure 8). This result clearly indicates the presence of LRD. The presence of long
memory is significant at the 0.05 level once the value of the statistic is at least 19. Thus,
the MRS-statistic supports the LRD-hypothesis for X0H0. Lo's statistic and classical R/S
also indicate long-range dependence for the index X0H0, but this was based on the assumption
that the underlying process of the time series follows a Gaussian law, i.e., that
 = 2. However, the MRS-statistic is independent of .
The second bond index that exhibits strong LRD in its returns with the former tests,
was the J0A3-index (C rated corporates). Its empirical mean is -3.2 e-4. We observe a
sharp increase of W() for   [0,6.5 e-4] up to a value of about 15, and it finally drops
to a level of about 1 as well. Thus, the hypothesis of long-range dependence can also be
confirmed for the J0A3-series as the MRS-statistic also exhibits significant values (see
Figure 9). However, the significance is not as strong as for the X0H0-series.
37 See Lo (1991).
434 B. Martin et al.
Fig. 3. Plot of log(R/S)­log(n) for X0H0.
Fig. 4. Plot of log(R/S)­log(n) for J0A3.
Ch. 10: Stable Non-Gaussian Models 435
Fig. 5. Plot of log(R/S)­log(n) for C0A0.
Fig. 6. Plot of log(R/S)­log(n) for C8B0.
436 B. Martin et al.
Fig.7.TheplotsofLo'sstatisticforX0H0,J0A3,C0A0,andC8B0.
Ch. 10: Stable Non-Gaussian Models 437
Fig. 8. Plot of W()­ for X0H0.
Fig. 9. Plot of W()­ for J0A3.
438 B. Martin et al.
Fig. 10. Plot of W()­ for C0A0.
Fig. 11. Plot of W()­ for C8B0.
Ch. 10: Stable Non-Gaussian Models 439
The returns of the two other indices, C0A0 and C8B0, do not exhibit long-range dependence
with the W()-statistic, and this is consistent with the results of the formerly
applied tests. The returns of the C8B0-index show a higher probability for the LRDhypothesis
than the returns of C0A0, however, both are not significant. Thus, for both
indices C0A0 and C8B0, there is no significant indication for long-range dependence with
the MRS-statistic (see Figures 10 and 11).
5. Conclusion
In the first part of this chapter we have shown the predominant performance of Value-atRisk
models based on stable distributions compared to Gaussian models. Furthermore, we
have presented a modified model for credit returns which makes practical implementation
easier.
In the second part of the chapter we have studied long-range dependence in credit return
data.
In Section 3 we have illustrated that the stable distribution much better approximates the
tail of the empirical distribution of credit returns. This is especially important for Value-atRisk
(VaR) applications. VaR has become increasingly important for risk management. The
stable VaR exhibits excellent performance for the high quantiles (i.e., 99% VaR). While
the Gaussian 99% VaR underestimates the empirical VaR, the stable 99% VaR slightly
overestimates it. Thus, the heavy-tailedness property of time series of credit returns is
captured very uniquely by the application of non-Gaussian stable distributions, as well as
the skewness property. The stability indices of the fitted corporate bond returns lie in the
range of 1.5­1.6, which clearly indicates heavy-tailedness.
In this context, a slightly modified model for credit returns has also been presented which
can be implemented without the building of yield curves for various rating grades. It makes
a practical application less burdensome.
The other phenomenon that has been analyzed in this work is the long-memory property
of credit returns (Section 4). A sign of long memory is the "burstiness" of plotted time series.
Long-range dependence is characterized by hyperbolically decaying autocorrelations
and the property that large (small) representations are more likely followed by large (small)
representations than small (large) representations.
While three of the chosen tests measure the Hurst-Exponent, the other two are
hypothesis-tests checking the significance of the LRD-hypothesis.
Applying the methods "Aggregated Variance" and "Absolute Values of Aggregated Series",
all four analyzed time series exhibit a Hurst-Exponent H greater than 0.5, which
means long-range dependence under the Gaussian assumption. For two of the four credit
return series, the modified R/S statistic of Lo confirms LRD to be significant. This is remarkable
because Lo's test tends to confirm the null-hypothesis "no LRD" for large sample
sizes and increasing lag q, even when the actual process is strongly long-range dependent.38
Also allowing infinite variance ( < 2), we apply the MRS-statistic. It analyzes a
38 Teverowsky, Taqqu and Willinger (1998).
440 B. Martin et al.
process for LRD without relying on the tail-index. For the X0H0 and J0A3 series which
have been confirmed for LRD by Lo's test, the MRS-statistic W() also indicates significant
long memory. This is probably the strongest result of our LRD studies which states
that long-range dependence in credit returns is also found to be significant in combination
with the non-Gaussian stable assumption.
Our examinations have only focused on the returns. However, for other financial series
­ such as stock prices ­ LRD has also been discovered in the trading time process, as
demonstrated by Marinelli et al. (1999).
The use of bond indices for the empirical examination, instead of individual bonds, is
advantageous in two respects: First, each index incorporates numerous bonds of a certain
market segment. Thus, the obtained results can then be considered a widespread phenomenon.
If only a small number of bonds within the observed indices would exhibit such an
effect, it would probably fade away. Second, LRD-analysis requires large samples which
are more readily available for indices than for single bonds.
Finally we can conclude that the issue of long memory cannot be neglected for time
series of credit returns. The increments of the underlying stochastic process are not i.i.d.
With the proven LRD in the time series of credit returns and by demonstrating that the
distribution of credit returns is better captured with stable non-Gaussian models, we obtain
a powerful tool to generate accurate forecasts of Value-at-Risk for longer horizons.
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University, New Haven.
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Chapter 11
MULTIFACTOR STOCHASTIC VARIANCE MODELS
IN RISK MANAGEMENT: MAXIMUM ENTROPY APPROACH
AND LÉVY PROCESSES
ALEXANDER LEVIN
Group Risk Management, TD Bank Financial Group, Toronto
e-mail: Alex.Levin@td.com
ALEXANDER TCHERNITSER
Bank of Montreal, Toronto
e-mail: Alexander.Tchernitser@bmo.com.
Contents
Abstract 444
1. Review of market risk models 445
1.1. Market risk management and Value-at-Risk 445
1.2. Statistical properties of the market risk factors 447
1.3. A short review of stochastic volatility models 448
2. Single-factor stochastic variance model 450
2.1. Maximum entropy approach and Lévy processes 450
2.2. Generalized Gamma Variance model 456
2.3. Mean-reverting stochastic variance model 460
3. Multifactor stochastic variance model 463
3.1. Requirements for multifactor VaR models 463
3.2. "Nave" multifactor model 464
3.3. Elliptical stochastic variance model 465
3.4. Independent stochastic variances for the principal components 467
3.5. A model with correlated stochastic variances 468
3.5.1. Example 1. Joint distribution for DEM/USD and JPY/USD FX rates 471
3.5.2. Example 2. Twenty risk factors 471
3.6. Calibration for the GSV model 472
Acknowledgment 477
References 477
 The views expressed in this chapter are those of the authors and not necessarily of the Bank of Montreal.
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
444 A. Levin and A. Tchernitser
Abstract
This chapter investigates a class of multifactor non-normalmodels for Market Risk Management,
and, specifically, for Value-at-Risk (VaR) calculations, with stochastic variance (SV)
driven by Lévy processes. Relevant statistical and dynamic properties for the risk factors
are discussed. A short review of the Market Risk Management requirements and stochastic
models for VaR is presented.
In the case of one asset, a broad class of pure jump Generalized Gamma processes for
the SV is derived from the Maximum Entropy principle. The corresponding family of Lévy
processes for the risk factors (RF) possesses skewed leptokurtic marginal distributions with
a wide range of heavy tails, from exponential and sub-exponential (stretched exponential)
to polynomial. The introduced extended Generalized Gamma Variance family is a two
shape parameter class of conditionally normal symmetric distributions (there is the third
shape parameter in the case of non-zero skewness) with the SV represented as an arbitrary
power (positive, zero or negative) of a gamma distribution. It includes normal, Variance
Gamma (Generalized Laplace), Student t, and Weibull Variance Mixture distributions as
special cases. Ornstein­Uhlenbeck type processes for the SV driven by positive Lévy noise
and the corresponding term structure of the RF kurtosis and quantiles are considered for
the purpose of modelling non-linear dependence in the asset returns.
A general framework for constructing multidimensional conditionally Gaussian stochastic
processes with the correlated multivariate stochastic variances that follow Lévy
processes is considered. This methodology allows for different shape and tail behavior of
the marginal RF and linear sub-portfolio distributions, exact fit into the RF correlation
structure, and proper non-linear scaling of VaR for different holding periods. Presented
empirical evidence for different markets confirms a good agreement between the model
and historical RF distributions. Effective numerical calibration and Monte Carlo simulation
procedures are developed.
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 445
1. Review of market risk models
1.1. Market risk management and Value-at-Risk
Market Risk Management deals with the risk of potential portfolio losses due to adverse
changes in the price of financial instruments caused by stochastic fluctuations of the market
variables (JP Morgan, 1996; Basle Committee on Banking Supervision, 1997; Jorion,
2001; Crouhy, Galai and Mark, 2001). The are many types of general market and specific
risk factors (RF) with different distributional properties and stochastic behavior in the foreign
exchange, interest rate, commodity and equity markets. Market variables include, for
example, stock prices, equity indices, spot foreign exchange rates, commodity prices, as
well as complex aggregate structures: interest rate curves, commodity futures price curves,
credit spread curves, implied volatility surfaces (e.g., European option implied volatility as
a function of strike and maturity) or "cubes" (e.g., swaption implied volatility as a function
of underlying swap tenor, swaption maturity and strike). Also, there are such "wild"
and "exotic" market variables as, for example, electricity prices and interest rate or foreign
exchange rate cross-correlations (the changes of latter variables effect the spread and
cross-currency option prices).
Proper modelling of the multivariate future RF distributions is important for financial
institutions for the purpose of accurate estimation of the market risk, identification of the
risk concentration, developing of trading and hedging strategies, portfolio optimization,
consistent measurement of the risk adjusted performance for different units (Risk Adjusted
Return On Capital (RAROC) and Capital-at-Risk methodologies), setting up the trading
limits, calculating of the regulatory capital (Basle Committee on Banking Supervision,
1997), back-testing of the market risk models required by regulators (Basle Committee
on Banking Supervision, 1996). Many financial institutions need to consistently estimate
market risk for large portfolios and sub-portfolios (aggregation levels) that comprise hundreds
of thousands of instruments dependent on thousands of risk factors in all markets.
These portfolios usually include sub-portfolios of options, which magnify and non-linearly
transform deviations of the underlyings. Modern Market Risk Management is interested in
comprehensivemodelling of the multidimensional risk factor stochastic processes and marginal
distributions for different time horizons rather than static multivariate distributions for
some fixed holding period. This interest comes from the requirements to capture liquidity
risk for many instrument types with varying liquidation periods [see Crouhy, Galai and
Mark (2001)], estimate intraday risk for some frequently rebalanced positions, consistently
evaluate VaR for one-day and ten-day time horizons prescribed by BIS documents (Basle
Committee on Banking Supervision, 1996, 1997) for back-testing and regulatory capital
calculations respectively, and actively dynamically manage risk. This problem points out
on the importance of adequate modelling of a non-linear dependence in the underlying
returns observed in the market to capture a proper VaR term profile.
Along with the RF volatilities (standard deviations of daily changes) and correlations
combined with the portfolio sensitivities [Greeks, Hull (1999)], the most widely accepted
methodology for measuring market risk is the Value-at-Risk approach. The VaR can be
446 A. Levin and A. Tchernitser
defined as the worst possible loss in the portfolio value over a given holding period (1 or
10 days) at the 99% confidence level (Jorion, 2001; Crouhy, Galai and Mark, 2001). Essentially,
a mathematical model for VaR consists of two main parts: (1) modelling of proper
multivariate risk factor distributions (processes) for the required time horizons; (2) evaluation
of the portfolio (linear instruments, options and other derivatives) changes for the risk
factor scenarios to produce a portfolio distribution. The evaluation part can be based on a
full revaluation for the prices of instruments or partial revaluation methodologies [for example,
Delta­Gamma­Vega approximation (Hull, 1999)]. Regulators also require complementing
the VaR analysis with stress testing (scenarios for crashes, extreme movements in
the market, stresses of volatilities and correlations, etc.). Traditional methods of the VaR
calculation are analytical (variance­covariance)method (JP Morgan, 1996), historical simulation
[combined with some bootstrapping procedures or other non-parametric methods
(Crouhy, Galai and Mark, 2001)], and parametric Monte Carlo simulation approach [see
Duffie and Pan (1997)]. Primarily developed for the "normal" market conditions (multivariate
Gaussian distribution for the risk factors), the variance­covariance method can be
applied only for linear portfolios. The variance­covariance method can be extended from
multivariate normal to the non-normal elliptical RF distributions (see Section 3.3). VaR for
option portfolios is usually calculated based on simulation approaches. In this chapter, we
concentrate on the parametric modelling of the RF distributions based on the Monte Carlo
simulation procedures given an appropriate portfolio valuation methodology.
There are some market risk measures other than VaR closely related to the tails of the RF
probability distributions, for example, Expected Shortfall [see Mausser and Rosen (2000)].
The Expected Shortfall is defined as an average loss calculated from the losses that exceed
VaR. The Expected Shortfall, as a conditional mathematical expectation, is an example of
so-called coherent risk measures [see Artzner et al. (1999)] that, contrary to VaR, possess a
natural subadditivity property (total risk of entire portfolio should be less or equal to a sum
of risks of all sub-portfolios). In some cases, Expected Shortfall reflects the market risk
better than VaR (it gives an answer to the question, what is the average of the worst case
losses that occur at the corresponding confidence level). This market risk measure is more
sensitive to the tail behavior than VaR. In general, it is wrong to say that only tails of the
underlying RF distributions are important for the VaR or other risk measures. For example,
a left tail for the portfolio of some barrier options or even European near at-the-money
options may mostly depend on the central part of the underlying distribution. Therefore, it
is a necessity to accurately model all parts of the RF distributions, including peaks at the
origin and tails.
Due to short time horizons utilized in Market Risk Management (1­10 business days)
contrary to Credit Risk Management with usual time horizons of years (Crouhy, Galai
and Mark, 2001; Duffie and Pan, 2001), the market risk factors are defined as daily logreturns,
relative or absolute changes in the underlying prices, rates or implied volatilities,
rather than these underlyings themselves. Such long-term effects as mean-reversion in the
interest rate, commodity price, and implied volatility dynamics (with characteristic times
1­20 years) are not taken into account in the VaR modelling. Most of financial variables
are positive (although, spreads and interest rate differentials may be negative). Except some
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 447
rare situations (e.g., Japanese interest rates), daily changes for the underlyings are much
less than 100% of the notional values, and, therefore, there is no need to apply any positive
transformations to the market variables, like exponential or square transformations.
Heuristically, this means that in most cases one can use "linear" RF simulation models for
the VaR calculation.
1.2. Statistical properties of the market risk factors
There is extensive empirical evidence that historical daily return distributions for different
underlyings in the foreign exchange, interest rate, commodity, and equity markets have
high peaks, "fat" tails (excess kurtosis, Figures 1 and 2) and skewness (right graph on
Figure 2) contrary to the normal distribution [see, for example, Mandelbrot (1960), Fama
(1965), Duffie and Pan (1997), Müller, Dacorogna and Pictet (1998), Barndorff-Nielsen
and Shephard (2000b), Rachev and Mittnik (2000), Bouchaud and Potters (2000), Cont
(2001)]. Also, it is well known that the volatility of these financial variables varies stochastically
with clustering (Bollerslev, Engle and Nelson, 1994) (see Figure 3). These distributional
properties have significant impact on Risk Management, specifically on VaR.
A standard methodology usually used for the VaR calculation (JP Morgan, 1996) exploits
a multivariate normal distribution as a proxy for the RF distributions. The standard model
corresponds to stable market conditions when one can neglect large jumps of the underlyings
and volatility fluctuations. This results in underestimating of the actual VaR by the
standard methodology and breaching the back-testing. A comprehensive RF simulation
model should additionally capture the following important features observed in the mar-
ket:
­ different distributional shapes for different risk factors and markets (for example, short
interest rates have much heavier tails, higher peaks and kurtosis than long term rates even
for the same interest rate curve, Figure 1; some commodity price distributions deviate
more from normal than others);
­ anomalously small normalization effect for large diversified portfolios contrary to the
one predicted by the Central Limit Theorem (for example, S&P 500 Industrial Index
or TSE 300 Index (Figure 2), viewed as large portfolios of stocks, have markedly nonnormal
distributions with kurtosis about ten). This phenomenon points to a non-linear
dependence between different risk factors [see also Embrechts, McNeil and Straumann
(1999)];
­ normalization of the risk factor distributions for longer holding periods [for example,
ten-day return distributions are significantly closer to normal than daily return distributions,
on the other hand, intraday change distributions are clearly more distant from normal
than daily ones (Müller, Dacorogna and Pictet, 1998; Cont, Potters and Bouchaud,
1997; Mantegna and Stanley, 2000)]. A decreasing term structure of kurtosis points out
to the same effect (Duffie and Pan, 1997; Bouchaud and Potters, 2000);
­ volatility clustering and non-linear time dependence in risk factor returns (for example,
statistically significant autocorrelation in squares of virtually uncorrelated daily returns,
see top graph on Figure 3 and Figure 10 in Section 2.3).
448 A. Levin and A. Tchernitser
Fig. 1. Variety of distributional shapes for CAD BA interest rate daily returns.
Fig. 2. Distributions for the CAD/USD FX and TSE 300 daily log-returns.
1.3. A short review of stochastic volatility models
In this chapter we restrict consideration of the SV models to the case of continuous time
models. Time series approaches (ARCH, GARCH, etc.) (Bollerslev, 1986; Bollerslev, Engle
and Nelson, 1994) are beyond the scope of the chapter.
L. Bachelier introduced the normal distribution and Brownian motion in finance in his
Ph.D. Thesis (Bachelier, 1900) more than one hundred years ago. Brownian motion [that
corresponds to a standard model for VaR (JP Morgan, 1996)] was rediscovered in finance
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 449
Fig. 3. Volatility clustering and large deviations in CAD/USD FX rate daily returns.
in Osborne (1959), and then replaced by a Geometric Brownian motion for modelling
of the stock dynamics (Samuelson, 1965). Without any doubt, the Black­Scholes­Merton
(Black and Scholes, 1973) option pricing model has become a main tool in modern finance.
Since well-known investigations of Mandelbrot (1960, 1963) and Fama (1965) on stable
processes in the market, researchers have developed different approaches for modelling
the abnormal behavior of the market variables. Fat-tailed distributions and jumps in the
risk factors have been usually modelled by jump-diffusion processes (Merton, 1976, 1990;
Bates, 1996; Kou, 2000), processes with diffusion stochastic volatility (Hull and White,
1987; Heston, 1993; Stein and Stein, 1991; Bates, 1991; Melino and Turnbull, 1990),
mixtures of normal and other distributions (Duffie and Pan, 1997; Rachev and SenGupta,
1993; Albanese, Levin and Ching-Ming Chao, 1997), and other methods (Hull and White,
1998; Sornette, Simonetti and Andersen, 2000). Also, different types of non-Gaussian
Lévy processes were used to describe the dynamics of underlyings [we refer to Bertoin
(1996), Feller (1966), Lukacs (1970) and Sato (1999) for the theory of infinitely divisible
distributions and Lévy processes]. Stable Paretian models in Finance were considered
in Madelbrot (1960, 1963), Fama (1965), McCulloch (1978, 1996), Mittnik and Rachev
450 A. Levin and A. Tchernitser
(1989), Willinger, Taqqu and Teverovsky (1999), Rachev and Mittnik (2000), and other
works [see also Samorodnitsky and Taqqu (1994), Janicki and Weron (1994), Nolan (1998)
for the theory, simulation and estimation of stable processes]. Since pioneering 1973 paper
of Clark (1973), there have been a lot of research works on subordinated Lévy processes in
finance: VG model (Madan and Seneta, 1990; Madan and Milne, 1991; Madan, 1999); Hyperbolic
and Generalized Hyperbolic models (Barndorff-Nielsen, 1977, 1978, 1997, 1998;
Eberlein and Keller, 1995; Embrechts, McNeil and Straumann, 1999; Eberlein and Raible,
1999) [see also Marinelli, Rachev and Roll (1999), Rachev and Mittnik (2000)]. A fine
structure of asset returns from a Lévy process point of view was considered in Carr et
al. (2000), Geman, Madan and Yor (1999, 1998) (CGMY model), Mantegna and Stanley
(2000), Bouchaud and Potters (2000), Boyarchenko and Levendorskii (2000), BarndorffNielsen
and Levendorskii (2001) (Truncated Lévy Flight). A general theory of conditionally
normal stochastic variance and stochastic time change models is considered in Steutel
(1970, 1973), Rosínski (1991), Maejima and Rosínski (2000), Barndorff-Nielsen and
Pérez-Abreu (2000).
Most papers discuss a one-dimensional case with applications to option pricing. However,
multidimensional models with a large number of risk factors are of significance for
Risk Management. This chapter presents a new class of multivariate VaR models with the
SV driven by Lévy processes.
2. Single-factor stochastic variance model
2.1. Maximum entropy approach and Lévy processes
Let a risk factor X denote a t-day absolute return, relative return, or log-return of the underlying
market variable. Value-at-Risk over a given holding period t with a specified confidence
level q (usually, q = 1%) is defined as a q-quantile of the distribution for the portfolio
changes during the period t. For the standard model, a RF probability density function
is normal with given constant mean and variance. We consider a class of conditional normal
models where the variance V of the risk factor X is stochastic rather than constant.
The stochastic variance of the underlying returns is not directly observable in the market.
Generally, the most reliable information about the SV is its average value over some period
of time. It can be estimated from the sampling variance of the underlying returns. Under
conditions of uncertainty, it is reasonable to adopt a conservative approach, i.e., choose a
probability distribution for the SV that provides the most uncertain outcomes given only
information about the average value. A well-known measure of uncertainty associated with
probability distributions is entropy (Kagan, Linnik and Rao, 1973). Therefore, it is reasonable
to determine the SV distribution from the Maximum Entropy principle.
A proposed single-factor SV model is based on the following assumptions (Levin and
Tchernitser, 1999a):
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 451
Assumption 1. The density function, pX(x,T ), of the risk factor X = X(T ) for some
holding period T is normal conditional upon the stochastic variance V = V (T ) that possesses
a probability density function pV (v,T ), v 0, i.e.,
pX(x,T ) =

0
1

2v
exp (x
- v - T )2
2v
pV (v,T )dv. (1)
Parameter T specifies a constant part of the mean for the conditional normal distribution,
and parameter  defines a shift in the mean proportional to the SV. As is shown
later,  determines the correlation between the RF and SV that results in a skewed RF
distribution. The case  = 0 corresponds to a symmetric distribution. Linear dependence of
the shift term v from v in the mean of normal density is important for further construction
of a Lévy process for the RF. The stochastic representation for X is as follows:
X(T ) = V (T )Z + V (T ) + T, Z  N(0,1), (2)
with Z being a standard normal random variable independent of V (T ).
Assumption 2. The average variance E{V (T )} for the holding period T is known and
equal to V :
E V (T ) =

0
vpV (v,T )dv = V . (3)
Assumption 3. The probability density function pV (v,T ) of the stochastic variance V (T )
is defined by the Maximum Entropy principle given the average variance (3):
H(pV ) = -

0
pV (v)lnpV (v)dv  max
pV (v) 0
. (4)
The optimization problem (4) for the SV density pV (v) subject to the constraint on
the average variance (3) and standard normalization constraint

0 pV (v)dv = 1 has the
exponential density
pV (v) =
1
V
exp -
v
V
as a solution calculated by the Lagrange multiplier method (Kagan, Linnik and Rao, 1973).
According to the Law of Total Probability, the unconditional density (1) of the risk factor
X(T ) has the following density:
pX(x,T ) =

V
exp |x
- T |

+ (x - T ) ,  =
V
2 + 2V
. (5)
452 A. Levin and A. Tchernitser
Distribution (5) is known as the skewed double exponential (Laplace) distribution (Kotz,
Kozubowski and Podgórski, 2001). This distribution has a sharp peak, exponential tails and
non-zero skewness for  = 0. Kurtosis of a symmetric Laplace distribution is always equal
to 6, in contrast to 3 for a normal distribution. Historical distributions of daily returns for
many market variables, such as CAD/USD FX rate (Figure 2), JPY/USD FX rate, S&P 500
Index, TSE 300 Index (Figure 2), NYMEX Natural Gas futures prices, some LIBOR rates,
etc., have a similar leptokurtic shape (Levin and Tchernitser, 1999a; Kotz, Kozubowski
and Podgórski, 2001).
In the case of a linear portfolio and symmetric Laplace distribution for the RF, the impact
of non-normality on VaR can be estimated as
VaRLaplace
VaRNormal
=
ln(2q)

2zq
,
where zq is a standardized normal quantile for the confidence level q. For the case q = 1%
(zq = 2.3263), VaRLaplace for a linear portfolio is 19% higher than the standard VaRNormal.
The impact on VaR is even more pronounced for non-linear instruments. For example, for
a non-linear perfectly delta-hedged option portfolio, (x), within Delta­Gamma approximation
for the portfolio changes, (x) = 0.5 x2, the corresponding formulas for VaR
are as follows:
VaRLaplace = -V

4
ln2
(q), VaRNormal = -V

2
(zq/2)2
.
This results in 60% higher VaRLaplace number than VaRNormal (Levin and Tchernitser,
1999a).
The exponential distribution for the SV was derived from the Maximum Entropy principle
for some unspecified holding period T . To calculate VaR for different holding periods t,
a stochastic process for the risk factor X is required. The standard normal model assumes
that the risk factor X follows a Wiener process with independent stationary Gaussian increments.
The simplest extension of this assumption is that the RF follows a Lévy process, i.e.,
a stochastic process with independent stationary (not necessarily Gaussian) increments. It
can be shown (Rosínski, 1991) that within the class of conditionally normal models (2) this
assumption is equivalent to the following assumption on the SV:
Assumption 4. The total stochastic variance V (t) in (2) follows a positive increasing Lévy
process.
The exponential distribution for the V (T ) is infinitely divisible. It uniquely determines
a positive increasing pure jump Gamma process [see Sato (1999)] for the total stochastic
variance V (t), t > 0, with a Gamma probability density function
pV (t)(v) =
vt-1
(t)t
exp -
v

, (6)
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 453
where  = 1/T ,  = V . Assumptions 1­4 define the corresponding Lévy process for the
risk factor X(t) with the following probability density function:
pX(x,t) =
2

2t-1 exp(y)
(t)t
|y|t-1/2
Kt-1/2 |y| , y =
x - t

. (7)
Here  is defined in (5), () is a gamma function, and K(y) is a modified Bessel function
of the third kind of the order , K-(y) = K(y) (Abramowitz and Stegun, 1972). Distribution
(7) is known as a Bessel K-function distribution (Johnson, Kotz and Balakrishnan,
1994) or as a Generalized Laplace distribution (Kotz, Kozubowski and Podgórski, 2001).
Essentially, the SV model derived from the Maximum Entropy principle is equivalent to
the Variance Gamma (VG) model [Gamma stochastic time change model, see Madan and
Seneta (1990), Madan and Milne (1991), Geman and Ané (1996)]. The tail asymptotic behavior
and behavior at the origin for the density (7) follows from known asymptotics for
the modified Bessel function K(y) (Abramowitz and Stegun, 1972)
K(y) 
y

2y
e-y
, K(y) 
y0
()2-1
y,
 > 0, K0(y) 
y0
-ln(y).
The RF density (7) has exponential tails for all t and a wide range of shapes at the origin,
from almost normal "bell" shape (for large  1) to a highly peaked (0.5 <  1) and
even unbounded shape (0 <  0.5) (see Figure 4). A skewed Laplace density (5) is a
special case of (7) for t = T . The Bessel K-function family of distributions possesses
finite moments of all orders. The characteristic function for the Gamma process has a
simple form
Fig. 4. Probability densities for the Gamma SV model.
454 A. Levin and A. Tchernitser
X(t)() = E eiX(t)
=
exp(it)
(1 - i + 2/2)t
. (8)
The Lévy density from the Lévy­Khintchine representation of X(t)() that characterizes
the intensity of jumps of different sizes x has the following closed form [see Sato (1999)]:
k(x) =

|x|
exp -
2

+ 2|x| + x .
The RF distribution (7) tends to a normal distribution for t  +. This normalization
effect is important for a proper VaR scaling from short holding periods to longer ones.
The total variance DX(t) is proportional to time, as it is for any Lévy process with finite
variance (Feller, 1966) (a "square root of time" rule for the volatility is valid). However,
contrary to the Gaussian case, the ratios of q-quantiles and standard deviation for the RF
distributions (7) are not constant for different holding periods t. For example, the standardized
1%-quantile (VaRVG) is higher for shorter holding period than the same 1%-quantile
for longer holding period (Figure 5).
The entropy for the SV distribution standardized by time t (the mean of a standardized
SV is equal to 1 for all t) has the maximum at t = T (Figure 6) that corresponds to the
exponential distribution. This property may be explained by transition of the standardized
Gamma density from the delta-function at 0 to the delta-function at 1 as time t passes.
Heuristically, this evolution of shape for the SV density corresponds to a transition from
the state of maximum certainty at time 0 to the limiting state of maximum certainty at
t =  (with the limiting normal density for the standardized RF).
The following expressions provide a connection between the first four moments of the
RF distribution and those of the SV distribution (Levin and Tchernitser, 2000a):
mX(t) = t + mV (t),
DX(t) = mV (t) + 2
DV (t),
(9)
m3,X(t) =  3DV (t) + 2
m3,V (t) ,
m4,X(t) = 3m2
V (t) + 3DV (t) + 62
mV (t)DV (t) + 62
m3,V (t) + 4
m4,V (t).
Fig. 5. VG model 1%-VaR term structure with respect to 1% Normal VaR = 2.33.
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 455
Fig. 6. Evolution of standardized Gamma SV density and entropy.
The expressions (9) for the moments are valid for conditional normal models of the form
(1) provided that the distribution pV (t)(v) for the stochastic variance V (t) possesses moments
up to the fourth order. Parameter  controls skewness of the RF distribution and
defines the correlation X,V between the risk factor X and its stochastic variance V :
X,V = 
DV
mV + 2DV
.
A parameter estimation procedure (model calibration), with respect to the four parameters,
, , , and  can be based either on the Maximum Likelihood approach or the
method of moments given four sampling central moments for the T1-day underlying returns
and analytical expressions for the moments of the Gamma stochastic variance (Johnson,
Kotz and Balakrishnan, 1994)
mV (T1) = T1, DV (T1) = T12,
m3,V (T1) = 2T13, m4,V (T1) = 3T14(T1 + 2).
Equations (9) can be used for the model calibration by the method of moments. Note that
time T , corresponding to the maximum entropy for the SV density, can be recovered from
the calibrated parameter  as T = 1/.
It follows from (6) and (9) that the term structure of the RF variance and kurtosis for the
symmetric case of the Gamma-SV model ( = 0) is:
DX(t) = t, KurtX(t) - 3 =
3
t
. (10)
456 A. Levin and A. Tchernitser
2.2. Generalized Gamma Variance model
Some market variables exhibit jumps as large as 5 to 10 daily standard deviations (Fama,
1965; Bouchaud and Potters, 2000; Mantegna and Stanley, 2000; Cont, 2001). Such events
have significantly lower theoretical probability to occur for the corresponding periods of
observations not only for the normal model, but even for the Gamma SV model with exponential
tails. Extremely large jumps in the risk factors have often been described by distributions
with polynomial tails, specifically by stable distributions (Mandelbrot, 1960, 1963;
Mittnik and Rachev, 1989, 2000). However, stable Paretian distributions do not have finite
variance (volatility). This contradicts the majority of empirical observations [see Müller,
Dacorogna and Pictet (1998)]. Also, volatility is a main tool in financial risk management
and pricing. Therefore, heavy tailed distributions with finite variance are of considerable
interest for the finance applications. An example of such distribution widely discussed in
the financial literature is Student t-distribution (Platen, 1999; Albanese, Levin and ChingMing
Chao, 1997; Rachev and Mittnik, 2000). A new family of the RF distributions introduced
below includes t-distribution as a special case. The symmetric Gamma SV model
considered above has only one shape parameter, , that controls both the tails and central
part of the distribution. It seems that one shape parameter is insufficient to distinguish
between sources of high kurtosis: whether it comes from heavy tails or high peak. It is
possible to show that for a class of conditional normal models the tail asymptotics of the
RF distribution depends upon the tail asymptotics of the corresponding SV distribution.
Therefore, a more general SV model that allows for separate control for the tails and peak
should more successfully describe large deviations of the risk factors.
Note, that the Gamma SV density (6) can be formally derived from the Maximum Entropy
principle (4) without Assumption 4. Instead, one can use a constraint on the logarithmic
moment E{ln(V )} in addition to the condition on the average variance E{V } (Kagan,
Linnik and Rao, 1973). Essentially, this logarithmic constraint defines a power behavior
of the SV density at the origin, while the constraint on E{V } defines the exponential tail
behavior. The condition on average variance can be replaced by a more flexible condition
to accommodate information on a generalized moment of any power for the SV (Levin
and Tchernitser, 2000a, b). For example, one can assume that the average volatility is
known instead of average variance. This approximately corresponds to a constraint on the
fractional moment E{

V } instead of E{V }. Hence, we can formally define the entropy
maximization problem (4) with two essential constraints

0
ln(v)pV (v)dv = c0,

0
v1/
pV (v)dv = c1 (11)
and a standard normalization constraint for a probability density function. The use of
the Maximum Entropy approach with a constraint on the generalized moment E{V 1/},
  R1, allows for a desirable generalization of the Gamma SV model to a broad class of
models with a wide range of heavy tails, from exponential and sub-exponential (stretched
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 457
exponential) to polynomial (Levin and Tchernitser, 2000a, b). A solution of the maximization
problem (4), (11) is the Generalized Gamma density for the stochastic variance V :
pV (v) =
v/-1
|| ()
exp -
v1/

. (12)
The corresponding stochastic representation for V is a -th power of the Gamma distributed
random variable  with the density (6) [see Johnson, Kotz and Balakrishnan (1994)]:
V =  
. (13)
Stochastic representations (2) and (13) allow for an effective Monte Carlo simulation procedure
for the RF given well-known simulation procedures for normal and gamma random
variables (Fishman, 1996).
The Generalized Gamma distribution is a very flexible class of distributions with two
shape parameters  and . This class includes Gamma ( = 1), Inverse Gamma ( = -1),
and Weibull ( = 1,  > 0) distributions as special cases. It is known that the Generalized
Gamma distribution is infinitely divisible for these three representatives [see Grosswald
(1976), Ismail (1977), Sato (1999)] and for positive  max(,1) (Ismail and Kelker,
1979). Therefore, for these cases the Generalized Gamma distribution produces Lévy
processes for the SV. We do not know if the Generalized Gamma distribution is infinitely
divisible for arbitrary values of   R1, nor whether there is a closed form representation
for the characteristic function. Hence, we apply the distribution (12) to describe the returns
for the shortest holding period available, say one day, and then construct an additive SV
process for a longer holding period, say 10 days, by summing up the independent Generalized
Gamma distributed random variables. An analytical formula for the moments of the
Generalized Gamma distribution is readily available
E V k
=
k ( + k)
()
(the condition for the k-moments to exist is ( + k) > 0).
The corresponding RF density pX(x) is given by the integral (1) with SV density pV (v)
being of the form (12). We call this density a Generalized Gamma Variance density (GGV).
Unfortunately, in the general case there is no closed analytical form for the density pX(x).
However, we consider an effective numerical procedure for calculating the integral (1) to be
as good as, for example, a "closed form" formula (7) involving special K-Bessel functions.
Effective asymptotic expansion methods (Olver, 1974; Abramowitz and Stegun, 1972) can
be applied for the numerical calculations.1 In the case of a symmetric GGV density, there
is an analytical formula for the moments of any fractional order k (finite for  +k/2 > 0):
1 Effective numerical procedure and software for the GGV density calculation was developed by Xiaofang Ma.
458 A. Levin and A. Tchernitser
E |X|k
=
2k/2k/2 ((k + 1)/2) ( + k/2)

 ()
. (14)
The moments cease to exist for some combinations of negative values of  and  > 0
because of polynomial tails for the GGV density.
Below, we provide some results for a symmetric density pX(x). There are some known
special analytical cases for pX(x):
(i)  = -1 corresponds to the t-distribution with 2 degrees of freedom;
(ii)  = 0 corresponds to the Gaussian distribution;
(iii)  = +1 corresponds to the K-Bessel function distribution (7).
Table 1 presents a summary of results for the Generalized Gamma Variance model,
including a constraint on the generalized moment in Maximum Entropy principle (column
1), SV stochastic representation (column 2), corresponding RF density (column 3),
and asymptotics for the tails of the RF density (column 4).
Some market variables are better described by distributions with polynomial tails, while
others are better described by distributions with semi-heavy tails (exponential and subexponential)
[see Platen (1999), Rachev and Mittnik (2000), Duffie and Pan (1997)]. The
GGV model is capable of accommodating both types of behavior. A range of values  < 0
corresponds to a power low tails. GGV density is finite at zero for all  < 0. A range of
values  > 0 corresponds to exponential and sub-exponential tails. In this case, tails are
far lighter and the moments of all orders exist. The range  > 1 corresponds to a class
of stretched exponential densities pX(x). The specific class of the stretched exponential
distributions based on a modified Weibull density was considered in Sornette, Simonetti
and Andersen (2000). Figure 7 shows the RF GGV density pX(x) for different values of
parameters  and . Parameter  brings an extra flexibility to the GGV density: it is seen
that GGV model can accommodate a wide variety of shapes and tail behavior.
A statistical investigation of different SV models from a Generalized Hyperbolic family
based on historical data for 15 stock market indices was presented in the paper by Platen
(1999). The class of Generalized Hyperbolic distributions developed in Barndorff-Nielsen
(1978, 1998), Eberlein and Keller (1995), Eberlein, Keller and Prause (1998) is also a two
shape parameter family in symmetric case. All members of this family have exponential
Table 1
GGV model summary
Constr. E{V 1/} SV density & Stoch. rep. RF density RF asymptotics x  
E{V },  = 1 Gamma, V =  K-Bessel x-1 e-cx
E{

V },  = 2 Square of Gamma, V =  2 GGV(2,) x2/3-1 e-cx2/3
E{1/V },  = -1 Inverse Gamma, V = 1/ t-Distribution x-(2+1)
E{V 1/},  > 0 Generalized Gamma, V =   GGV(,) x2/(1+)-1 e-cx2/(1+)
E{V 1/},  < 0 Generalized (Inv.) Gamma, V =   GGV(,) x-(2/||+1)
 = 0 SV degenerates to V  1 Normal e-x2/2
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 459
Fig. 7. Generalized Gamma Variance densities.
Fig. 8. Historical and calibrated GGV densities for the CAD 3-month BA interest rate daily log-returns.
tails except the Student t-distribution, which has polynomial tails. For this specific case,
the class of Generalized Hyperbolic distributions collapses to a one shape parameter (number
of degrees of freedom) family. Four representatives from a Generalized Hyperbolic
class (Student t-distribution, Normal Inverse Gaussian, Variance Gamma, and Hyperbolic
distributions) were compared based on the Maximum Likelihood criteria. The last three
of these distributions have exponential tails. Results presented in Platen (1999) show that
all distributions having exponential tails fail to satisfy the Pearson 2 test. In contrast, the
t-distribution has not been rejected on a 99% confidence level for ten of the fifteen indices.
Two-parameter Paretian tail GGV distributions perform better than the t-distribution. As an
460 A. Levin and A. Tchernitser
Fig. 9. GGV model log-likelihood surface for S&P 500.
example, Figure 8 demonstrates a fit for the Canadian 3-month BA interest rate daily return
density (1992­1998) by normal, Student-t, and GGV densities calibrated using Maximum
Likelihood approach. It is seen that GGV(,) with optimal parameters  = -5.5 and
 = 15 outperforms t-distribution, and both GGV and t-distributions significantly outperform
normal. The 2 value for the GGV(15,-5.5) is about 80% less than 2 value for the
calibrated t-distribution. It is interesting to note, that during the period 1992­1998, Canadian
3-month BA interest rate exhibited 14 large daily moves greater than four standard
deviations (about 1% of all observations).
Another example (Figure 9) shows a GGV model log-likelihood surface for S&P 500
Index as a function of parameters  and . A deep minimum for  = 0 corresponds to the
normal distribution, while two wings correspond to the power law ( < 0) and stretched
exponential ( > 1) tailed distributions. For this example, a stretched exponential sub-class
produces almost the same maximum likelihood value as a power law sub-class.
2.3. Mean-reverting stochastic variance model
So far, we have considered a class of the SV models driven by Lévy processes with independent,
identically distributed, but not necessarily Gaussian increments. The model
explains non-normality of the RF distributions. For any conditional normal SV model, expressions
(9) provide an exact answer for the term structures of the risk factor variance and
kurtosis
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 461
DX(t) = mV (t), KurtX(t) - 3 = 3
DV (t)
m2
V (t)
. (15)
Here V (t) is a total variance. Since mV (t) and DV (t) are linear functions of time for any
Lévy process for V (t), the above expressions predict linear increase of the RF variance
and hyperbolic decrease of its excess kurtosis.
However, empirical investigations show that the underlying returns are almost uncorrelated,
but not independent [see Bouchaud and Potters (2000), Cont (2001), Müller, Dacorogna
and Pictet (1998)]. The easiest way to demonstrate this dependence is to consider
the empirical correlations for the absolute values or squares of the returns. It is seen (Figure
10) that autocorrelations of squares are statistically significant. This phenomenon is
connected with a known volatility clustering effect (Figure 3). Also, it is known that empirical
term structure of kurtosis decreases slower than is predicted by a "Lévy term structure"
model (15) [so called "anomalous decay", see Bouchaud and Potters (2000), Cont
(2001)]. All this suggests that a better model for the instantaneous stochastic volatility is
not a "white noise" kind of process, but rather a process with autocorrelation. One way
to account for the autocorrelation structure of the SV is to consider regime-switching SV
processes [see Konikov and Madan (2000)]. We will follow another approach to introduce
the SV autocorrelation by considering Ornstein­Uhlenbeck (OU) type processes for
the instantaneous SV (Levin and Tchernitser, 1999a, 2000a). Such class of non-Gaussian
OU type processes driven by positive Lévy noise was investigated in detail in BarndorffNielsen
and Shephard (2000a, b). In this section we will only demonstrate that the empirically
observed term structure of kurtosis can be consistently described by such models.
Consider a stationary non-negative process (t) with autocorrelation function R () that
describes the instantaneous stochastic variance. For the total variance V (t) being V (t) =
t
0 (t )dt , it follows that
mV (t) = mV (1)t, DV (t) = 2
t
0
(t - )R ()d.
The above expressions in conjunction with (15) can be used to calculate a term structure
of the RF kurtosis. In particular, assume a mean-reverting process for the instantaneous
stochastic variance (t) be a Ornstein­Uhlenbeck type process
d(t) = -(t)dt + dG(t), (16)
where G(t) is, for example, a Gamma process,  > 0 is a mean-reversion speed parameter.
Expressions for R () and variance DV (t) are as follows
R () =
2
2
e-||
, DV (t) =
2t
2
1 -
1 - e-t
t
.
462 A. Levin and A. Tchernitser
Fig. 10. Autocorrelation in the squared CAD/USD FX daily returns and in the SV.
The autocorrelation function R () is an exponential function for any OU model (16). It
is seen that DV (t) is not a linear function of time contrary to the Lévy case (10). Previous
formulas and formulas (15) result in the following term structure of the RF kurtosis:
KurtOU
X(t) - 3 =
3
t
1 -
1 - e-t
t
.
Figure 11 shows a term structure of the RF kurtosis for different values of the meanreversion
speed parameter . As expected, the OU stochastic variance process provides
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 463
Fig. 11. Term structure of the RF kurtosis for the model with autocorrelated SV.
slower decay of kurtosis vs. Lévy SV process. Reduction in decay can be significant depending
upon the mean-reversion speed . This is equivalent to slower "normalization"
effect. The bottom graph in Figure 10 presents the time series for the simulated SV and
empirical CAD/USD FX rate squared daily log-returns. The bottom graph in Figure 3
presents the simulated RF time series. It is evident that the model produces large deviations
for the FX rate and volatility clustering effect that is very similar to the one observed
in the market (top graph in Figure 3).
3. Multifactor stochastic variance model
3.1. Requirements for multifactor VaR models
A realistic multifactor VaR model should consistently describe not only the correlation and
volatility structure for the risk factors, but also different shapes of the marginal risk factor
distributions and distributions in other "diagonal" directions. Also, a principal component
analysis for daily returns in different markets (interest rate curves, commodity futures
prices, implied volatility curves and surfaces), clearly indicates the presence of non-linear
dependence between risk factors (principal components). For example, the squared daily
changes of the principal components are significantly correlated, while daily changes themselves
are uncorrelated. This non-linear dependence breaks conditions of the Central Limit
Theorem and has an important impact on VaR calculation: even for well-diversified linear
portfolios with a large number of instruments there is no full normalization of the portfolio
return distributions (Levin and Tchernitser, 1999a, b). An example of such large diversified
portfolio is the S&P 500 Index. Its distribution is quite far from normal despite the
portfolio averaging effect. Hence, a comprehensive model for multiple risk factors should
additionally capture the following important features observed in the market:
464 A. Levin and A. Tchernitser
* exact match of a given volatility and correlation structure of the risk factors;
* approximate match of shapes, kurtosis, and tails for different risk factors (marginal dis-
tributions);
* approximate match of shapes, kurtosis, and tails for different linear sub-portfolios (marginal
distributions in diagonal directions).
The model should also allow for an effective Monte Carlo simulation procedure. To facilitate
further multivariate analysis, in the sequel, we shall consider the case of symmetric
joint probability distributions for the RF returns.
3.2. "Nave" multifactor model
A very simple idea for constructing a multivariate conditionally Gaussian stochastic variance
model is to define a distribution for the vector of risk factors X(t)  RN as a multivariate
normal with some fixed correlation matrix R and independent stochastic variances
Vi(t), i = 1,...,N. A symmetric multivariate probability density function for the vector
of risk factors is represented as:
pX(t)(x) =
V1

VN
1
(2)N det(C)
× exp x
C-1x
2
pV (V1,...,VN )dV1 dVN , (17)
C = R ,  = diag V1(t),..., VN (t) . (18)
Here pV (V1,...,VN ) = N
i=1 pVi (Vi) is a probability density for independent stochastic
variances Vi(t), x is transpose of x. The corresponding stochastic representation for
the risk factors X(t) is
X(t) = diag V1(t),..., VN (t) AZ, AA = C, Z  N(0,I), (19)
where Z  N(0,I) is independent of V standard normal vector with identity covariance
matrix I. This representation allows for modelling marginal distributions with different
leptokurtic shapes.
However, it can be shown that this "nave" approach reduces the correlations between
risk factors because of "randomization" for the covariance matrix (Levin and Tchernitser,
1999a). Due to independence of the stochastic variances Vi, absolute values of the model
correlations Corr(Xi,Xj ) are less than absolute values of the correlations Rij used in (17):
Cov(Xi,Xj ) = xixj pX(x)dxi dxj = Rij Vi Vj pV (Vi,Vj )dVi dVj
= Rij VipVi (Vi)dVi Vj pVj (Vj )dVj
= fij X,iX,j Rij , i = j. (20)
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 465
Table 2
Correlation reduction factors
i = j 0.5 1 2 5 10
f (i,j ) 0.64 0.79 0.88 0.95 0.98
Reduction factors fij , i = j, are less than one, because
Vi1pVi (Vi)dVi < VipVi (Vi)dVi 1pVi (Vi)dVi = E{Vi} = X,i.
It means that the sampling correlation matrix cannot be used as the matrix R in (17).
For example, the reduction factors fij < 1, i = j, calculated explicitly for the case of the
Gamma stochastic variances (6) are as follows:
Corr(Xi,Xj ) = fij Rij , fij = f (i,j ) =
(i + 1/2) (j + 1/2)
(i) (j )

ij
, i = j.
The underestimation of the correlations can be significant for some values of parameters
i, j as it is shown in Table 2.
The randomization effect exists for any probability density functions pVi (Vi) for independent
stochastic variances. Usually, equations Corr(Xi,Xj ) = fij Rij cannot be resolved
with respect to correlations Rij given sampling correlations Corr(Xi,Xj ) while preserving
the necessary conditions |Rij | 1 or non-negative definite matrices R. Hence, this "nave"
model does not allow to preserve historical correlations between the risk factors.
Remark. Equation (20) and the inequality
Vi Vj pV (Vi,Vj )dVi dVj < Cov(Vi,Vj ) + E{Vi}E{Vj }
imply that the class of the SV models with the stochastic representation (18) for the covariance
matrix preserves the RF correlation structure only if
Vi Vj pV (Vi,Vj )dVi dVj = E{Vi} E{Vj },
which requires dependent stochastic variances with positive correlations. We do not investigate
this direction in the chapter.
3.3. Elliptical stochastic variance model
The simplest extension of a single-factor SV model to the multifactor case is an elliptical
stochastic variance model. Elliptical models are widely used for representing non-
466 A. Levin and A. Tchernitser
normal multivariate distributions in finance [see Eberlein, Keller and Prause (1998), Kotz,
Kozubowski and Podgórski (2001)]. This class of models preserves the observed RF correlation
structure. The model is similar by construction to the one-dimensional variance
mixture of normals. An elliptical N-dimensional symmetric process XE(t) for N risk factors
has a stochastic representation as a single variance mixture of multivariate normals
with a given covariance matrix C:
XE
(t) = V (t)ZC
, ZC
 N(0,C). (21)
Here V (t) is a univariate SV process, ZC is a multivariate normal N-dimensional vector
independent of V (t). The covariance matrix C is estimated from historical T1-day returns
(e.g., daily returns), while the SV is normalized to satisfy a condition mV (T1) =
E{V (T1)} = 1. The unconditional density for the random vector of risk factors XE(t) is:
pXE(t)(x) =

0
1
(2V )N det(C)
exp x
C-1x
2V
pV (t)(V )dV.
As an example, consider the case of Gamma stochastic variance V (t). A closed analytical
form for the unconditional elliptical Bessel K-function density for XE(T ) is available
in Kotz, Kozubowski and Podgórski (2001). A characteristic function XE(t)() for the
elliptical Lévy process XE(t) is represented as:
XE(t)() = 1 +

2
 C
-t
, (22)
where  is N-dimensional vector,  is a vector transposed to . Due to known properties
of elliptical distributions [see Fang, Kotz and Ng (1990)], all marginal one-dimensional
distributions for the risk factors are univariate Bessel K-function distributions with the
same shape parameter t and the same kurtosis. They differ only by the standard deviations.
The same property holds for all one-dimensional distributions of linear combinations
X =  XE(t) of the risk factors. These linear combinations correspond to the
linear portfolios defined by . The kurtosis of X(T1) for any arbitrary  is equal to
k = 3(1 + DV (T1)/m2
V (T1)) = 3(1 + DV (T1)). Therefore, within the class of elliptical
models there is no normalization effect at all for the distributions of large diversified portfolios.
This is a result of violation of the conditions for the Central Limit Theorem: the
risk factors are dependent through the common stochastic variance V . Such property is a
drawback for all elliptical models. It is clear that the actual RF fluctuations are not driven
by a single stochastic variance ("global market activity"). More realistic SV model should
include a multidimensional processes for the SV to model different distributional shapes
for the risk factors and linear sub-portfolios. Since sampling marginal RF distributions
have different shapes, the calibration of elliptical model is restricted to fitting a distribution
of some preselected portfolio. Hence, the calibration of elliptical models is portfolio
dependent.
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 467
3.4. Independent stochastic variances for the principal components
One of the possible ways to model different shapes for the RF distributions while preserving
a given correlation structure was considered in Levin and Tchernitser (1999a, b). An
N-dimensional vector of the risk factors is represented as a linear combination of principal
components (PC) with independent one-dimensional stochastic variances. The corresponding
stochastic representation is as follows:
XL
(t) = ~AZI
(t), ZI
i (t) = Vi(t)Zi, Zi  N(0,1), i = 1,...,M.
Here Zi are independent standard normal variables, Vi(t) are independent SV processes
with a unit mean and some variances DV i for a specified time horizon T1. The columns
of a constant matrix ~A are the principal components of a given covariance matrix C. The
covariance matrix C is estimated from the historical T1-day returns. Matrix ~A is calculated
based on eigenvalue decomposition of the covariance matrix C [see Wilkinson and Reinsch
(1971)]:
C = UDU , U = U-1
, D = diag(d1,...,dN ), (23)
~A = UMD
1/2
M , DM = diag(d1,...,dM), M N, C = ~A ~A . (24)
Matrix UM consists of the first M columns of the orthogonal matrix U, which correspond
to the first M largest eigenvalues d1,...,dM of the matrix C. Number M may be chosen
less than N if the matrix C is singular and has only M non-zero eigenvalues. Some numerical
issues related to singularity of the matrix C were considered in Kreinin and Levin
(2000). It follows from the construction of the process XL(t) that Cov(XL(T1)) = C. This
ensures an exact match of the sampling covariance matrix C. One can keep even a smaller
number M of the principal components in (24) and recover the matrix C with the required
accuracy.
A characteristic function for the model is a product of the characteristic functions of onedimensional
processes for the PCs. For example, a characteristic function for the Gamma
SV model with independent SV has a form
XE(t)() =
M
i=1
1 +
( ( ~A)i)2
2
-i t
,
where ( ~A)i is i-th column of the matrix ~A.
The matrix ~A can be defined up to an arbitrary orthogonal transformation H without
change of the covariance matrix C since Zi are independent standard normal variables, Zi
and Vi(T1) are independent and E{Vi(T1)} = 1. Hence, E{ZI (T1)ZI (T1) } = I and
E XL
(T1)XL
(T1) = ~AHE ZI
(T1)ZI
(T1) H ~A = ~AHH-1 ~A = C
468 A. Levin and A. Tchernitser
for any orthogonal matrix H . However, the matrix H influences a matrix of the fourth
moments of XL(t), Kij = E{(XL)2
i (XL)2
j }. The orthogonal matrix H and shape parameters
for the Vi can be determined to approximate a given sampling matrix {Kij } of the
fourth moments for the RF distribution (all moments E{(XL)3
i (XL)j } are equal to zero for
symmetric distributions). An explicit calculation yields:
Kij = E XL 2
i
XL 2
j
= 3
M
k=1
a2
ika2
jkDV k + CiiCjj + 2C2
ij ,
i,j = 1,...,N, (25)
where aik are the elements of the matrix A = ~AH . An effective method for calculating the
matrix H and shape parameters is discussed in Section 3.6 below.
The model provides an exact match of the RF correlation and volatility structures and
approximates different shapes and kurtosis of the marginal RF distributions contrary to
the Elliptical model. However, there is a significant drawback for this model. Since the
stochastic variances Vi are independent, there is a strong normalization effect in any "diagonal"
direction. This means that some linear portfolios X(t) =  XL(t) have almost
normal distributions whenever the portfolio Delta, , is not a marginal direction and the
number of principal risk factors M is large enough. Described effect presents a real danger,
because the non-normal marginal RF distributions may be well-approximated, while
the modelled portfolio distributions (contrary to the actual sampling distributions) may be
almost normalized and the VaR underestimated.
3.5. A model with correlated stochastic variances
As it was pointed out above, a more general and realistic market model should incorporate
the correlated stochastic variances that can correct the deficiencies of both Elliptical
model and the model with independent SV for the principal components. The correlated SV
structure should allow modeling of some general economic factors as well as idiosyncratic
components that drive the SV processes for different risk factors and markets.
The model is defined via stochastic representation of the following form (Levin and
Tchernitser, 2000a, b):
XCV
(t) = AZI
(t), ZI
i (t) = Vi(t)Zi, Zi  N(0,1), i = 1,...,M. (26)
Here Zi are independent standard normal variables, Vi(t) are the correlated stochastic variance
processes with a unit mean for a specified time horizon T1. The matrix A  RN×M is
defined as in the previous section through the eigenvalue decomposition for the covariance
matrix C up to an arbitrary orthogonal transformation H  RM×M :
C = Cov XCV
(T1) = AA = ~A ~A , A = ~AH, H = H-1
.
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 469
Stochastic variances Vi(t) are correlated to each other due to the following stochastic rep-
resentation:
Vi(t) =
L
k=1
bikk(t),
L
k=1
bik = 1, bik 0, B  RM×L
, (27)
where k(t) are independent positive increasing Lévy processes with unit mean for the time
horizon T1 and different shape parameters, and B is a constant matrix with non-negative
elements. The processes k(t) are the drivers for the SV processes Vi(t). For example, each
driver k can be a Gamma process or Generalized Gamma process. Linear structure in (27)
with bik 0 ensures that Vi(t) are positive increasing Lévy processes. The normalization
conditions E{k(T1)} = 1 and bik = 1 ensure, as in Section 3.4, exact recovering of the
sampling covariance matrix for the risk factors. It follows, that the vector of stochastic
variances V (T1) has covariance matrix CV equal to
CV = Cov V (T1) = BD B ,
D = diag(D1,...,DL), Dk = Var k(T1) . (28)
The multivariate Generalized Stochastic Variance (GSV) model (26), (27) has two levels
of correlations. First level defines usual correlations across the risk factors described by
the covariance matrix C. Second level defines the correlations across the stochastic variances
described by the covariance matrix CV . The second level of correlations provides
a possibility to obtain an approximate, but consistent match of the higher order moments
and shape of the RF multivariate distribution. The elliptical model and the model with
independent stochastic variances are the special cases of the above GSV model. Elliptical
model corresponds to the matrix B being equal to one column with all unit entries,
B = [1,...,1] . The model with independent SV corresponds to the case when the matrix
B is equal to the identity matrix, B = I.
There is no analytical form for the probability density function of the vector XCV(t)
even for the Gamma drivers k(t). However, a characteristic function XCV (t)() can be
calculated as
XCV (t)() =
RL
+
exp -
1
2
 Adiag(B)A  p(t)()d
=
L
j=1
+
0
exp -
j
2
M
i=1
bij
N
k=1
Akik
2
pj (t)(j )dj .
The expression for the characteristic function above is equivalent to
XCV (t)() =
L
j=1
+
0
exp -
j
2
 Cj  pj (t)(j )dj ,
470 A. Levin and A. Tchernitser
where Cj , j = 1,...,L, are certain positive semi-definite matrices. The latter expression
for the characteristic function allows for a different interpretation of the GSV model. It
shows that the process for the risk factors XCV(t) can be presented as a sum of L independent
elliptical Lévy processes. In turn, each of these elliptical processes has a multivariate
conditional normal distribution with a covariance matrix proportional to Cj and the corresponding
stochastic variance j (t).
The kurtosis k of a linear combination of the risk factors X(T1) =  XCV (T1) for
any given direction  can be calculated analytically:
k - 3 =
E{X4
(T1)}
E{X2
(T1)}2
- 3 = 3 CV  = 3 BD B ,
  RM
, k =
( A)2
k
 A 2
, k = 1,...,M. (29)
The above expression provides a link between the covariance matrix CV and the kurtosis
k, that characterizes the shape of the RF multivariate distribution for the linear portfolio
with Delta equal to . Another useful quantity that clarifies the role of the correlated
variances Vi is a standardized matrix of the fourth moments. This matrix, {kij }, is a multidimensional
analog for kurtosis
kij =
E{(XL)2
i (XL)2
j }
E{(XL)2
i }E{(XL)2
i }
. (30)
The matrix {kij } incorporates kurtosis in all marginal and all pair-wise diagonal directions
in the original risk factor space. It is expressed as
kij - 1 + 22
ij =
M
k=1
M
l=1
2
ik2
jl Cov(VkVl) + 2
M
k=1
M
l=1
ikjkiljl Cov(VkVl),
ik =
aik
ai
, ai
2
=
M
k=1
a2
ik, (31)
where ij is a correlation between i-th and j-th risk factors. Formulas (29) and (31) clearly
indicate that the correlation structure of the SV is embedded into the correlation structure
of the fourth moments of the RF distribution. This connection will be used as the base
for the GSV model calibration. A number L of the SV drivers can be chosen significantly
smaller than a number of stochastic variances M and risk factors N. These SV drivers may
be thought as "stochastic activities" for different countries, industries, sectors, etc.
The GSV model with the correlated stochastic variances is, in fact, a general framework.
It can incorporate any reasonable processes to represent the SV drivers k(t), k = 1,...,L.
Some examples of suitable one-dimensional SV driver distributions are: Inverse Gaussian
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 471
distribution (Barndorff-Nielsen, 1997), Gamma distribution (Madan and Seneta, 1990;
Levin and Tchernitser, 1999a), Lognormal distribution (Clark, 1973), or considered above
class of Generalized Gamma distributions. The GSV model is practical in terms of effective
Monte Carlo simulation: it is based on the simulation of one-dimensional SV processes
and standard multivariate normal variables.
3.5.1. Example 1. Joint distribution for DEM/USD and JPY/USD FX rates
The first example presents a bivariate GSV model applied to the foreign exchange market
data. Four bivariate models were examined for DEM/USD and JPY/USD FX rate daily
returns: Standard Gaussian model, Elliptical Gamma Variance model, model with independent
stochastic variances for PCs, and the model with correlated SV. Figures 12 and
13 show a 3-D plot and a contour plot of the joint probability density for the historical
data and four types of the models considered. All three SV models provide a far better fit
than the Gaussian distribution. However, the most convincing fit is provided by the GSV
model with the correlated stochastic variances. Marginal distributions for DEM/USD and
JPY/USD FX rates have kurtosis 5.2 and 6.9 respectively. Figures 14 and 15 show that
the latter model is able to capture kurtosis and shape of marginal distributions in different
directions.
3.5.2. Example 2. Twenty risk factors
The second example examines a 20-dimensional GSV model with correlated SV applied
to the data from the interest rate, FX rate, and equity markets. The USD and CAD zero
interest rate curves each consisting of nine interest rates, CAD/USD FX rate, and S&P 500
Index were chosen as a representative set of the risk factors. There were 5 years (1994­
1999) of daily historical data used for the model calibration (about 1,250 data points).
Figure 16 presents statistical results for principal component analysis and the correlation
matrix for squares of the first three PCs. These results indicate that uncorrelated PCs neither
are normal nor independent. The first three "largest" PCs per zero curve were used for the
GSV model calibration and simulation. Three Gamma distributed drivers k, k = 1,2,3,
with different shape parameters were utilized to represent each stochastic variance Vi, i =
1,2,3, for PCs. Therefore, the following values for parameters were assigned: number of
risk factors N = 9×2+1+1 = 20, number of principal risk factors M = 3×2+1+1 = 8,
number of SV drivers L = 3.
The model was calibrated to match kurtosis (in the least squares sense) for all 20 risk
factors and kurtosis for 15 additional linear sub-portfolios. Sampling kurtosis varies within
a wide range from 5 to 25. Typically, kurtosis for short-term interest rates is much higher
than kurtosis for long-term rates. It is seen (Figure 17) that the GSV model reproduces
this typical decreasing kurtosis term structure quite well. It is also seen that the model
adequately matches kurtosis of the FX rate and S&P 500 Index, as well as kurtosis of different
linear sub-portfolios. To compare, the standard multi-dimensional Gaussian model
produces a flat kurtosis term structure identically equal to three.
472 A. Levin and A. Tchernitser
Fig. 12. Joint density for DEM/USD and JPY/USD FX rate.
3.6. Calibration for the GSV model
The GSV model is a two-level model that incorporates a traditional variance­covariance
structure of the risk factors and novel variance­covariance structure of the RF stochastic
variances. The GSV model with correlated SV automatically preserves the RF covariance
matrix C. At the second level, it is necessary to calibrate the SV covariance matrix CV to
approximate the fourth moments of the multivariate RF distribution.
The main steps of the model calibration procedure are as follows:
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 473
Fig. 13. Contour plot for the DEM/USD­JPY/USD FX rate joint density.
1. Calculate a sampling covariance matrix C  RN×N for a given set of risk factors X. The
time window usually used for calibration of the covariance matrix C is about 1­2 years.
Exponentially weighted averaging or uniform sliding window are the usual methods for
the covariance matrix calculation (JP Morgan, 1996).
2. Decompose the sampling covariance matrix C using a standard eigenvalue decomposition
procedure and form a matrix ~A  RN×M from a set of M eigenvectors corresponding
to M largest eigenvalues. Number M has to be chosen to recover the matrix C with
a required accuracy.
474 A. Levin and A. Tchernitser
Fig. 14. DEM/USD and JPY/USD FX marginal distributions.
Fig. 15. Fit of the kurtosis for different sub-portfolios.
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 475
Fig. 16. PCA for USD zero curve.
Fig. 17. Fit of the kurtosis.
476 A. Levin and A. Tchernitser
3. Calculate sampling fourth order moments for the risk factors X (the matrix kij in (30))
and kurtosis k for any preselected set of directions (linear sub-portfolios) {}. The
time window typically required for calculation of the fourth moments is of the order 5­
10 years. This period of observations has to be much longer than the one for the second
order moments. This is necessary to incorporate relatively rare extreme events into the
calibration. Longer time horizon allows for an adequate approximation of the tails and
general shape of the multivariate RF distribution.
4. Calculate matrices H , B, and D using the least squares approach:
i
wi ^ki - ki(H,B,D )
2
+
i j
wij ^ke
ij - ke
ij (H,B,D )
2
 min
H,B,D
, (32)
where wi, and wij are some predefined weights (these weights may be chosen depending
on the importance of particular risk factors and sub-portfolios), ^ki is the sampling
kurtosis for the direction i, ki(H,B,D ) is the analytical estimate (29), ^ke
ij is the
sampling matrix of the fourth moments, and ke
ij (H,B,D ) is its analytical estimate
(31).
The minimization problem above is a subject to constraints imposed on the matrices
H , B, D . The most difficult condition to satisfy is orthogonality of the matrix H . It
follows from the analysis of expressions (29) and (31) that M × M orthogonal matrix
H can be constructed as a product of M × (M - 1)/2 elementary rotation matrices
(Wilkinson and Reinsch, 1971). It can be shown that for the problem (29), a representation
for the orthogonal matrix H does not require reflections. The diagonal matrix D is
subject to simple non-negativity constraints. The matrix B is subject to constraints (27).
Hence, the non-linear optimization problem (32) can be re-formulated with respect to
M × (M - 1)/2 angles m for the elementary rotation matrices with simple constraints
- m  and elements of the matrices B and D with mentioned above simple
constraints.
5. If the Gamma Variance model for the SV drivers k is adopted, the diagonal matrix
D and conditions E{k(T1)} = 1 determine the shape and scale parameters k and k
in (6). For the GGV model, the powers k  R1 have to be additionally specified. As
a practical approach, the following methodology has been adopted: a set of parameters
{k} is fixed in such a way that it covers a reasonably wide range of values k. For
example, the set of k can be chosen as
{k} = {-2,-1,+1,+2}.
This choice is justified by the fact that the SV drivers k with negative values of k
will produce the RF probability density function with heavy polynomial tails. On the
other hand, positive values of k can produce the RF distributions with semi-heavy
exponential and sub-exponential tails, but with unbounded peaks at the origin. However,
it is quite possible that a more flexible and adjustable structure for the set of parameters
{k} is more beneficial for the model calibration.
Ch. 11: Multifactor Stochastic Variance Models in Risk Management 477
Acknowledgment
Authors thank C. Albanese, O. Barndorff-Nielsen, D. Duffie, P. Embrechts, H. Geman,
D. Madan, J. Nolan, and, especially, M. Taqqu for interesting discussions and useful comments
related to presented models.
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Willinger, W., Taqqu, M., Teverovsky, V., 1999. Stock market prices and long-range dependence. Finance and
Stochastics 3, 1­13.
Chapter 12
MODELLING THE TERM STRUCTURE OF MONETARY RATES
LUISA IZZI
Credit Risk Department of Banca Nazionale del Lavoro, Department of Mathematics,
Statistics and Information Technology, Faculty of Economics, and Centro Interdipartimentale Vito Volterra,
University of Rome Tor Vergata, Italy
Contents
Abstract 482
1. Introduction 483
2. The mathematical framework 484
2.1. Model setup and notation 484
2.2. Regularity conditions on the jump-diffusion process 486
2.3. Interest rates with non-identically distributed jumps 487
2.4. The smile effect and infinitely divisible distributions 490
2.5. The discrete-time process 492
3. The tree representation 494
4. The econometric analysis 500
4.1. Data 500
4.2. Estimation results 501
5. Conclusions 506
References 507
 The analysis and conclusions set forth in this chapter are those of the author and do not indicate concurrence
by other members of the Banca Nazionale del Lavoro.
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
482 L. Izzi
Abstract
The chapter addresses the general problem of modeling and estimating the term structure
of interest rates by adopting the use of jump-diffusion mean-reverting and stable Paretian
models. The chapter proposes a new procedure to recursively compute interest rates subject
to both Brownian and Poissonian noises. This procedure is consistent with the absence
of arbitrage, non-negativity of interest rates, the mean-reverting hypothesis and the recombining
condition, and can be calibrated with respect to any term structure which can be
observed in the market. The numerical study shows that the proposed model is particularly
suited to describe the behavior of European money market rates.
Ch. 12: Modelling the Term Structure of Monetary Rates 483
1. Introduction
The term structure of interest rates provides a characterization of interest rates as a function
of maturity, which is mainly used in the pricing of fixed-income securities and for the
valuation of contingent claims. The chapter addresses the general problem of the modeling
and estimation of the term structure of interest rates. The problem is indeed of great importance,
in view of its possible application to the analysis of Economic and Monetary Union
monetary policy. It is this specific application that makes the problem of modeling interest
rates difficult; in fact, the narrowness of the time horizons of interest rates time series has
been one of the main obstacles in developing a solid quantitative analysis of the short-term
interest rates in the Euro-zone.
The term structure has been modeled extensively using primarily pure-Gaussian models.
Notable examples of these kinds of models are those of Merton (1973), Cox (1975), Cox
and Ross (1976), Vasicek (1977), Dothan (1978), Brennan and Schwartz (1980), Cox, Ingersoll
and Ross (1980, 1985) and Black, Derman and Toy (1990). Recent empirical works
show that models which accommodate skewness and kurtosis, such as stochastic volatility
[see, e.g., Bates (1996) and Britten-Jones and Neuberger (2000)] and jump-diffusion models
(Ahn and Thompson, 1988; Ho, Perraudin and Sorensen, 1996; Scott, 1997), appear
to fit the time series of interest rates better than pure-Gaussian type ones.1 In fact, unlike
jump-diffusion processes, stochastic volatility models are not capable of generating high
levels of skewness and kurtosis at short maturities under reasonable parameterizations.
As a consequence, stochastic volatility models cannot generate as sharp implied volatility
"smiles" as those typically observed empirically.2
It should also be noted that conditional skewness and kurtosis in stochastic volatility
models are always hump-shaped in the length of horizon: indeed, for plausible parameter
values, both quantities must be increasing over short to moderate maturities. This implies
that the smile does not flatten out appreciably as maturity increases. Moreover, the kurtosis
of changes in interest rates increases when the data is sampled daily instead of monthly,
and interest rates often display discontinuous behavior, partly on account of the discrete
changes the Central Bank makes in the official rates [see Baz and Das (1996), Das and
Foresi (1996), Björk, Kabanov and Runggaldier (1997), Attari (1999)]. Modeling the term
structure using a jump-diffusion process captures these market features better, because with
these processes the variance shrinks with the time interval, yet the size of jumps remains
1 The class of jump-diffusion models augments the Black­Scholes (1973) return distribution with a Poissondriven
jump process; while the class of stochastic volatility models extends the Black­Scholes model by allowing
the volatility of the return process itself to evolve randomly over time.
2 The "smile" is the plot of implied volatilities from a range of options of the same maturity across different
strike prices. It can be observed that at-the-money options seem to trade at the lowest implied volatilities and the
in-the-money and out-of-the-money options trade at higher volatilities. Since the options are all written on the
same underlying variable, there should be no plausible reason for this, other than the fact that the model is inexact.
Since the observed distribution of interest rates has fatter tails than that assumed by a pure-Gaussian model, such
an effect is intuitively obvious. When plotted against the strike price, the graph of implied volatilities appears
U-shaped like a smile. Hence the terminology.
484 L. Izzi
the same, enhancing kurtosis (which is the relative size of the outliers to the variance of
the process).
In this chapter a jump-diffusion mean-reverting model for estimating monetary rates is
introduced and related to the class of models driven by infinitely divisible processes [see
Gnedenko (1992)] to which Gaussian, Poissonian and Stable ones [see Samorodnitsky and
Taqqu (1994)] belong. Relating interest rate models to the class of Stable processes is
an attempt to unify, from the probabilistic point of view, the extensive literature on purediffusion,
pure-Poissonian and jump-diffusion processes. It completes recent works on the
stock [see Schumacher (1997), Rieken (1999)] and exchange markets [see Akgiray and
Booth (1988), Rachev and Mittnik (2000)] concerning the same topic.
We also propose a new numerical procedure to recursively compute interest rates subject
to both Brownian and Poissonian noises. This procedure ­ which generalizes the trinomial
approach proposed by Hull and White (1994, 1996) in the presence of pure-diffusion stochastic
components ­ is consistent with the absence of arbitrage, non-negativity of interest
rates, the mean-reverting hypothesis and the recombining condition. It can be calibrated
with respect to any term structure which can be observed in the market, for instance, the
present one.
Numerical experiments demonstrate how the jump-diffusion mean-reverting model is
particularly suited to describe the behavior of European money market rates. Interest rates
controlled by the monetary authorities behave like jump processes and the term structure,
at short maturity, is contingent upon the levels of these official rates.
The rest of the chapter is organized as follows. Section 2 introduces the interest rate
model used to describe the behavior of monetary rates both in continuous and in discrete
time. To provide a direct application of the model, a new numerical procedure is proposed
in Section 3. The new method determines a unique fully specified hexanomial tree, which is
consistent with respect to the risk neutral probability. Econometric estimations are related
in Section 4. The estimation of the mean-reverting jump-diffusion process is obtained via
the indirect inference method. Finally, Section 5 concludes the chapter.
2. The mathematical framework
2.1. Model setup and notation
Let  be a (rich enough) sample space, representing the state-of-the-world, and let
Xt :  be a stochastic process on  such that
Xt = Bt +
n
i=1
aiN
(i)
t + , (1)
where t  [0,T ] and T is a finite horizon; Bt is a Brownian motion;  0 is constant;
n 1 is fixed; N(i)
t are mutually independent Poisson processes, also independent of Bt ,
with parameters i > 0, for i = 1,...,n; ai  Bin(i, 2
i ) is independent of any other
Ch. 12: Modelling the Term Structure of Monetary Rates 485
variable, with i and i  , and represents the jump size of the corresponding Poisson
process, for i = 1,...,n;  is a shift parameter. Let F = {Ft}0 t T be the filtration
generated by {Xt}0 t T (the information structure on ), that is Ft is the -field generated
by {Xs}0 s t , where the initial filtration F0 is trivial, and P a probability measure
on (,FT ). Let rt :  + be a stochastic process on the filtered probability space
(,{Ft},P) described by the stochastic differential equation:
drt = t dt + t dXt (t  [0,T ]), (2)
where t = k(t - rt ) is the drift function (or conditional mean); t is a positive bounded
real function of time; k is the calling back strength of the t rate on the instantaneous rate rt ;
t is the diffusion function of the interest rate change (2
t is the conditional variance).
So, we can redefine the model as follows:
drt = k(t - rt )dt + t[ dBt + a dNt] (t  [0,T ]), (3)
where a dNt = n
i=1 ai dN(i)
t ; a = [a1,a2,...,an]; Nt = [N(1)
t ,N(2)
t ,...,N(n)
t ]T.
Remark 2.1. Let us consider the particular case in which n = 2 and a1  Bin(1, 2
1 )
and a2  Bin(2, 2
2 ) are independent of any other random variable, with 1 > 0, 2 < 0,
1,2  . Then, the stochastic process Xt has the form:
Xt = Bt + a1N
(1)
t + a2N
(2)
t +  (t  [0,T ])
and the two Poisson processes N(1)
t and N(2)
t could represent the actions of the European
Central Bank (N
(1)
t "up" and N
(2)
t "down") on the official rates. Moreover, the stochastic
differential equation:
drt = k(t - rt )dt + t  dBt + a1 dN
(1)
t + a2 dN
(2)
t (t  [0,T ])
is subject to the constraint: |t - rt | ut - dt, where t is the rate of the main refinancing
operation of the European Central Bank (ECB); ut is the rate of the marginal lending
facility; dt is the rate of the deposit facility (see Figure 1). In fact, money market rates
should not exceed the corridor fixed by ut (upper bound) and dt (lower bound), because
under normal conditions the ECB gives banks all the money they need at the ut rate. The
only limit is determined by the availability of collateral to be presented to the ECB as a
guarantee of the credit received. In such a way, if the collateral available to banks has no
limits, also the credit that banks receive from the ECB at the ut rate has no limits. Similarly,
at the end of every working day, banks can deposit an unlimited amount of money in the
ECB accounts at the dt rate.3
3 EONIA (the European overnight rate which is computed as an average value by the ECB) has gone up to the
ut rate only in very few occasions due to market inefficiency problems.
486 L. Izzi
Fig. 1. Official rates during the ten policy periods (percentage values).
2.2. Regularity conditions on the jump-diffusion process
The jump-diffusion process defined in (3) can be used to approximate a wide range of
Markovian or non-Markovian processes. In particular, the weak convergence of a driving
noise process to a Lévy process implies the weak convergence of the driving noise process
to a jump-diffusion process [see, for example, Blasikiewicz and Brown (1996)]. In the
general case, the Brownian motion Bt and the compound Poisson process
t
0 a dN () are
infinitely divisible in time, when appropriately scaled, and have independent increments.
It is noted, however, that even though Bt is a martingale process, a compound Poisson
process is in general not a martingale. Define the compensated Poisson process:
dNt() = dNt() - dt
since
E
t+ t
t
a dN () = E
t+ t
t
a dN () - d = 0,
i.e.,
t
0 a dN () is a martingale process. Therefore, without loss of generality, we only
need to consider the case in which the compound Poisson process is a martingale process.4
To ensure that the generalized stochastic differential equation defined in (3) is well behaved,
it is necessary to make two additional assumptions.
4 For further details, see Zacks et al. (1999).
Ch. 12: Modelling the Term Structure of Monetary Rates 487
Assumption 2.1. The coefficient functions () and () are measurable in the product
-algebra B × F, where B is the -field of the Borel sets on , and furthermore
T
0
|t|dt <  a.s.,
T
0
2
t dt <  a.s.
and
E
T
0
2
t dt <  a.s.
To ensure the existence and uniqueness of a solution to the stochastic differential equation
(3), it is necessary to impose:
Assumption 2.2. The coefficient functions () and () satisfy both Lipschitz and linear
growth conditions, i.e., there exists a positive constant K for which
(r,t) - (r ,t) + (r,t) - (r ,t) K|r - r |,
(r,t) - (r,t ) + (r,t) - (r,t ) K|t - t |
and
2
(r,t) + 2
(r,t) K 1 + r2
.
Let us consider the model:
drt = k(t - rt )dt +  dBt + a dNt (t  [0,T ]), (4)
where the diffusion function t is assumed to be equal to 1;  0 is the volatility of the
interest rate, provided that no jumps occur; Nt() is a Poisson process with an intensity
parameter ; a is the jump size; dBt and dNt are statistically independent.
2.3. Interest rates with non-identically distributed jumps
Let us focus our attention on the stochastic part of the interest rate process, i.e., on Xt ,
defined by
Xt = Bt +
n
i=1
aiN(i)
t +  (t  [0,T ] and n 1). (5)
Following Rachev and Rüschendorf (1994) and Schumacher (1997), we can study Xt as
an infinitely divisible random variable.5
5 For further details, see Rachev and Mittnik (2000) and Rieken (1999).
488 L. Izzi
Definition 2.1. A sequence {Xn,k} of random variables (with n 1 and k = 1,...,n) is
uniformly asymptotically negligible (u.a.n.) if
Xn,k
P
 0 (n  ),
uniformly in k, or equivalently if for every  > 0
max
1 k n
P |Xn,k|   0 (n  ).
Let Fn,k denote the cumulative distribution function of Xn,k and n,k its characteristic
function. It can be shown [see Rachev and Mittnik (2000) and the references therein] that
the u.a.n. condition is equivalent to
max
1 k n
x2
1 + x2
dFn,k  0 (n  )
or
max
1 k n
|n,k - 1|  0 (n  ),
uniformly on every finite interval.
The use of u.a.n. random variables restricts the class of summands, but this restricted
class, known as the class of infinitely divisible random variables, is that investigated in
the classical central limit problem. It is of primary interest in modeling interest rates and
includes the most commonly models, i.e., Normal, Poisson, Stable and others.
Definition 2.2. A random variable X, its distribution FX, and its characteristic function
X are said to be infinitely divisible (I.D.) if, for every n 1, there are independent and
identically distributed (i.i.d.) random variables 1,2,...,n, with distribution function
F and characteristic function  each, such that
X
d
= 1 +  + n,
or equivalently if, for every n 1, FX is the n-fold convolution of F, i.e.,
FX = F    F = (F)n
,
or equivalently if, for every n 1, its characteristic function X is the n-th power of the
characteristic function .
From Definition 2.2 it follows immediately that the Normal, the Poisson and the degenerate
distributions, i.e., all possible limits of n
k=1 Xn,k in the i.i.d. case, belong to this class.
Ch. 12: Modelling the Term Structure of Monetary Rates 489
The I.D. law is uniquely defined by the Lévy­Khintchinerepresentation of its characteristic
function [see Rachev and Mittnik (2000)]:
X(u) = E eiuX
= exp iu +
+
-
eiux
- 1 -
iux
1 + x2
1 + x2
x2
d(x) , (6)
where   and  is a distribution function up to a multiplicative constant. The characteristic
function (6) describes the stochastic process which drives the interest rate in the
natural world. Hence,  and  can be estimated from the observed data. In the case in
which the variance of X is finite, the characteristic function (6) simplifies to:
X(u) = exp iu +
+
-
eiux
- 1 - iux
1
x2
d(x) . (7)
The finite variance I.D. model with characteristic function (7) contains the Normal and the
Poisson distributions as special cases. In fact, suppose that
(x) = I[0,)(x),
where I[a,)(x) denotes the indicator function of the interval [a,), i.e.,
I[a,)(x) =
0, if x < a,
1, if x a,
with a,x  . In this case, the integral in (7) becomes -2t2/2 and the characteristic
function reduces to:
X(u) = exp iu -
2t2
2
, (8)
which is the characteristic function of a Normal random variable with mean  and variance
2. The Poisson distribution is obtained from (7) by setting
(x) = K2
I[K,)(x),
with  > 0 and K = 0. In this case, the characteristic function becomes:
X(u) = exp  eiuK
- 1 - iuK . (9)
The relation (9) defines the characteristic function of a scaled and shifted Poisson random
variable of the form K (N - ), where N is a Poisson random variable with parameter ,
i.e.,
P{N = k} =
e-
k!
,
490 L. Izzi
with  > 0 and k = 1,2,.... Yet, the characteristic function of N is:
N (u) = exp  eiu
- 1 . (10)
Hence, the finite-variance I.D. distribution defined by (7) can be interpreted as an infinite
mixture of independent Poisson random variables and one independent Gaussian random
variable. For practical purposes, the I.D. model can, for example, be specified as a Normal
random variable and a finite number n of independent Poisson random variables, as defined
in (5), i.e.,
Xt = Bt +
n
i=1
aiN
(i)
t + , (11)
where t  [0,T ]; n 1;  0; Bt is a standard Normal random variable; N(i)
t are independent
Poisson random variables (independent from Bt ), respectively, of parameters i,
for i = 1,...,n; ai  Bin(i, 2
i ) are independent of any other variable, with i,i  ,
for i = 1,...,n;   .
2.4. The smile effect and infinitely divisible distributions
If the Black­Scholes (1973) model of pricing contingent claims were true, the implied
volatility would be constant for all claims. This, however, is not observed in practice. Instead,
the phenomenon of a volatility smile is observed on call options on increasing strike
prices. That is, deep-in-the-money call options (options with a strike price K X, where
X is the current price of the underlying asset) and deep-out-of-the-money options have an
implied volatility parameter  (the scale parameter in the Normal distribution) which exceeds
the implied volatility parameters of at-the-money options. Said another way, (K)
appears to be a convex function of K. Said still a third way, the Black­Scholes formula
underprices deep-in-the-money and deep-out-of-the-money call options, whereas it overprices
at the money options. A related issue is the leptokurtic nature of financial asset
values. That is, it is also widely observed that asset values tend to be more highly peaked
and have fatter tails than is implied by the Normal model. The use of infinitely divisible
distributions may help to explain the phenomenon of leptokurtosis as well as the smile
effect.6
Definition 2.3. The coefficient of kurtosis 4 of the random variable X is defined as
4 =
4
2
2
,
where 4 and 2 are the fourth and second central moments of X, respectively.
6 See Schumacher (1997) and Rachev and Mittnik (2000) for further details.
Ch. 12: Modelling the Term Structure of Monetary Rates 491
Remark 2.2. The existence of a moment generating function of X is enough to ensure the
finiteness of the central moments.
The coefficient of kurtosis is a measure of the peakedness of the distribution. It can be
shown that for a Normally distributed random variable X, the coefficient of kurtosis 4 = 3.
Given the leptokurtotic nature of asset prices, we might find it desirable that our model of
interest rate behavior have a coefficient of kurtosis 4 3.
Theorem 2.1. Let 4(X) be the coefficient of kurtosis of an infinitely divisible random
variable X. Then 4(X) 3.
Proof: Let Yn = n
i=1(aiNi + bi) where Ni are independent Poisson random variables
with rate i . The log characteristic function of Yn is then
logYn (u) =
n
i=1
i eiaiu
- 1 + iubi .
The k-th derivative (for k 2) of the log characteristic function is
d(k) logYn (u)
du(k)
=
n
i=1
i(iai)k
eiaiu
.
Hence, we obtain the semi-invariants of order j, for j 2 for the finite sum of scaled and
shifted Poisson random variables:
j =
n
i=1
a
j
i i.
The semi-invariants can be expressed in terms of the central moments as
2 = 2 = 2
, 3 = 3, 4 = 4 - 32
2,
and so on. From this it follows that the coefficient of kurtosis for the given sum is given by
4 =
4
2
2
=
n
i=1 a4
i i + 3( n
i=1 a2
i i)2
( n
i=1 a2
i i)2
3.
Now by a remark of Gnedenko (1992), all infinitely divisible random variables are limits
of sums of scaled and shifted Poisson random variables; so the theorem follows.
492 L. Izzi
2.5. The discrete-time process
Let us divide the time interval [0,T ] into M (M 1) sub-intervals of equal length t,
i.e., [ti,ti+1), where i = 0,1,...,M - 1, ti = i t and t = T/M. Then, for i = 0,1,...,
M - 1, the stochastic differential equation (4) admits the discretization:
r(M)
ti+1
- r(M)
ti
= k (M)
ti
- r(M)
ti
t +  B(M)
ti+1
- B(M)
ti
+ a N(M)
ti+1
- N(M)
ti
, (12)
where {B
(M)
ti+1
- B
(M)
ti
}M-1
i=0 are i.i.d. random variables, distributed as N(0, t);
a N
(M)
ti+1
- N
(M)
ti
=
n
j=1
aj N
(M),(j)
ti+1
- N
(M),(j)
ti
;
{N
(M),(j)
ti+1
- N
(M),(j)
ti
}M-1
i=0 are i.i.d. random variables, distributed as Poisson ((j) t), for
any j = 1,...,n. As M  , r
(M)
ti
(i = 0,1,...,M), as defined in (12), converges in
weak sense to the process rt (t  [0,T ]), satisfying (4). In the following, the index M will
be omitted for simplicity.
Let  and   and set, for notational convenience,  =  t. Following Das (1997),
let us define the four sequences of random variables:
X
(1)
ti+1
=
1

t
(Bti+1 - Bti ) (i = 0,1,...,M - 1),
where {X
(1)
ti
}M
i=0 are i.i.d. standard Normal random variables;
X(2)
ti
=



 + , w. prob. 
2 ,
0, w. prob. 1 - ,
 - , w. prob. 
2 ,
i.i.d. for i = 0,1,...,M and with {X(2)
ti
}M
i=0 and {X
(1)
ti
}M
i=0 mutually independent;
X
(2)
ti
=
 + , w. prob. 1
2 ,
 - , w. prob. 1
2 ,
i.i.d. for i = 0,1,...,M;
Bin()ti =
1, w. prob. ,
0, w. prob. 1 - ,
i.i.d. for i = 0,1,...,M and with {Bin()ti }M
i=0 and {X
(2)
ti
}M
i=0 mutually independent. It is
easy to show that {X
(2)
ti
}M
i=0
d
= {X
(2)
ti
Bin()ti }M
i=0.
Ch. 12: Modelling the Term Structure of Monetary Rates 493
Let D be the metric space of right continuous real-valued functions on [0,T ], with left
limits. Let the interest rate process be defined on the path space D[0,T ]. Let C[0,T ] be
the subspace of D[0,T ] of all real-valued continuous functions on [0,T ]. The space D is
restricted to the Skorokhod topology which, when restricted to the space C[0,T ], is the
topology of uniform convergence.7
By the Functional Limit Theorem for stochastic differential equations, to approximate
the spot rate process as defined in (4), in weak sense in D[0,T ], with the process rti
(i = 0,1,...,M) as defined in (12), we can replace { X
(1)
ti

t}M
i=0 by the sequence
of i.i.d. random variables {X
(1)
ti
}M
i=0 distributed as follows:
X
(1)
ti
=
+

t, w. prob. 1
2,
-

t, w. prob. 1
2,
for any i = 0,1,...,M. Then, the discrete spot rate process rti (i = 0,1,...,M) as defined
in (12) is equivalent to
rti+1 = rti + k(ti - rti ) t + X(1)
ti+1
+ X(2)
ti+1
(i = 0,...,M - 1). (13)
Remark 2.3. The discrete process above mimics the behavior of a continuous-time jumpdiffusion
process. Hence for i = 0,...,M, the first noise term, X
(1)
ti
, represents the diffusion
component, while the second one, X
(2)
ti
, represents the jump, which assumes values
 +  or  -  . Therefore, the jump has mean  and variance  2.  is the probability
parameter of a jump in unit time and hence the probability of a jump in any time interval
approximates  =  t.
In our process, jumps occur "rarely", which is achieved by choosing a low value for
  (0,1). Moreover, as the time interval t decreases, the probability of a jump goes
to zero. Finally, we may choose the parameters  and  to provide the necessary skewness
and kurtosis. In particular, the parameter  governs skewness, while  drives kurtosis.
Remark 2.4. The moments of the discrete process provide the intuition for why the choice
of jump form injects the necessary skewness and kurtosis into the model. The first four
moments are as follows:
* mean: rt + k(t - rt) + ;
* variance: (2 +  2) - 22 + 2 t;
* skewness: (1 - )(2 + 3 2 - 22), the sign of which clearly depends on the sign
of ;
7 See Billingsley (1995).
494 L. Izzi
* kurtosis (expressed for  = 0 and  = 0, to keep the expression simple and to observe
the effect of  ):  4, which demonstrates that the magnitude of kurtosis depends on the
size of  . When  = 0, kurtosis is equal to: (1 - )44 + 
2 ( -  - )4 + 
2 ( +
 - )4, which means that the sign and size of  also affect the degree of kurtosis.
However, here too, the degree of kurtosis increases with  .
3. The tree representation
The diffusion process X
(1)
ti
(i = 0,...,M - 1), as defined above, can be represented on the
binomial tree:
X
(1)
ti
X(1)
ti+1
= X(1)
ti
+ 

t, w. prob. 1
2 ,
X
(1)
ti+1
= X
(1)
ti
- 

t, w. prob. 1
2 .
The random term representing the jump in Equation (13) is X(2)
ti
and can be represented on
the trinomial tree:
X(2)
ti
X(2)
ti+1
= X(2)
ti
+  + , w. prob. 
2 ,
X(2)
ti+1
= X(2)
ti
, w. prob. 1 - ,
X(2)
ti+1
= X(2)
ti
+  - , w. prob. 
2 .
Let us consider, at first, only the discrete diffusion process:
rti+1 = rti + k(ti - rti ) t + X(1)
ti+1
(i = 0,...,M - 1), (14)
and, following the methodology of Hull and White (1994), let us construct a binomial recombining
tree for a variable r
ti
(i = 0,...,M), which is initially zero, follows the process:
r
ti+1
= r
ti
- kr
ti
t + X
(1)
ti+1
(i = 0,...,M - 1) (15)
and is symmetrical about r
ti
= 0. The variable [r
ti+1
- r
ti
] is Bernoulli distributed. If terms
of higher order than t2 are ignored, it holds:
E r
ti+1
- r
ti
= -kr
ti
t, Var r
ti+1
- r
ti
= 2
t + k2
r
ti
2
t2
.
Let ji be a positive or negative integer, varying in the range (- 1
k t ,+ 1
k t ) and depending
on i, with i = 0,...,M. Recalling that ti = i t, let denote r
D(i,) the corresponding value
of r
ti
on a lattice scheme. For any (integer) i  [0,M] and (any integer) ji  (- 1
k t ,+ 1
k t )
Ch. 12: Modelling the Term Structure of Monetary Rates 495
the value of r
D(i,) at "level" ji, i.e., at node (i, ji), will be denoted with r
D(i,ji) and
similar notation applies to rti .
Assumption 3.1. The binomial tree is recombining.
For every i = 0,...,M, let ni be the number of nodes of the tree generated at time ti = i t.
The binomial tree is assumed to be recombining, then, the number of nodes on the tree will
be
i
k=0
nk =
i
k=0
(k + 1) =
i+1
k=1
k.
Let Ji denote the set of indices ji generated at time ti = i t, with i = 0,...,M. Then
|Ji| = ni = i + 1. Assuming J-2 = J-1 =  and recalling that the tree is recombining, we
can calculate recursively the set Ji as: Ji = Ji-2  {i,-i}, for any i = 0,...,M.
For any integer ji  [0,+ 1
k t ), let (ji) denote the spacing between interest rates on
the tree by moving from level ji - 1 to level ji ("up" movement), or from level ji to
level ji - 1 ("down" movement). In both cases, "up" or "down" movement, the width
of spacing between rates depends on the maximum (starting or ending) level reached by
moving between the two adjacent nodes, i.e., if the interest rate moves (up or down) from
node (i,ji) to node (i + 1,ji+1), with ji 1, then
r
D(i + 1,ji+1) - r
D(i,ji) = max{ji,ji+1} .
On the contrary, for any integer ji  (- 1
k t ,0) set:
r
D(i + 1,ji+1) - r
D(i,ji) = min{ji,ji+1} .
Then
(ji) = (-ji), ji  -
1
k t
,+
1
k t
{0}.
Remark 3.1. Note that, by construction, the quantity (0) = 0 will never be employed in
the tree building, i.e., in deriving the interest rate position on the tree.
For any i = 1,...,M - 1 and any integer ji  (- 1
k t ,+ 1
k t ), let u(ji,ji+1) and
d(ji,ji+1) denote, respectively, an "up" and a "down" movement on the binomial tree
from level ji to level ji+1. Then
u(ji,ji+1) = - d(-ji,-ji+1).
496 L. Izzi
Assumption 3.2. Let us suppose that, for any i = 1,...,M and ji  Ji  (- 1
k t ,+ 1
k t ),
the spacing between interest rates on the tree is such that
(ji) =
|ji|

t
1 - j2
i k2 t2
.
Remark 3.2. The spacing between nodes is not constant, but assumes values increasing
with ji. This makes our model more flexible than the standard Cox, Ross and Rubinstein
(1979) binomial tree, which consists of a set of nodes ­ representing possible future stock
prices ­ with constant logarithmic spacing between them. This spacing is a measure of the
future volatility, itself assumed to be constant in the Cox, Ross and Rubinstein framework
and in the Black and Scholes model (1973) to which a Cox, Ross and Rubinstein tree, with
an "infinite" number of time steps, converges.
The constancy of volatility cannot easily be reconciled with the observed structure of
implied volatilities for options traded in most financial markets. Volatility varies with both
strike price and expiration. This variation, known as implied volatility "smile", is currently
a significant and persistent feature of option markets. Nevertheless, the constant
local volatility assumption in the Black and Scholes theory and in the Cox, Ross and Rubinstein
tree leads to the absence of a volatility smile, at least as long as market frictions
are ignored [see Derman, Kani and Chriss (1996)]. Our tree refers to implied tree theories
that extend the Black and Scholes model to make it consistent with the shape of the smile
[see Derman and Kani (1997a, b) and Rubinstein (1994)].
Because r
ti
is initially zero, then r
D(0,0) = 0. The dynamics of r
ti
incorporate a mean
reversion to zero, where the strength of the mean reversion is proportional to the value
of r
ti
. Therefore, there exist an upper and a lower bound to the interest rate process. Let us
formalize this condition as follows:
Assumption 3.3. There exist two integer numbers ju  (0,+ 1
k t ) and jd  (- 1
k t ,0) such
that
* jd ji ju for any ji  Ji (i = 0,...,M);
* if ji = ju at step i, then ji+1 = ju - 1 at step i + 1 (i = 0,...,M - 1);
* if ji = jd at step i, then ji+1 = jd + 1 at step i + 1 (i = 0,...,M - 1).
The parameters ju and jd may be estimated from the real values assumed by rti , otherwise
we can set:
ju =
1
k t
- 1 and jd = -ju,
where [ 1
k t ] denotes the integer part of 1
k t .
Let pu(i,ji) and pd(i,ji) denote, respectively, the transition probabilities of "up" and
"down" movements starting from node (i,ji), with i = 0,...,M -1 and ji  Ji (jd,ju).
Ch. 12: Modelling the Term Structure of Monetary Rates 497
The two probabilities must be chosen to match the expected change and variance of the
change in r
ti
over the next time interval t. The probabilities must also be non-negative
and sum to unity. This leads to the equations below:



pu(i,ji) u(ji,ji+1) + pd(i,ji) d(ji,ji+1) = -kr
D(i,ji) t,
pu(i,ji) 2
u(ji,ji+1) + pd(i,ji) 2
d(ji,ji+1) = 2 t + k2r
D(i,ji)2 t2,
pu(i,ji) + pd(i,ji) = 1,
0 pu(i,ji) 1,
0 pd(i,ji) 1.
Using Assumptions 3.2 and 3.3, the solution to the system above is given by the following
cases:
* for any i = 0,...,M - 1 and ji  Ji  [+1,ju):
pu(i,ji) = pu(ji) =
1
2
-
1
4
-


t
u(ji,ji+1) - d(ji,ji+1)
2
,
pd(i,ji) = pd(ji) =
1
2
+
1
4
-


t
u(ji,ji+1) - d(ji,ji+1)
2
;
* for any i = 0,...,M - 1 and ji  Ji  (jd,-1]:
pu(i,ji) = pu(ji) =
1
2
+
1
4
-


t
u(ji,ji+1) - d(ji,ji+1)
2
,
pd(i,ji) = pd(ji) =
1
2
-
1
4
-


t
u(ji,ji+1) - d(ji,ji+1)
2
;
* for any i = 0,...,M - 1 and ji = 0:
pu(i,0) = pu(0) =
1 - k t
2
, pd(i,0) = pd(0) =
1 + k t
2
;
* for any i = 0,...,M - 1 and ji = ju:
pu(i,ju) = pu(ju) = 0, pd(i,ju) = pd(ju) = 1;
* for any i = 0,...,M - 1 and ji = jd:
pu(i,jd) = pu(jd) = 1, pd(i,jd) = pd(jd) = 0.
498 L. Izzi
Remark 3.3. At each node (i,ji), for any i = 0,...,M - 1 and ji  Ji  [jd,ju], with
jd = -ju and ji = 0, the transition probabilities depend only on ji, so the resulting binomial
tree, besides being recombining, is also symmetrical around ji = 0, i.e.,
pu(ji) = pd(-ji), pu(ju) = pd(jd) and pu(jd) = pd(ju).
Thus, it is not necessary to save all transition probabilities in one large array and the loss
in computing time is very small.
Remark 3.4. For every i = 0,...,M - 1 and ji  Ji  [jd,ju], the transition probabilities
pu(ji) and pd(ji) reflect the mean reversion of the rate r
ti
and, thus, of r
D(i,ji). In fact,
* if ji > 0, r
ti
lies above its medium term tendency, pu(ji) < 1
2 and pd(ji) > 1
2 ; moreover,
for any integer k > 0, pu(ji) > pu(ji + k) while pd(ji) < pd(ji + k), i.e., the larger
the ji, the greater the calling back strength;
* if ji < 0, r
ti
lies below its medium term tendency, pu(ji) > 1
2 and pd(ji) < 1
2 ; moreover,
for any integer k < 0, pu(ji) < pu(ji + k) while pd(ji) > pd(ji + k), i.e., the smaller
the ji, the greater the calling back strength.
Following Hull and White (1994), we have to convert the tree for r
ti
(i.e., for r
D(i,)) into
a tree for rti (i.e., for rD(i,)). This is accomplished by displacing the nodes on the r
ti
-tree
so that the initial term structure is exactly matched. Define:
ti = rti - r
ti
(i = 0,...,M)
and let
(i) = rD(i,) - r
D(i,)
its lattice version. For a given "time level" i = 0,...,M, all nodes are shifted by the same
amount (i). By the definitions of rti and r
ti
(14), (15), the variation of ti in the interval
t is:
ti = k[ti - ti ] t.
Then, as t  0 it holds:
dt = k[t - t ]dt.
The estimation of t , i.e., the calibration of the tree, could be done via the "spline­wavelet"
method proposed in Cattani and Izzi (2000)8 or, in turn, by following the procedure pro-
8 Cattani and Izzi (2000) propose a spline Haar-wavelet interpolation as a flexible tool for interest rate and term
structure estimation. The main advantage in employing wavelet functions is that they are a very well-localized
representation of the process: the accuracy is reached with only a few sets of data and singularities do not influence
the estimation of the entire process.
Ch. 12: Modelling the Term Structure of Monetary Rates 499
posed by Hull and White (1994). The solution t can be used to create a tree for rti from
the corresponding tree for r
ti
. The approach is to set the interest rates on the rD-tree at time
i t to be equal to the corresponding interest rates on the r
D-tree plus the value of () at
time i t while keeping the probabilities the same. Then, for any i = 0,...,M - 1 and
ji  Ji  [jd,ju], it holds that:
r
D(i,ji) = sgn(ji) 
|ji |
l=0
|l|

t

1 - l2k2 t2
and
rD(i,ji) = r
D(i,ji) + (i),
where
sgn(ji) =
+1, if ji 0,
-1, if ji < 0.
Remark 3.5. For any i = 0,...,M - 1 and ji  Ji  [jd,ju], r
D(,), pu() and pd() do
not depend on i, a property referred to as stationarity: the behavior of r
D depends on its
current value via j but not on the date.
In the next instant of time, ti+1 = (i + 1) t, r
D() could assume the following values:
r
D(ji+1) =
r
D(ji) + u(ji,ji+1), w. prob. pu(ji),
r
D(ji) + d(ji,ji+1), w. prob. pd(ji),
while
rD(i + 1,ji+1) =
r
D(ji) + u(ji,ji+1) + (i + 1), w. prob. pu(ji),
r
D(ji) + d(ji,ji+1) + (i + 1), w. prob. pd(ji).
Let us consider now the discrete jump process:
X
(2)
ti
=



 + , w. prob. 
2 ,
0, w. prob. 1 - ,
 - , w. prob. 
2 .
For i = 0,...,M - 1, let rJD(i,) denote the value corresponding to
rti+1 = rti + k(ti - rti ) t + X
(1)
ti+1
+ X
(2)
ti+1
(i = 0,...,M - 1), (16)
500 L. Izzi
on a lattice scheme, while r
JD(i,) that of
r
ti+1
= r
ti
- kr
ti
t + X
(1)
ti+1
+ X
(2)
ti+1
(i = 0,...,M - 1). (17)
The convolution between the jump and the diffusion processes provides the outcome for
representing the evolution of the entire term structure. The product space resulting from
this convolution could be represented by a hexanomial tree.
With reference to r
JD(i,), i.e., to Equation (17), we have:
r
JD(i + 1,ji+1) = r
JD(ji+1)
=



r
D(ji) + u(ji,ji+1) +  + , w. prob. 
2 pu(ji),
r
D(ji) + u(ji,ji+1), w. prob. (1 - )pu(ji),
r
D(ji) + u(ji,ji+1) +  - , w. prob. 
2 pu(ji),
r
D(ji) + d(ji,ji+1) +  + , w. prob. 
2 pd(ji),
r
D(ji) + d(ji,ji+1), w. prob. (1 - )pd(ji),
r
D(ji) + d(ji,ji+1) +  - , w. prob. 
2 pd(ji).
Then, the hexanomial tree representing the jump-diffusion process (16) is defined by
rJD(i + 1,ji+1) = r
JD(ji+1) + (i + 1),
for any i = 0,...,M - 1 and ji  Ji  (jd,ju).
4. The econometric analysis
4.1. Data
The short-term rate used in estimating our model is the one-month Euribor (EURo InterBank
Offered Rate). Data come from the European Central Bank and Datastream and have
been sampled daily from 1 January 1999 to 5 March 2001 (567 working days). Weekends
and holidays have not been treated specifically (Monday is taken as the next day after Friday).
Whereas weekend effects have been documented for stock prices, there does not seem
to be a conclusive weekend effect on money market instruments. While similar theoretical
and empirical work could be performed on zero-coupon bond yield, the liquidity and
default characteristics of such securities are different from those of interbank instruments
and the time-varying effects introduced by such features would blur our analysis.
Before defining our stochastic model for the interbank interest rates, a statistical analysis
on a wide set of data has been done. The short-term rates used in this preliminary step are
the daily interbank euro rates: overnight, 1 week, 1­12 months. The overnight rate is EONIA
(Euro OverNight Index Average) daily calculated by the European Central Bank; the
Ch. 12: Modelling the Term Structure of Monetary Rates 501
Table 1
Descriptive statistics of the short-term interbank rates
Rate Mean Standard deviation Skewness Kurtosis
o / n 3.549 0.906 0.256 -1.295
1 w 3.604 0.872 0.257 -1.512
1 m 3.653 0.872 0.216 -1.524
2 m 3.712 0.875 0.175 -1.508
3 m 3.768 0.876 0.133 -1.498
4 m 3.809 0.870 0.126 -1.499
5 m 3.846 0.867 0.114 -1.499
6 m 3.875 0.867 0.099 -1.509
7 m 3.900 0.868 0.083 -1.511
8 m 3.927 0.870 0.062 -1.510
9 m 3.955 0.873 0.036 -1.503
10 m 3.983 0.877 0.015 -1.484
11 m 4.007 0.877 -0.016 -1.469
12 m 4.034 0.881 -0.042 -1.450
interbank rates, from 1 week to 12 months, are the Euribor. Table 1 reports the descriptive
statistics of the fourteen rates.9
4.2. Estimation results
In the European money market, jumps may arise from intervention by the European Central
Bank (ECB) on the official rates. Short rates tend to track these rates ­ and, above all, the
main refinancing operation rate ­ rather closely. The aim of estimations is to examine
the impact of changes in the main refinancing operation rate level t on monetary rates.
Equation (4) is specified as follows:
rt = k[ti - rti ] t +  Bt + a , 2
Nt(),
at+ t = 0[t+ t - t].
The process is modelled as if the main refinancing operation rate were actually implemented
and perceived by the market. Main refinancing operation rate changes are assumed
to be independent from the short-rate process and to be infrequent, with  1 the probability
of the main refinancing operation rate change on any day t. In other words, the
main refinancing operation rate changes are modelled here as outcomes in Bernoulli trials,
where the probability of an event ­ main refinancing operation rate change on any given
day ­ occurring in a single trial is constant.
9 It should be noted that monetary policy actions influence the three policy rates, beginning from different dates;
however, for the statistical analysis, we consider as simultaneous the ECB influence on the three rates, because of
the immediate effect of the announcement on the short-term structure.
502 L. Izzi
The jump-diffusion process is estimated using indirect inference method. The problem
of the optimal estimator is solved with respect to both the generalized method of moments
(GMM) and the full information maximum likelihood estimation (FIML). In the latter case,
parameter estimates are obtained by numerically maximizing the sample log-likelihood
function. The standard error is estimated using the Hessian matrix at the optimum point.
The optimization algorithm is the Gauss­Newton method.
In a second stage, the generalized method of moments is used to obtain efficient estimates
of the parameters of the Poissonian part of the process. These are derived as follows:
 = E{t+ t - t},  = E{at},  2 = Var{at}. In particular, the estimate for the daily probability
 is given by the empirical frequency of main refinancing operation rate changes.
There are 8 main refinancing operation rate changes in our sample of 567 working days.
Equation (4) has been estimated with respect to all the interbank short rates, from
overnight to twelve-month maturity. Estimation results indicate that the presence of jump
components superimposed on the diffusion process is significant in any of the fourteen interest
rates, so that the assumption is consistent with the empirical evidence. Estimations
also show that the impact of monetary policy actions is particularly evident in those rates
whose maturity lies between one and three months. The estimates performed on these rates
produce ­ from the statistical point of view ­ the most significant results.
The model shows a good fit with respect to both the GMM and the FIML estimation
method. The best performance is obtained with respect to the two-month Euribor (see
Figure 2). A T-test on the forecast error has also been applied. The test results show how,
in the case of two-month Euribor, the prevision error is in mean statistically equal to zero.
In the case of one-month and three-month Euribor, even if the error is not so large, the
T-test refuses the hypothesis that the observed and estimated series are statistically equal
in the average (see Tables 2 and 3).10
10 The usual T-statistic for testing the equality of averages Xn and Ym from two independent samples with n and
m observations is:
Tn,m =
|Xn - Ym|
(n - 1)S2
n + (m - 1)S2
m
nm(n + m - 2)
n + m
 T(n+m-2),
where S2
n and S2
m are the sample variances of the two groups. The T-statistic assumes that n = m, where n
and m are the population variances of the two groups.
We use the folded form of the F (Fisher) statistic, F , to test the assumption that the variances are equal, where
F =
max[S2
n,S2
m]
min[S2
n,S2
m]
.
Under the assumption of equal variances, the T-statistic is computed with the formula given above. Under the
assumption of unequal variances, the approximate T-statistic is computed as
Tn,m =
|Xn - Ym|
S2
n/n + S2
m/m
.
Ch. 12: Modelling the Term Structure of Monetary Rates 503
Fig. 2. A comparison among the main refinancing operation rate and actual and fitted two-month Euribor (percentage
values).
Table 2
T-test between the actual series of one-, two- and three-month Euribor and their estimates with respect to the full
information maximum likelihood method
Series 1m Euribor 2m Euribor 3m Euribor
Mean value of the actual series 3.65312898 3.71241696 3.76756714
Mean value of the fitted series 3.43574164 3.64401606 3.51284076
T-test 4.2619 1.2725 5.0116
Prob > |T | 0.0001 0.2035 0.0001
Table 3
T-test between the actual series of one-, two- and three-month Euribor and their estimates with respect to the
generalized method of moments
Series 1m Euribor 2m Euribor 3m Euribor
Mean value of the actual series 3.65312898 3.71241696 3.76756714
Mean value of the fitted series 3.42677354 3.63484966 3.50756698
T-test 4.4117 1.4274 5.0721
Prob > |T | 0.0001 0.1538 0.0001
504 L. Izzi
Table 4
Parameter estimates of the jump-diffusion process with respect
to the generalized method of moments
One-month Euribor
Parameter Estimate T-statistic
k -0.010033 -22.08
 0.993974 395.68
 0.003086 1.70
 0.019877 4.41
 0.013522 4.41
Fig. 3. A comparison between the main refinancing operation rate changes and the shock related to the estimation
of the one-month Euribor (percentage values).
The estimates of the parameters of the Poissonian part of the process, which are calculated
with respect to the one-month Euribor and become asymptotically more significant
as the length of the horizon grows, suffer the narrow size of the sample, which does not
allow us to make rolling estimations. This problem could be overcame using Monte Carlo
simulations. Parameter estimates are summarized in Table 4.
As a final observation, let us consider the shock of the model, derived from a stochastic
perturbation process. As shown in Figures 3­5, the shocks related to the three monetary
rates (from one- to three-month Euribor) capture the effects of the approach main refinancing
operation rate changes, thus giving a measure of market expectations.
It has to be noted that in all the three cases, the shocks (Figures 3­5, peaks in plain)
anticipate the interventions of the ECB (peaks in bold). In particular, in the case of the
first intervention ­ the only downward one in our sample period ­ the shock magnitude
Ch. 12: Modelling the Term Structure of Monetary Rates 505
Fig. 4. A comparison between the main refinancing operation rate changes and the shock related to the estimation
of the two-month Euribor (percentage values).
Fig. 5. A comparison between the main refinancing operation rate changes and the shock related to the estimation
of the three-month Euribor (percentage values).
is smaller than that observed in the case of upward interventions, taking into account that
upward interventions are closer to each other.
The downward movements observed in the three shock series at the end of 1999 (see the
plain peaks) show a behavior only apparently contrary to monetary policy indications (bold
506 L. Izzi
peaks). In fact, with the intention of contributing to a smooth transition to the year 2000,
the European Central Bank injected, via longer-term and main refinancing operations, an
amount of liquidity greater than that demanded by the market. Normal liquidity conditions
in the money market were restored with the fine-tuning operation conducted by the ECB
on 5 January 2000.
Some large positive peaks observed in the shock series far from the main refinancing operation
rate changes are due to technical reasons like, first of all, the end of the maintenance
period.
5. Conclusions
In this chapter a jump-diffusion mean-reverting model for estimating monetary rates is
introduced and related to the class of models driven by infinitely divisible processes to
which Gaussian, Poissonian and Stable ones belong. Relating the interest rate models to
the class of Stable processes is an attempt to unify, from the probabilistic point of view, the
extensive literature of pure-diffusion, pure-Poissonian and jump-diffusion processes and
completes recent works on the stock and exchange markets on the same topic.
We also propose a new numerical procedure to recursively compute interest rates subject
Ch. 12: Modelling the Term Structure of Monetary Rates 507
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Chapter 13
ASSET LIABILITY MANAGEMENT: A REVIEW AND SOME NEW
RESULTS IN THE PRESENCE OF HEAVY TAILS
YESIM TOKAT
Department of Economics, University of California, Santa Barbara, USA
SVETLOZAR T. RACHEV
Department of Statistics and Applied Probability, University of California, Santa Barbara, USA
Institute of Statistics and Mathematical Economics, University of Karlsruhe, Germany
e-mail: rachev@lsoe-4.wiwi.uni-karlsruhe.de
EDUARDO S. SCHWARTZ
Anderson School of Management, University of California, Los Angeles, USA
Contents
Abstract 510
1. Introduction 511
Part I: Review of the stochastic programming ALM literature 513
2. Stochastic programming ALM models 513
2.1. Chance-constrained model 513
2.2. Dynamic programming 515
2.3. Sequential decision analysis 516
2.4. Stochastic Linear Programming with Recourse (SLPR) 518
2.5. Dynamic generalized networks 520
2.6. Scenario optimization 521
2.7. Robust optimization 522
3. Multistage stochastic ALM programming with decision rules 523
4. Scenario generation 524
4.1. Discrete time series model 524
4.1.1. Multivariate approach 524
4.1.2. Cascade approach 525
4.2. Continuous time model 526
Part II: Stable asset allocation 528
5. Stable distribution 528
5.1. Description of stable distributions 529
5.2. Financial modeling and estimation 530
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
510 Y. Tokat et al.
5.2.1. Maximum likelihood estimation 531
5.2.2. Comparison of estimation methods 532
6. Multistage stable asset allocation model with decision rules 532
6.1. Scenario generation 534
6.2. Valuation of assets 536
6.3. Computational results 537
7. Conclusion 542
References 543
Abstract
Asset and liability management is the simultaneous consideration of assets and liabilities in
strategic investment planning. In this chapter, asset and liability management models that
use stochastic programming framework are reviewed. Most of these models describe the financial
uncertainty by a set of representative scenarios. We propose to replace the classical
assumption of Gaussian returns in the scenario generation with the stable Paretian distribution,
which can capture the leptokurtic nature of financial data. A multistage stochastic
asset allocation model with decision rules is analyzed. Optimal asset allocation under the
Gaussian and stable Paretian returns are compared. Our computational results suggest that
asset allocation may be up to 20% different depending on the utility function and the risk
aversion level of the investor. Certainty equivalent return can be increased up to 0.13% and
utility can be improved up to 0.72% by switching to the stable Paretian model.
Ch. 13: Asset Liability Management 511
1. Introduction
Managing assets and liabilities is a concern for banks, pension funds and insurance companies.
Before the deregulation of interest rates, the market value of liabilities changed very
little from year to year. However, after interest rates were deregulated in 1979, they showed
much more volatility. This lead the institutional investors mentioned above to consider
assets and liabilities simultaneously during their strategic planning. Strategic investment
planning is the allocation of portfolio across broad asset classes such as bonds, stocks,
cash and real estate considering the legal and policy constraints facing the institution. Empirical
evidence by Brinson, Hood and Beebower (1986) suggests that asset allocation is
the most important factor in determining investment performance.
Most of the early models in this field are either myopic or represent deterministic formulations
of multiperiod problems. Hakansson (1971) show that solving a sequence of single
period models optimizes investor's long-run wealth or the expected utility of wealth.1 They
assume absence of transaction costs, market impact costs, and liquidity considerations.
However, these assumptions are not justifiable in many situations. Myopic models cannot
capture long-term investment goals in the presence of transaction costs. There is considerable
evidence of predictability in asset returns2 and the myopic models do not take this
empirical finding into account. These models tend to produce high portfolio turnovers and
opportunistic asset trades.
There has been a growing interest in the development of multiperiod stochastic models
for asset and liability management (ALM). Kusy and Ziemba (1986) developed a multiperiod
stochastic linear programming model for Vancouver City Savings Credit Union
for a 5-year planning period. Their work suggests that their stochastic ALM model is superior
to 5-year deterministic models. Another successful application of multistage stochastic
programming is the Russell­Yasuda Kasai model by Carino et al. (1994). The investment
strategy suggested by the model resulted in extra income of $79 million during the first two
years of its application (1991 and 1992). An ALM model designed by Mulvey (1994) has
been implemented by the Pacific Financial Asset Management Company. Boender (1997)
reported the success of a hybrid simulation/optimization scenario model for ALM of pension
funds in the Netherlands. The application of the model to a particular pension fund
lead to a reduction of the yearly expected contributions of $100 million.
The ALM models that have gained applicability are based on stochastic programming
with or without decision rules. In these models, the future economic uncertainty is modeled
using discrete scenarios. Most of the models assume that the variables or the innovations
of these variables follow normal distribution or the continuous time counterpart, Brownian
1 Merton (1969) and Samuelson (1969) show that a constant relative risk aversion investor chooses the same
asset proportions independent of the investment horizon if the market is frictionless and returns are independent
over time.
2 See for example Hodrick (1992), Bekaert and Hodrick (1992), Kandel and Staumbaugh (1996), and Brandt
(1999).
512 Y. Tokat et al.
motion. In response to the empirical evidence about the heavy tails, high peak and possible
skewness in financial data, Fama (1965) and Mandelbrot (1963, 1967) propose stable
Paretian distribution3 as an alternative model. Among the alternative non-Gaussian distributions
in the literature,4 stable distribution has unique characteristics that make it an ideal
candidate. The stable laws are the only possible limit distributions for properly normalized
and centered sums of independent identically distributed random variables (Embrechts
et al., 1997; Rachev and Mittnik, 2000). If a financial variable can be regarded as the result
of many microscopic effects, then it can be described by a stable law. Stable distributions
are leptokurtotic. When compared to normal distribution, they typically have fatter tails
and higher peak around the center. Asymmetry is allowed. Due to its flexibilities, stable
model fits the empirical distribution of the financial data better [see Mittnik et al. (2000)].
Gaussian distribution is a special case of stable distribution. In fact, it is the only distribution
in the stable family with a finite second moment. Although autoregressive conditional
heteroskedastic models driven by normally distributed innovations imply unconditional
distributions that possess heavier tails, there is still considerable kurtosis unexplained by
this model. Mittnik, Paolella and Rachev (2000) present empirical evidence favoring stable
hypothesis over the normal assumption as a model for unconditional, homoskedastic
conditional, and heteroskedastic conditional distributions of several asset return series.
The purpose of this chapter is to review the stochastic programming models in the ALM
literature and to analyze an asset allocation problem in the presence of heavy tails. In
the first part of the chapter, we review ALM models that utilize stochastic programming
methodology. In the second part, a multistage asset allocation model with decision rules
is analyzed under the Gaussian and stable returns scenarios. Our computational results
suggest that if the investor has very high or low risk aversion, then the normal and stable
scenarios result in similar asset allocations. However, when the risk aversion level is
between the two cases, the two distributional assumptions may result in considerably different
asset allocations depending on the utility function and the risk aversion level of the
decision maker. The investor may reduce his equity allocation up to 20%, increase his certainty
equivalent wealth up to 0.13% and improve his utility by 0.72% by switching to
stable model.
Section 2 reviews the stochastic programming ALM models without imposing any decision
rules. Section 3 describes stochastic programming models that assume the investor
uses the same decision rule to update the asset allocation every period. In Section 4, we
present the scenario generation methods available in the ALM literature. In the second
part of the chapter, a stable asset allocation model is described. Section 5 states the reasons
for desirability of the stable model, describes the distribution and presents estimation
methods. In Section 6, we set up the asset allocation model and report the computational
results. Section 7 concludes.
3 We will call it stable distribution from now on.
4 A well known alternative to stable model is Student-t distribution. A major drawback of Student-t distribution
is its lack of stability with respect to summation, i.e., a portfolio of Student-t distributed asset returns does not
have Student-t distribution. It is not supported by a central limit theorem. Student-t distribution is a symmetric
distribution and it cannot capture the possible skewness in financial data.
Ch. 13: Asset Liability Management 513
Part I: Review of the stochastic programming ALM literature
2. Stochastic programming ALM models
This method provides a general-purpose modeling framework that conveniently addresses
real world concerns such as transaction costs, taxes, legal and policy constraints. The number
of decision variables becomes very large resulting in large scale optimization problems.
The computational costs make it impractical to test the recommendations out of the sample.
We describe various modeling approaches developed within this framework:
2.1. Chance-constrained model
Charnes and Kirby (1966) develop a chance-constrained model that expresses future deposits
and loan payments as jointly distributed random variables, and capital adequacy
formula by chance-constraints on meeting withdrawal claims. A drawback of the model is
that constraint violations are not penalized according to their magnitude.
The methodology has found applications in various areas: Charnes, Gallegos and Yao
(1982) applies this methodology to balance sheet management, Li (1995a, b) uses chanceconstrained
programming in portfolio analysis of insurance companies, and Dert (1998)
develops a multistage chance-constrained ALM model for a defined benefit pension fund.
As opposed to the original approach of Charnes and Kirby, Dert models the uncertainty
using scenarios rather than making distributional assumptions.
Derťs model minimizes the cost of funding while ensuring the stability of contributions
and ability to make benefit payments timely with an acceptable level of insolvency risk.
The solvency requirement is the asset level being at least equal to the product of required
funding level with the value of the remaining liabilities (constraint (7)). The asset value
falling below the required level is modeled as a probabilistic constraint. Since uncertainty is
modeled through scenarios, binary variables are needed to formulate the chance constraint
explicitly (constraints (8)­(10)). It is assumed that remedial contributions are made in case
of under-funding (constraint (6)).
The ALM model is formulated as follows:
minA01 +
T -1
t=1
St
s=1
P(t,s)tsYts + 
T
t=1
St
s=1
P(t,s)tsZts subject to
Yl
ts Yts Yu
ts, (1)
yl
ts
Yts
Wts
yu
ts, (2)
Yts
Wts
-
Yt-1,^s
Wt-1,^s
t, (3)
514 Y. Tokat et al.
Ats + Yts - lts =
N
i=1
Xits, (4)
xl
its(Ats + Yts - lts) Xits xu
its(Ats + Yts - lts),
t = 0,...,T - 1, s = 1,...,St , (5)
Ats = Zts +
N
i=1
erits
Xi,t-1,^s, (6)
Ats Lts, (7)
Zts ftsMts, (8)
St
s=1
P (t,s)|(t - 1, ^s) fts t-1,^s, (9)
fts  {0,1}, t = 0,...,T - 1, s = 1,...,St , (10)
where,
t = 0,1,...,T is the time period,
s = 1,2,...,St is the status of the world,
i = 1,2,...,N is the asset class,
 is the demanded funding level,
t is the maximal raise in contribution per period as a fraction of the cost of wages at
time t,
ts is the discount factor for a cash flow at time t in state s,
lts is the benefit payments and costs to the fund at time t in state s,
Lts is the actuarial reserve at time t in state s,
 is the penalty parameter to penalize remedial contributions,
rits is the continuous return on investment of each asset class i during period t in
state s,
Mts is the large constant at time t in state s,
Wts is the cost of wages during period t in state s,
Ats is the total asset value before receiving regular contributions and making benefit
payments at time t in state s,
fts is the binary variable for remedial contributions at time t in state s,
ts is the probability of under-funding at time t + 1 given the world was in state s at
time t,
Xits is the amount of money invested in asset class i at time t in state s,
xits is the fraction of asset value invested in asset class i at time t in state s,
Yts is the regular contribution during period t in state s,
yts is the regular contribution as a fraction of the cost of wages during period t in
state s,
Zts is the remedial contribution at time t in state s.
Ch. 13: Asset Liability Management 515
The first three constraints, namely (1)­(3), limit the regular contribution amount, regular
contribution as a fraction of wages and maximal raise in contribution as a fraction of cost
of wages, respectively. After receiving regular contributions and making benefit payments,
the assets are reallocated (4) considering the upper and lower bounds on the asset mix (5).
The price inflation, wage inflation, and asset return scenarios are generated using vector
autoregressive model. The characteristics of participants are modeled using a Markov
chain. More detailed description of a similar model is given in Dert (1998).
2.2. Dynamic programming
The main idea behind dynamic programming is to solve the problem by dealing with one
stage at a time. The procedure produces one solution per possible state in each stage. If
there are many state variables or the objective function depends in an arbitrary way on the
whole history up to the current period, this method is not very appropriate. It can handle
small number of financial instruments simultaneously. Therefore it is of limited use in
practice.
Eppen and Fama (1971) model a three-asset portfolio problem using this approach.
At any point in time, they assume that state of the system is described by two variables:
m being the level of cash balance (m  N), and b being the level of bond account
(b  {N - N-}). Decisions concerning the state of the system are made at equally spaced
discrete points in time. The stochastic changes in the cash balance between the periods
are a sequence of independent identically distributed random variables with the discrete
probability mass function p(d). The function p(d) is positive only on a finite state space,
i.e., there is a finite K such that p(d) = 0 if |d| > K.
The notation is as follows:
T (m,b;m ,b ) is the minimum transfer cost involved in changing the state from
(m,b) to (m ,b ),
ch is the marginal opportunity cost of starting a period with an additional dollar of
cash,
cp is loss of being a dollar short on cash which is incurred at the beginning of the
period,
L(m ) is the penalty cost of carrying cash:
L(m ) =
chm , m 0,
-cpm , m < 0,
 is the discount factor,
fn(m,b) is the discounted expected cost for an n period problem whose state at the
beginning of period n is (m,b).
The recursive relationship for fn(m,b) is given by:
fn(m,b) = min
m ,b
T (m,b;m ,b ) + Gn(m ,b ) ,
516 Y. Tokat et al.
where
Gn(m,b) = cb
h  b + L(m) + 
K
d=-K
fn-1(m + d,b)  p(d).
Gn(m,b) is the current expected holding penalty cost (the first two terms) plus the discounted
expected cost if a decision is made to start period n in state (m,b) and an optimal
policy is followed in period n - 1 and all future periods.
2.3. Sequential decision analysis
This approach uses implicit enumeration to find an optimal solution. It results in extremely
large equivalent linear programming problems since it enumerates all possible portfolio
strategies for all scenarios in all periods of consideration. The method ensures feasibility of
the first period for every possible scenario, this shrinks the feasible set and gives substantial
importance to scenarios with low probabilities of occurrence.
Stochastic decision tree model by Bradley and Crane (1972) overcomes the computational
difficulties of the approach by using a decomposition algorithm. The objective is
the maximization of expected terminal wealth of the firm. Constraint (11) guarantees that
the firm cannot purchase assets that cost more than it has funds available. The second set
of constraints balance the inventory. The net realized capital losses in a period are controlled
by some pre-specified upper bound using (13). Constraint (14) limits the holding of
a particular asset.
Their linear programming formulation5 is
max
eN EN
p(eN )
K
k=1
N-1
m=0
yk
m(em) + uk
m,N (eN ) hk
m,N (eN )
+ yk
N (eN ) + uk
N,N (eN ) bk
N (eN ) subject to
K
k=1
bk
n(en) -
K
k=1
n-2
m=0
yk
m(em)hk
m,n-1(en-1) + yk
n-1(en-1)bk
n-1(en-1)
-
K
k=1
n-1
m=0
1 + gk
m,n(en) sk
m,n(en) = fn(en), (11)
-hk
m,n-1(en - 1) + sk
m,n(en) + hk
m,n(en) = 0, m = 0,1,...,n - 2, (12)
5 The formulation is taken from Kusy and Ziemba (1986).
Ch. 13: Asset Liability Management 517
-bk
n-1(en-1) + sk
n-1,n(en) + hk
n-1,n(en) = 0,
hk
0,0(e0) = hk
0,
-
K
k=1
n-1
m=0
gk
m,n(en)sk
m,n(en) Ln(en), (13)
kKi
bk
n(en) +
n-1
m=0
hk
m,n(en) Ci
n(en), i = 1,2,...,I, (14)
yk
m,n(en) 0,
sk
m,n(en) 0,
hk
m,n(en) 0, m = 1,...,n - 1,
where
en  En, n = 1,2,...,N,
k = 1,2,...,K,
en is an economic scenario from period 1 to n having probability p(en),
En is the set of possible economic scenarios from period 1 to n,
K is the total number of assets,
Ki is the number of assets of type i,
N is the number of time periods,
yk
m(em) is the income yield per dollar of purchase price in period m of asset k conditional
on scenario en,
uk
m,N (eN ) is the expected terminal value per dollar of purchase price in period m of
asset k held at period N, conditional on scenario en,
bk
n(en) is the dollar amount of asset k purchased in period n conditional on scenario
en,
hk
m,n(en) is the dollar amount of asset k purchased in period m and held in period n
conditional on scenario en,
sk
m,n(en) is the dollar amount of asset k purchased in period m and sold in period n
conditional on scenario en
gk
m,n(en) is the capital gain or loss per dollar of purchase price of asset k purchased
in period m and sold in period n conditional on scenario en,
fn(en) is the incremental increase or decrease of funds available for period n,
Ln(en) is the dollar amount of maximum allowable net realized capital losses in period
n,
Ci
n(en) is the upper bound in dollars on the amount of funds invested in asset type i
in period n.
They use a decomposition algorithm to breakdown the problem and use an efficient technique
to solve the sub-problems of the overall portfolio. However, the solution is still computationally
intractable for real life problems.
518 Y. Tokat et al.
2.4. Stochastic Linear Programming with Recourse (SLPR)
The basic formulation of the general T -stage SLPR model is
min
x1
c1x1 + Ew1 min
x2
c2(w1) x2 +  + EwT -1 min
xT
cT (wT -1) xT
subject to
A1x1 = b1,
B2(w1)x1 + A2(w1)x2 = b2(w1),
B3(w2)x2 + A3(w2)x3 = b3(w2),
...
BT (wT -1)xT -1 + AT (wT -1)xT = bT (wT -1),
lt xt ut, t = 1,2,...,T,
where
wt is the random vector that generates the coefficients bt, ct , At, and Bt of the decision
problem at the t-th stage, t = 2,...,T ,
lt , ut are the vector of deterministic bounds on xt at stage t, t = 2,...,T ,
b1, c1, and A1 are the deterministic first stage coefficient vectors or matrices, and
xt is the vector decision variable.
The objective formalizes a sequence of optimization problems corresponding to different
stages. At stage 1, the outcome completely depends on future realizations of the
uncertainty. After the first period, decisions are allowed to be a function of the observed
realization (xt-1,wt) only. One first decides on x1, then observes w1, then decides on x2,
then observes w2, and so on. The recourse decisions depend on the current state of the system
as determined by previous decisions and random events. The uncertainty is modeled
by using finite scenarios which have pre-assigned probabilities. In this case, the problem
reduces to a large linear program of a special structure:
min c1x1 +
K2
k2=2
pk2 ck2
xk2 +
K3
k3=K2+1
pk3 ck3
xk3 +  +
KT
kT =KT -1+1
pkT ckT
xkT
subject to
A1x1 = b1,
Bk2 x1 + Ak2 x2 = bk2 , k2 = 2,...,K2,
Bk3 xa(k3) + Ak3 xk3 = bk3 , k3 = K2 + 1,...,K3,
...
BkT xa(kT ) + AkT xkT = bkT , kT = KT -1+1,...,KT ,
lt xkt ut , kt = Kt-1 + 1,...,Kt, t = 1,2,...,T.
The scenarios used determine the size, form and optimal solution of the linear program.
There are finitely many sequences of possible realizations of the random coefficients
Ch. 13: Asset Liability Management 519
(ckt , Akt , Bkt , bkt ) with path probabilities pkt of the subsequences of these realizations,
pkt > 0, kt , Kt
kt =Kt-1+1 pkt = 1, t = 2,...,T , that identify the discrete joint probability
distribution of w = {w1,...,wT -1}. In the program, a(kt) denotes the immediate predecessor
of kt, for example a(k2) = k1.
An important application of stochastic linear programming with simple recourse model
is given by Kusy and Ziemba (1986). The model was developed for the Vancouver City
Savings Credit Union for a 5-year planning period. The formulation has the following
features:
(1) Changing yield spreads across time, transaction costs associated with selling assets
prior to maturity, and synchronization of cash flows across time are incorporated in a
multiperiod context.
(2) Assets and liabilities are considered simultaneously to satisfy basic accounting principles
and match liquidities.
(3) Transaction costs are included.
(4) Uncertainty of withdrawal claims and deposits is reflected in uncertain cash flows.
(5) Uncertainty of interest rates is explicitly recognized.
(6) Legal and policy constraints are taken into account.
Their two-stage model did not contain end effects. Three possible scenarios that are independent
over time were considered to keep the computations tractable. Their results indicate
that their model generates policies that are superior than stochastic decision analysis.
Another milestone after the Kusy and Ziemba model is the Russell­Yasuda Kasai model
by Carino et al. (1994). The model builds on the previous research to design a large scale
SLPR model with possibly dependent scenarios, end effects, and all the relevant institutional
and policy constraints. We present their model next.
Decision variables are
Vt : total fund market value at time t,
Xnt : market value in asset n at time t,
wt+1: income shortfall at time t + 1, and
vt+1: income surplus at time t + 1.
Random coefficients are
RPnt+1: price return of asset n from end of t to end of t + 1,
RInt+1: income returns of asset n from end of t to end of t + 1,
Ft+1: deposit inflow from end of t to end of t + 1,
Pt+1: principal payout from end of t to end of t + 1,
It+1: interest payout from end of t to end of t + 1,
gt+1: rate at which interest is credited to policies from end of t to end of t + 1,
Lt : liability valuation at t.
The objective is to maximize the expected market value of the firm at the horizon net of
penalties for the shortfalls. Expected amount by which goals are not achieved is a more tangible
risk measure than variance. The penalty costs of shortfalls may be based on expected
financial impact or psychological costs. The piecewise linear convex cost function for the
shortfall is denoted by ct(wt). (15) is the budget constraint. The return on assets and inflow
of deposits net of principal and interest payout gives the total fund market value (16).
520 Y. Tokat et al.
Liability balances and cash flows are computed to model liability accumulation (18). If
Yasuda does not achieve adequate income, recourse action must be taken at a cost. The
income generation is modeled as a soft constraint (17), which permits surpluses or deficits.
maxE Vt -
T
t=1
ct (wt) subject to
n
Xnt - Vt = 0, (15)
Vt+1 -
n
(1 + RPnt+1 + RInt+1)Xnt = Ft+1 - Pt+1 - It+1, (16)
n
RInt+1Xnt + wt+1 - vt+1 = gt+1Lt, (17)
Lt+1 = (1 + gt+1)Lt + Ft+1 - Pt+1 - It+1, (18)
Xnt 0, wt+1 0, vt+1 0.
The abbreviated formulation does not include some elements of the model. There are
additional types of shortfalls, indirect investment types, regulatory restrictions, multiple
accounts, loan assets, tax effects and end effects that are included in the original model. See
Carino and Ziemba (1998) for the details of the formulation. Carino, Myers and Ziemba
(1998) discusses the concepts, technical issues and uses of the model.
Korhonen (1987) applies SLPSR to multicriteria decision making. Oguzsoy and Guven
(1997) use the SLPSR methodology for a bank ALM model in Turkey. Geyer et al. (2002)
describes a pension fund planning model that utilizes this approach.
Some authors argue against linearizing the objective function. Bai, Carpenter and Mulvey
(1997) demonstrates that nonlinear programs are not much more difficult than their
linear counterparts. Zenios et al. (1998) applies multistage stochastic nonlinear programming
with recourse to fixed income portfolio management.
2.5. Dynamic generalized networks
Multistage stochastic nonlinear programs with recourse can be represented by generalized
network formulations. This framework can be used to account for the dynamic aspects of
ALM problems while considering uncertainty in all relevant parameters and accommodating
random parameters by means of a moderate number of scenarios.
The network structure is exploited in the solution procedure.The problem is decomposed
into its constituent scenario subproblems. Preserving the network structure of each subproblem
is challenged by the existence of non-anticipativity constraints. These constraints
dictate that scenarios that share common information history up to a specific period must
yield the same decision up to that period, i.e., dependence on hindsight is avoided.
Ch. 13: Asset Liability Management 521
The desired decomposition is achieved by dualizing the non-anticipativity constraints.
The algorithm by Rockafeller and Wets (1991) operates on the split-variable form of the
original problem. The problem is solved by progressively enforcing the non-anticipativity
constraints.
Mulvey (1994) utilizes this methodology in designing an asset allocation model for the
Pacific Financial Asset Management Company. The single period portfolio model is formulated
as a network model. The arcs can be constrained to impose legal or policy constraints.
The objective function is the expected utility of surplus at the end of the planning horizon.
The model was implemented in a PC environment with acceptable accuracy and efficiency.
Mulvey and Vladimirou (1989, 1991, 1992) present several aspects of stochastic generalized
network models. See also the review of Mulvey and Ziemba (1995) which discusses
the model in a general context.
2.6. Scenario optimization
According to the scenario optimization approach, one computes a solution to the deterministic
problem under all scenarios then solves a coordinating model to find a single feasible
policy. This approach can be compared to the scenario aggregation method suggested by
Rockafeller and Wets (1991). It handles multistage stochastic programming problems, and
allows for decisions to depend on future outcomes. On the other hand, scenario optimization
is designed for two-period models only. It is assumed that scenario probabilities are
functions of time, and estimates of the random parameters in the future stages are poor.
Hence one only selects a policy for the immediate future.
Suppose the scenario subproblem is
vs = mincT
s x subject to
Asx = bs, (19)
Adx = bd, (20)
x 0,
where the objective function is a particular realization of the uncertainty under scenario s,
(19) is also a particular realization of the uncertain constraints under scenario s, and (20)
is the deterministic constraints.
A possible coordinating model could be
min
s
ps cT
s x - vs
2
+
s
ps Asx - bs
2
subject to
Adx = bd,
x 0.
522 Y. Tokat et al.
The coordinating model tracks the scenario solution as close as possible while still maintaining
feasibility. Alternative coordination models are discussed in Dembo (1991, 1993).
Illustrative applications in portfolio immunization and dedication are also presented
therein.
2.7. Robust optimization
Robust optimization approach integrates goal programming formulations with a scenario
based description of the uncertainty in the data. The aim is to produce solutions that are
relatively less sensitive to the realizations of different scenarios. The objective function, in
its most general form, is composed of two terms: the first term trades off between mean
value and the variability in the mean; the second term is a feasibility penalty function.
Consider the following formulation.
min (x,y1,...,ys) + w(z1,...,zs) subject to
Ax = b,
Bsx + Csys + zs = es, s  ,
x 0, ys 0, s  ,
where
x is the vector of decision variables whose value cannot be adjusted once a specific
realization of the data is observed,
y is the vector of decision variables that are subject to adjustment once uncertain
parameters are observed,
z is the vector of decision variables that measure infeasibility allowed,
s   = {1,...,S} is the set of possible scenarios,
A, b, Bs, Cs, es are the coefficients related to the variables,
w is the goal programming weight that is used to derive a spectrum of answers that
trade-off the two objectives.
The inclusion of higher order moments in the objective function reduces the variability
of the solution. Hence, few adjustments become necessary as scenarios unfold. The model
recognizes that it may not always be possible to find a feasible solution to the problem under
all scenarios. The penalty function is used to detect the least amount of infeasibilities to
be dealt with outside the model. See Mulvey, Carpenter and Mulvey (1995) for possible objective
function choices and their applications. Bai, Vanderbrei and Zenios (1997) argues
that linear objective functions fail to identify robust solutions and concave utility functions
produce much better results for risk averse decision makers even when penalty term
is not used. Both papers compare robust optimization with stochastic linear programming
approach (SLP). Since SLP optimizes only the first moment of the distribution of the objective
value, more adjustment is needed as scenarios are realized. However, there is no mechanism
for choosing w, and the cost of the robust solution may be higher than that of SLP.
Ch. 13: Asset Liability Management 523
3. Multistage stochastic ALM programming with decision rules
In this method, time is discretized into n-stages across the planning horizon, and investments
are made using a decision rule, e.g., fixed mix, at the beginning of each time period.
The decision rule can easily be tested with out-of-sample scenarios and confidence limits
on the recommendations can be constructed. The use of this approach hinges on discovering
policies that are intuitive and that will produce superior results. Decision rules may
lead to non-convexities and highly nonlinear functions. Some decision rules used in the literature
are fixed mix, no rebalancing, life cycle mix (Berger and Mulvey, 1998), constant
proportional portfolio insurance (Perold and Sharpe, 1988), target wealth path tracking
(Mulvey and Ziemba, 1998).
Boender (1997) and Boender, van Aalst and Heemskerk (1998) describe an ALM model
designed for Dutch pension funds. Their goal is to find efficient frontiers of initial asset
allocations, which minimize the value of downside risk for given certain values of average
contribution rates. The scenarios are generated across time horizon of interest. The management
selects a funding policy, an indexation policy of the earned pension rights, and an
investment decision rule. These strategies are simulated against generated scenarios. Then,
the objective function of the optimization problem is a completely specified simulation
model except for the initial asset mix. The hybrid simulation/optimization model has the
following three steps:
(1) Randomly generate initial asset mixes, simulate them, and evaluate their contribution
rates and downside risks.
(2) Select the best performing initial asset mixes that are located at a minimal critical
distance from each other.
(3) Use a local search algorithm to identify the optimal initial asset mix.
Maranas et al. (1997) adopt another approach to stochastic programming with decision
rules. They determine the optimal parameters of the decision rule by means of a global
optimization algorithm. They propose a dynamically balanced investment policy which is
specified by the following parameters:
w0: initial dollar wealth,
rs
it : percentage return of asset i  {1,2,...,I} in time period t  {1,2,...,T } under
scenario s  {1,2,...,S},
ps: probability of occurrence of scenario s.
The decision variables are:
ws
t : dollar wealth at time t in scenario s,
i : fraction of wealth invested in asset category i (note that it is constant over time).
The model is a multiperiod extension of mean­variance method. The multi-period efficient
frontier is obtained by varying  (0  1). The formulation is as follows:
max
i,ws
t
 mean(wT ) - (1 - )var(wT ) subject to
ws
T = w0
T
t=1
I
i=1
(1 + rs
it)i , s = 1,...,S, (21)
524 Y. Tokat et al.
I
i=1
i = 1, (22)
0 i 1, i = 1,...,I.
The wealth accumulation is governed by (21). When (21) is substituted into the objective1
i = 1, (22)
0 i 1, i = 1,...,I.
The wealth accumulation is governed by (21). When (21) is substituted into the objective1
i = 1, (22)
0 i 1, i = 1,...,I.lat2 Tf.8-2..31lo8()].126r()T
Ch. 13: Asset Liability Management 525
There are a number of variables used to predict stock returns in various studies. Brennan,
Schwartz and Lagnado (1997) use Treasury bill rate, Treasury bond rate and dividend yield
as state variables in their model. Brandt (1999) uses lagged excess return on NYSE index
over Treasury bill rate as a state variable in addition to dividend yield, default spread and
term spread.
VAR may sometimes diverge from long-term equilibrium. Boender, van Aalst and
Heemskerk (1998) extend VAR model to a Vector Error Correction Model (VECM) which
additionally takes economic regime changes and long term equilibria into account. First,
a sub-model generates future economic scenarios. Then, a liability sub-model determines
the earned pension rights and payments corresponding to each economic scenario.
The economic scenario sub-module uses time series analysis. The vector of the lognormal
transformations of inflation, wage growth, bond return, cash return, equity return,
real estate return and nominal GNP growth is yt . Diagnostic tests revealed that the order of
the VAR process as 1.
yt  N  +   {yt-1 - }, ,
where N(,) denotes a Gaussian distribution with mean  and covariance matrix .
The extended VECM is given as
yt  N 1yt-1 + 2CT
(xt-1 - 1I{T1} - 2I{T2}), ,
where the 1 corresponds to the short term dynamics and the 2 corresponds to the long
term correction. The index set T1 specifies the period of an economic regime with growth
vector 1, and T2 gives the period of another economic regime with growth vector 2.
The second term, CT (xt-1 - 1I{T1} - 2I{T2}) generates the error correction to restore
violations of the equilibria, while 2 determines the speed of the response. They estimated
the model by row wise ordinary least squares and seemingly unrelated regression methods.
Then, scenarios are generated iteratively using the parameter estimates. They report that
the VECM improves the explanatory power of the model. The VECM has a more clear
economic interpretation which incorporates regime changes and long run equilibrium.
The liability sub-module uses a push Markov model to determine the future status of
each individual plan member depending on age, gender, and employee category. Given
this information, the pull part of the model is used to determine additional promotions and
new employees. Then, the pension rules are applied to compute the guaranteed pension
payments and earned pension rights.
4.1.2. Cascade approach
Wilkie (1986) suggests using a cascade structure rather than a multivariate model, in which
each variable could affect each of the others. He considers inflation, ordinary shares and
fixed interest securities as the main economic determinants of a stochastic investment
model. The model includes the following variables: inflation, an index of share of dividends,
the dividend yield (the dividend index divided by the corresponding price index)
526 Y. Tokat et al.
Fig. 1. Wilkie's scenario generation model.
on these share indices, and the yield on consols (as a measure of the general level of fixed
interest yields in the market).
Wilkie's investigations and actuarial experience lead him to the conclusion that inflation
is the driving force for the other investment variables. Figure 1,6 where the arrows indicate
the direction of influences, depicts the cascade structure of the model.
The inflation is described first using a first order autoregressive model. The dividend
yield depends on both the current level of inflation and the previous values of itself. The
index of share dividends depends on inflation and the residual of the yield model. The
consol yield also depends on inflation and the residual of the yield model along with the
previous values of itself. Then, the estimated parameters are used to generate future economic
scenarios. Wilkie (1986) improves this basic model.
4.2. Continuous time model
Mulvey (1996) designs an economic projection model for Towers Perrin using stochastic
differential equations. The model has a cascade structure as depicted in Figure 2.7 First
the Treasury yield curve, and then government bond returns, price and wage inflation, and
large cap returns are generated. Lastly, returns on primary asset categories such as small
cap stock and corporate bonds are projected.
It is assumed that short- and long-term interest rates (denoted by rt and lt , respectively)
are linked through a correlated white noise term. The spread between the two is kept under
control by using a stabilizing term. This variant of the two-factor Brennan and Schwartz
(1982) model is as follows:
drt = a(r0 - rt)dt + b

rt dz1,
dlt = c(l0 - lt)dt + e

lt dz2,
6 Source: Wilkie (1986).
7 Resource: Mulvey (1996).
Ch. 13: Asset Liability Management 527
Fig. 2. Mulvey's scenario generation model.
where a and c are functions that depend on the spread between the long and short rates,
b and e are constants, and dz1 and dz2 are correlated Weiner terms.
The price inflation rate is modeled as a diffusion process that depends on short term
interest rate:
dpt = ndrt + g(p0 - pt )dt + h(vpt)dz3,
where pt is the price inflation at time t, and vpt is the stochastic volatility at time t. Since
the volatility of inflation persists, it is represented using Autoregressive Conditional Heteroskedasticity
(ARCH) model. The equation for the stochastic volatility is given by:
dvpt = k(vp0 - vpt)dt + m

vpt dz4,
where g and k are functions that handle the independent movement of the underlying prices
at time t for the price inflation and stochastic volatility, respectively, and h and m are
constants.
Real yields are related to interest rates, current inflation, and expectations for future
inflation. The diffusion equation for long-term yield is
dyt = n(yu,yt,lu,lt ,pu,pl)dl + q(yu,yt,lu,lt ,pu,pl)dt + u(yt)dz5,
where yu is the normative level of real yields, n and q are vector functions that depend
upon various economic factors.
The wage inflation is connected to price inflation in a lagged and smoothed fashion.
The stock returns are broken down into two components: dividends and capital appreciation,
and they are estimated independently. Mulvey reports that the decomposed structure
provides more accurate linkages to the key economic factors such as inflation and interest
rates.
The parameters of the model are calibrated by considering the overall market trends in
the light of historical evidence and subjective beliefs of the management. This model has
been in use at Towers Perrin since 1992. Mulvey and Thorlacius (1998) extend the model
528 Y. Tokat et al.
to a global environment that links the economies of individual countries within a common
framework.
Modeling term structure of interest rates is a very essential part of scenario generation.
The use of binomial lattice models in the valuation of interest rate contingencies is prevalent.
However, the number of scenarios grows very large if the valuation is to be precise.
There are some sampling methods to reduce the size of the event tree such as Monte Carlo
simulation, antithetic sampling and stratified sampling. However, Klaasen (1997) points
out that even if the underlying description is arbitrage-free, a subset of it may include arbitrage
opportunities that may lead to spurious profits. Instead of sampling paths, Klaasen
(1998a) suggests an aggregation method that can be used to reduce the size of the event tree
preserving the arbitrage free description of uncertainty. In Klaasen (1998b), he presents a
solution algorithm which iteratively disaggregates the condensed representation towards a
more precise description of uncertainty.
Part II: Stable asset allocation
In this part of the chapter, we analyze asset allocation problem for an investor that maximizes
isoelastic utility function or an analog of mean-variance objective function at the
end of the investment horizon. Stochastic programming with decision rules is used as the
solution methodology. The financial uncertainty is represented by using a branching event
tree. Each node of the tree represents the joint outcome of all the random variables at
that decision stage and each path through the event tree represents a `scenario'. Financial
scenarios are generated by using asset return predictors documented in the literature. The
investor perceives the world as modeled by the scenario tree and chooses the fix mix proportions
that maximize his objective function. The asset allocation problem is solved under
Gaussian and stable return scenarios. Optimal allocations and utility function values under
these alternative sets of scenarios are compared.
We state the reasons for desirability of the stable model, describe the distribution and
present estimation methods in Section 5. Section 6 sets up the asset allocation model and
reports computational results. Section 7 concludes.
5. Stable distribution
There are several important reasons for modeling financial variables using stable distributions.
The stable law is supported by a generalized central limit theorem (Embrechts et al.,
1997; Rachev and Mittnik, 2000). Stable distributions are leptokurtotic. Since they can accommodate
the fat tails and asymmetry, they fit empirical distribution of the financial data
better.
Any distribution in the domain of attraction of a specified stable distribution will have
properties, which are close to the ones of stable distribution. Even if the observed data does
Ch. 13: Asset Liability Management 529
not exactly follow the ideal distribution specified by the modeler, in principle, the resulting
decision is not affected.
Each stable distribution has an index of stability, which remains the same regardless of
the sampling interval adopted. The index of stability can be regarded as an overall parameter
that can be employed in inference and decision making. However, we should note that
for some financial data empirical analysis shows that the index of stability increases as the
sampling interval increases.
It is possible to check whether a distribution is in the domain of attraction of a stable
distribution or not by examining the tails of the distribution. The tails dictate the properties
of the distribution.
This section describes the properties of stable distribution and addresses the estimation
issues.
5.1. Description of stable distributions
If the sums of linear functions of independent identically distributed (iid) random variables
belong to the same family of distributions, the family is called stable. Formally, a random
variable r has stable distribution if for any a > 0 and b > 0 there exists constants c > 0
and d  R such that
ar1 + br2
d
= cr + d,
where r1 and r2 are independent copies of r, and
d
= denotes equality in distribution.
The distribution is described by the following parameters:   (0,2] (index of stability),
  [-1,1] (skewness parameter),   R (location parameter), and   [0,) (scale parameter).
The variable is then represented as r  S,(,). Gaussian distribution is actually
a special case of stable distribution when  = 2,  = 0. The smaller the stability index is,
the stronger the leptokurtic nature of the distribution becomes, i.e., with higher peak and
fatter tails. If the skewness parameter is equal to zero, as in the case of Gaussian distribution,
the distribution is symmetric. When  > 0 ( < 0), the distribution is skewed to
the right (left). If  = 0 and  = 0, then the stable random variable is called symmetric
-stable (SS). The scale parameter generalizes the definition of standard deviation. The
stable analog of variance is variation, v, which is given by .
Stable distributions generally do not have closed form expressions for density and distribution
functions. They are more conveniently described by characteristic functions. The
characteristic function of random variable r, r() = E[exp(ir)], is given by
r() =



exp -
||
1 - i sign()tan

2
+ i , if  = 1,
exp -|| 1 - i
2

sign()ln + i , if  = 1.
530 Y. Tokat et al.
The p-th absolute moment of r, E|X|p =

0 P(|X|p > y)dy, is finite if 0 < p < ,
and infinite otherwise. Hence, when  < 1 the first moment is infinite, and when  < 2
the second moment is infinite. The only stable distribution that has finite first and second
moments is the Gaussian distribution.
In models that use financial data, it is generally assumed that   (1,2]. There are several
reasons for this:
(1) When  > 1, the first moment of the distribution is finite. It is convenient to be able to
speak of expected returns.
(2) Empirical studies support this parametrization.
(3) Although the empirical distributions of the financial data sometimes depart from normality,
the deviation is not "too much".
In scenario generation, one may need to use multivariate stable distributions. The extension
to the multivariate case is nontrivial. Although most of the literature concentrates
on the univariate case, recently some new results have become available. See for example
Samorodnitsky and Taqqu (1994), Rachev and Mittnik (2000).
If R is a stable d-dimensional stable vector, then any linear combination of the components
of R is also a stable random variable. However, the converse is true under certain
conditions (Samorodnitsky and Taqqu, 1994). The characteristic function of R is given by:
Y () =



exp -
Sd
T
s 1 - isign T
s tan

2
 (ds) + iT
 , if  = 1,
exp -
Sd
T
s 1 + i
2

sign T
s ln T
s  (ds) + iT
 , if  = 1,
where  is the spectral measure which replaces the scale and skewness parameters that
enter the description of the univariate stable distribution. It is a bounded nonnegative measure
on the unit sphere Sd, and s  Sd is the integrand unit vector. The index of stability is
again , and  is the vector of locations.
Stable distributions have infinite variances. The stable equivalent of covariance for SS
variables is covariation:
[R1,R2] =
Sd
s1s
-1
2  (ds),
where (R1,R2) is a SS vector (  (1,2)), and x -1 = |x|k sign(x). The matrix of
covariations determines the dependence structure among the individual variables.
5.2. Financial modeling and estimation
Financial modeling frequently involves information on past market movements. Examples
include technical analysis to derive investment decisions, or researchers assessing the efficiency
of financial markets. In such cases, it is not the unconditional return distribution
which is of interest, but the conditional distribution, which is conditioned on information
Ch. 13: Asset Liability Management 531
contained in past return data, or a more general information set. The class of autoregressive
moving average (ARMA) models is a natural candidate for conditioning on the past
of a return series. These models have the property that the conditional distribution is homoskedastic.
In view of the fact that financial markets frequently exhibit volatility clusters,
the homoskedasticity assumption may be too restrictive. As a consequence, conditional
heteroskedastic models, such as Engle's (1982) autoregressive conditional heteroskedastic
(ARCH) models and the generalization (GARCH) of Bollerslev (1986), possibly in combination
with an ARMA model, referred to as an ARMA-GARCH model, are now common
in empirical finance. It turns out that ARCH-type models driven by normally distributed innovations
imply unconditional distributions which themselves possess heavier tails. Thus,
in this respect, ARCH models and stable distributions can be viewed as competing hy-
potheses.
Mittnik, Rachev and Paolella (1997) present empirical evidence favoring stable hypothesis
over the normal assumption as a model for unconditional, homoskedastic conditional,
and heteroskedastic conditional distributions of several asset return series.
5.2.1. Maximum likelihood estimation
We will describe an approximate conditional maximum-likelihood (ML) estimation procedure
suggested by Mittnik et al. (1999). The unconditional ML estimate of  = (,,,)
is obtained by maximizing the logarithm of the likelihood function
L() =
T
t=1
S,
rt - 

-1
.
One needs to use conditional ML to estimate ARMA and ARMA-GARCH models.
The ML estimation is conditional, in the sense that, when estimating, for example,
an ARMA(p,q) model, one conditions on the first p realizations of the sample, rp,
rp-1,...,r1, and, when  > 1 holds, sets innovations p, p-1,...,p-q+1 to their unconditional
mean E(t ) = 0. The estimation of all stable models is approximate in the
sense that the stable density function, S,(,), is approximated via fast Fourier transformation
(FFT) of the stable characteristic function,

-
eitx
dH(x) =



exp -
|t|
1 - i sign(t)tan

2
+ it , if  = 1,
exp -|t| 1 + i
2

sign(t)ln|t| + it , if  = 1,
where H is the distribution function corresponding to S,(,).
This ML estimation method essentially follows that of DuMouchel (1973), but differs in
that the stable density is approximated numerically by an FFT of the characteristic function
rather than some series expansion. As DuMouchel shows, the resulting estimates are consistent
and asymptotically normal with the asymptotic covariance matrix of T 1/2( ^ - 0)
532 Y. Tokat et al.
being given by the inverse of the Fisher information matrix. The standard errors of the estimates
are obtained by evaluating the Fisher information matrix at the ML point estimates.
For details on stable ML estimation see Mittnik et al. (1999), Mittnik and Rachev (1993),
and Paulauskas and Rachev (1999).
5.2.2. Comparison of estimation methods
When the residuals of the ARMA model have Gaussian distribution, Least Squares (LS) estimation
is equivalent to conditional ML estimation. Furthermore, Whittle estimator is asymptotically
equivalent to LS and ML estimation methods. However, when the innovations
have stable distribution, the properties of conventional estimation methods may change due
to the infinite variance property. In the stable case, ML estimates are still consistent and asymptotically
normal (DuMouchel, 1973); LS and Whittle estimates are consistent but they
are not asymptotically normal. The LS and Whittle estimates have infinite variance limits
with a convergence rate that is faster than that of the Gaussian case (Mikosch et al., 1995).
Calder and Davis (1998) compare LS, Least Absolute Deviation (LAD), and ML methods
for the estimation of ARMA model with stable innovations. Their simulations reveal that
the difference between the estimates of the three methods is insignificant when the index
of stability of the residuals is 1.75. However, when  = 1 or  = 0.75, they report that
the LAD and ML estimation procedures are superior to LS estimation. ML estimation has
desirable properties in both the Gaussian and stable setting, but it is computationally very
demanding. Since the variables of interest in this paper have indices of stability greater
than 1.5, nonlinear LS estimation method has been utilized in this study. Our parameter
estimates are consistent, but they are not asymptotically normal. However, due to the high
index of stability, the parameter estimates are comparable to those that would be achieved
if ML estimation were to be used.
6. Multistage stable asset allocation model with decision rules
The asset allocation problem for an investor that maximizes isoelastic utility function or an
analog of mean-variance objective function at the end of the investment horizon is formulated
as follows:
maxE u ´Ri
s,T subject to
´Ri
s,T =
T
t=1
1 + Ri
s,t - 1,
Ri
s,t =
J
j=1
wi
j rjst,
wi
j 0,
Ch. 13: Asset Liability Management 533
where wi
j is the proportion of funds8 of portfolio i invested in asset j,
´Ri
s,T is compound return of allocation i in time period of 1 through T under scenario
s  {1,2,...,S},
Ri
s,t is the return of the portfolio i under scenario s  {1,2,...,S} in time period
t  {1,2,...,T }, and
rjst is the percentage return of asset j  {1,2,...,J} under scenario s in time period
t.
The restrictions on the model are that there are no short sales and the asset allocation is
updated every month according to fixed mix decision rule.9 In general, fixed mix strategy
requires the purchase of stocks as they fall in value, and the sale of stocks as they rise in
value. Fixed mix strategy does not have much downside protection, and tends to do very
well in flat but oscillating markets. However, it tends to do relatively poorly in bullish
markets (Perold and Sharpe, 1988).
We use two alternative objective functions: the first one is power utility function and the
second one is an analog of mean­variance analysis. The power utility function, which has
constant relative risk aversion, is calculated as follows:10
U Wi
=
1
S
S
s=1
1
(1 -  )
Wi
s
(1- )
,  > -1,
where  is the coefficient of relative risk aversion, and Wi
s is the final wealth. Assuming
that the initial wealth is 1, we compute the final wealth as follows: Wi
s = 1  (1 + ´Ri
s,T ).
A constant relative risk aversion investor chooses the same investment proportions independent
of the investment horizon if the market is frictionless and returns are independent
over time. Fix mix is the optimal portfolio choice in this setting. However, if the returns are
predictable, which is the conjecture of this paper, then the portfolio choice depends on the
investment horizon. Although the fix mix strategy is no longer optimal in this economic
environment, the investor is assumed to follow this decision strategy for computational
simplicity.
The second objective function trades off between mean final return and a measure of
risk:
U ´Ri
T = E ´Ri
T - c  MD ´Ri
T ,
where c is the coefficient of risk aversion.
8 Fix mix rule requires that wi
j does not depend on time.
9 Perold and Sharpe (1988) suggest constant proportion portfolio insurance as an alternative strategy. In this
strategy, one sells stocks as they fall in value and buy stocks as they rise in value.
10 Note that U(Wi ) is finite if (1 -  ) < 2.
534 Y. Tokat et al.
The mean compound portfolio return of fixed mix rule i  {1,2,...,I} at the final
date is:
E ´Ri
T =
1
S
S
s=1
´Ri
s,T .
We consider the following risk measure which gives less importance to outliers than
variance does:
MD ´Ri
T =
1
S
S
s=1
´Ri
s,T - E ´Ri
T
r
, where 1 < r < 2.
Notice that when r = 2, the above risk measure becomes the variance. Since variance
is not defined for non-Gaussian stable variables, we use those values of r < 2 for which
MD( ´Ri
T ) is finite, such as r = 1.5.
The scenario generation module generates asset return scenarios, rjst , for each time
period. At each stage, n new offspring scenarios are generated from the parent scenarios. If
the horizon of interest is T periods, then we produce nT alternative asset return scenarios
for the final date. Optimal asset allocation is calculated for this scenario tree. The scenario
tree is repeated 100 times and the sample average of optimal allocations is reported as the
optimal asset allocation.
6.1. Scenario generation
The portfolio we analyze is composed of Treasury bill and S&P 500. The monthly return
on Treasury bill is assumed to be constant at 6% annualized rate of return. The main challenge
is predicting the return scenarios for S&P 500. The financial variables that are used
to generate the return scenarios for S&P 500 are modeled in a cascade structure similar
to Mulvey11 (1996) (see Figure 3). However, the analysis is done in discrete time as in
Wilkie12 (1995). Monthly data from 2/1965 through 12/1999 is used for the estimation
of the time series models.3-month Treasury bill rate and 10-year Treasury bond rate are
modeled first as measures of short term and long term interest rates. The price inflation
depends on the Treasury bond rate and the previous values of inflation. Following Wilkie's
and Mulvey's approaches, stock returns are analyzed in two components: dividend growth
and dividend yield growth.13
The relationship of economic variables does not denote a one way casual relationship,
but rather indicates the sequencing of the modules. The economic variables are modeled
11 See Section 4.2 for a brief description.
12 See Section 4.1.2 for a brief review.
13 Tokat, Rachev and Schwartz (2002) gives the details of the time series analysis.
Ch. 13: Asset Liability Management 535
using Box­Jenkins methodology. The standard Gaussian Box­Jenkins techniques carry
over to the stable setting with some possible changes.
We do not model the time varying volatility of the economic variables. Fitting ARMAGARCH
models may reduce the kurtosis in the residuals. However, Balke and Fomby
(1994) show that even after estimating GARCH models, significant excess kurtosis and/or
skewness still remains. Mittnik, Rachev and Paolella (1997) present empirical evidence
favoring stable hypothesis over the normal assumption as a model for ARMA-GARCH
residuals. We postpone modeling the time varying volatility to another paper.
Future economic scenarios are simulated at monthly intervals. One set of scenarios is
generated by assuming that the residuals of each variable is identical normally distributed.
This is the classical assumption made in the literature. Another set of scenarios is generated
by assuming that the residuals are identical stable distributed. The estimated normal and
stable parameters14 for the innovations of the time series models are given in Table 1. See
Figures 4­8 for graphical comparison of stable and normal fit to the residuals.
Fig. 3. The scenario generation model.
Table 1
The estimated normal and stable parameters for the innovations
Innovations of Normal distribution Stable distribution
     
Price inflation
(Inf) 6.15e-06 0.0021 1.7072 0.1073 6.15e-06 0.0012
Dividend gr.
(Divg) 9.89e-4 0.0195 1.7505 -0.0229 9.89e-4 0.0114
Dividend yield
(d(Divy)) -0.002551 0.0407 1.8076 0.2252 -0.002551 0.0239
Treasury bill
(d(Tbill)) 0.000336 0.0579 1.5600 0 0 0.0308
Treasury bond
(d(Tbond)) 0.000818 0.0339 1.9100 0 0 0.0230
14 Stable parameters are estimated using maximum likelihood estimation method.
536 Y. Tokat et al.
Fig. 4. The empirical pdf of the residuals of inflation, the Gaussian fit and the stable fit.
The scenarios have a tree structure. At each stage (month) we generate n possible scenarios.
For each scenario, we first generate a normal or stable residual for Treasury bill,
and calculate the corresponding Treasury bill rate for the proceeding month. Then, given
this short rate, we generate Treasury bond rate, price inflation, dividend growth rate and
dividend yield for that month according to the cascade structure and the time series models
we have built. For instance, the inflation rate for next month is generated by using the
Treasury bond rate, inflation rate and the surprise to expected inflation this month, and the
normal or stable innovation of inflation rate next month. Note that we allow for innovation
of each economic variable in each simulated month.
At the next stage, n new offspring scenarios are generated from the parent scenarios.
This continues until the final time of interest. In this study, we generate 2 scenarios for each
month, so 512 possible economic scenarios are considered over the next three quarters.
6.2. Valuation of assets
The monthly return of S&P 500 is derived using the dividend yield and the dividend index.
Dividend index is calculated by multiplying price index with the dividend yield:
DIt = Pt × DYt,
Ch. 13: Asset Liability Management 537
Fig. 5. The empirical pdf of the residuals of dividend growth, the Gaussian fit and the stable fit.
where DIt is the dividend index for period t, Pt is the price index for period t, and DYt
is the dividend yield for period t. The dividend growth is just log differences of dividend
indices.
The dividend yield and dividend growth rate are simulated as explained in the previous
section. Hence, we can get back simulated future price index in period t under scenario s
from the simulated dividend growth and dividend yield indices by
Pst = DIst/DYst.
Then, we can calculate the return for holding S&P 500 for a month under scenario s as
rst = (Pst - Ps(t-1) + DIst)/Ps(t-1).
6.3. Computational results
We first present the mean annualized return of S&P 500 in 100 repetitions of the scenario
tree generated by using the Gaussian and stable distribution models (see Table 2).
538 Y. Tokat et al.
Fig. 6. The empirical pdf of the residuals of dividend yield, the Gaussian fit and the stable fit.
Table 2
Annualized return scenarios on S&P 500
Mean 1% 2.5% 25% 75% 97.5% 99%
Normal 9.07 -122.34 -103.03 -31.97 48.54 129.66 152.90
scenarios
Stable 10.20 -149.17 -107.29 -27.45 44.96 128.68 171.16
scenarios
The table also depicts the percentiles of these return scenarios. It should be noted that the
S&P 500 returns generated by stable scenarios have fatter tails than those of Gaussian scenarios.
Hence, stable scenarios consider more extreme scenarios than Gaussian scenarios
do. Khindanova, Rachev and Schwartz (2001) report similar observations in their paper
where they compute value at risk employing Gaussian and stable distributed daily returns.
They state that 5% percentile of normal and stable distribution are very close, but the 1%
percentile of stable distribution is greater than that of the Gaussian.
Ch. 13: Asset Liability Management 539
Fig. 7. The empirical pdf of the residuals of Treasury bill, the Gaussian fit and the stable fit.
The asset allocation problem has been solved for an investor that maximizes the power
utility of final wealth. The optimal asset allocation depends on the risk aversion level of
the agent. If his relative risk aversion coefficient is very low, such as 0.80, or very high,
such as 10.00, then the Gaussian and stable scenarios result in similar asset allocations
(see Table 3). The intuitive explanation for this is that, the investor who has very low risk
aversion, does not mind the risk very much. Therefore, his decision does not change when
the extreme events are modeled more realistically. Similarly, the investor who has very
high risk aversion, is already scared away from the risky asset. The fatter tails do not affect
his decision much either. On the other hand, an investor who would put 60% in S&P 500 if
he were to use normal scenarios, will put only 48% in S&P 500 if he uses stable scenarios.
The fact that stable scenarios model the extreme events more realistically, results in stable
investor putting less in the risky asset than Gaussian investor does.
The time series models which generate the Gaussian and stable scenarios are the same
except for the residuals being Gaussian or stable, respectively. In our computations, the
15 Note that when  = 1 the power utility function reduces to logarithmic utility function.
540 Y. Tokat et al.
Fig. 8. The empirical pdf of the residuals of Treasury bond yield, the Gaussian fit and the stable fit.
Table 3
Optimal allocations under normal and stable scenarios (T = 3 quarters)
 Normal scenarios Stable scenarios
Optimal percentage invested Optimal percentage invested
S&P 500 (%) Treasury Bill (%) S&P 500 (%) Treasury Bill (%)
0.80 100 0 100 0
1.0015 100 0 88 12
1.50 86 14 66 44
2.30 60 40 48 52
2.70 52 48 42 58
10.00 14 86 12 88
mean return of Gaussian S&P 500 scenarios came out to be less than stable S&P 500
scenarios. The equity premium is 3.07% in the normal scenarios and 4.20% in the stable
scenarios. Since the premium on equity is higher in stable scenarios, the equity is more
attractive. However, the fact that the stable scenarios also have heavier tails outweighs
Ch. 13: Asset Liability Management 541
this, and consequently the investor puts considerably less money in the stock index. If the
equity premium were the same in both sets of scenarios, we contemplate that the allocation
difference would be even more pronounced.
Table 4 depicts the change in the utility16 if the investor uses stable scenarios rather
than Gaussian scenarios. The improvement can be as large as 0.72% depending on the risk
aversion level of the investor. Table 5 reports the improvement in the certainty equivalent
final wealth (CEFW) if an investor uses stable scenarios rather than Gaussian scenarios.17
The computations show a 6 basis point improvement in the certainty equivalent wealth of
the investor who would put 60% in S&P 500. The difference could get larger or smaller
depending on the risk aversion level of the decision maker.
The other `utility' function we consider is an analog of mean-variance criterion. The
computational results achieved are very similar to the constant relative risk aversion utility.
The investor who has very low or very high risk aversion, does not gain much from using
Table 4
Comparison of utility achieved from normal and stable scenarios (T = 3 quarters)
 Normal scenarios Stable scenarios % Change in utility
% in S&P 500 Utility % in S&P 500 Utility
0.80 100 5.0633 100 5.0633 0.00
1.00 100 0.0600 88 0.0604 0.72
1.50 86 -1.9458 66 -1.9445 0.06
2.30 60 -0.7188 48 -0.7181 0.09
2.70 52 -0.5391 42 -0.5386 0.09
10.00 14 -0.0728 12 -0.0728 0.03
Table 5
Comparison of certainty equivalent wealth achieved from normal and stable scenarios (T = 3 quarters)
 Normal scenarios Stable scenarios % Change in CEFW
% in S&P 500 CEFW % in S&P 500 CEFW
0.80 100 1.0650 100 1.0650 0.00
1.00 100 1.0618 88 1.0623 0.04
1.50 86 1.0565 66 1.0579 0.13
2.30 60 1.0536 48 1.0543 0.07
2.70 52 1.0526 42 1.0532 0.05
10.00 14 1.0480 12 1.0481 0.00
16 Note that the utility value becomes negative when  > 1. Although negative utility does not make much sense,
it can be made positive by monotonic transformations.
17 Since Gaussian distribution is a special case of stable distribution, the stable model encompasses the Gaussian
model. Therefore, the certainty equivalency comparison is made under the assumption that stable is the correct
model.
542 Y. Tokat et al.
Table 6
Optimal allocations under normal and stable scenarios (T = 3 quarters)
c Normal scenarios Stable scenarios
Optimal percentage invested Optimal percentage invested
S&P 500 (%) Treasury Bill (%) S&P 500 (%) Treasury Bill (%)
0.35 100 0 100 0
0.40 90 10 80 20
0.52 60 40 54 46
0.59 50 50 44 66
1.00 20 80 18 82
Table 7
Percentage change in utility achieved from normal and stable scenarios (T = 3 quarters)
c Normal scenarios Stable scenarios % Change in utility
% in S&P 500 Utility % in S&P 500 Utility
0.35 100 0.0583 100 0.0583 0.00
0.40 90 0.0561 80 0.0562 0.28
0.52 60 0.0526 54 0.0527 0.10
0.59 50 0.0513 44 0.0514 0.12
1.00 20 0.0479 18 0.0480 0.08
the stable model. However, the stable model makes a difference for the investors in the
middle. Table 6 depicts that an investor who would put 60% in S&P 500 if he were to use
normal scenarios, will put only 56% in S&P 500 if he uses stable scenarios. Table 7 reports
the percentage improvement in the `utility' function18 if one uses stable model as opposed
to Gaussian model. If there is any percentage improvement in the utility function, an investor
can reduce the risk for a given level of mean return or increase the mean return for a
given level of risk. This can be achieved by switching from Gaussian scenario generation
to stable scenario generation.
7. Conclusion
The ALM models that are based on stochastic programming with or without decision rules
are starting to gain applicability in the industry. In these models, the future uncertainty is
modeled using discrete scenarios. A representative set of scenarios describes the possible
future economic situations facing the institution.
18 Since the risk corresponding to certainty equivalent return is zero, the certainty equivalent return is equal to
the utility of return. Hence, the percentage improvement in the utility of return is equivalent to the percentage
improvement in the certainty equivalent return.
Ch. 13: Asset Liability Management 543
Generating scenarios that realistically represent the future uncertainty is important for
the validity of the results of stochastic programming based ALM models. The assumption
underlying the scenario generation models used in the literature is the normal distribution.
The validity of normal distribution has been questioned in the finance and macroeconomics
literature. The leptokurtic (heavy tailed and peaked), and asymmetric nature of the economic
variables can be better captured by using stable distribution as opposed to normal
distribution.
We analyze the effects of the distributional assumptions on optimal asset allocation.
A multistage dynamic asset allocation model with decision rules has been set up. The optimal
asset allocations found under normal and stable scenarios are compared. The analysis
suggests that the normal scenarios greatly underestimate risks. Stable scenario modeling
leads to asset allocations that are up to 20% different than those of normal scenario mod-
eling.
Although the financial data exhibit time varying volatility and long range dependence as
well as heavy tails, this study has only considered explicit modeling of heavy tails in the
financial data. The conditional heteroskedastic models (ARMA-GARCH) utilizing stable
distributions can be used to describe the time varying volatility along with the asymmetric
and leptokurtic behavior. In addition to these, the long-range dependence can also be modeled
if fractional-stable GARCH models are employed. These aspects of financial data will
be considered in a later paper.
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Chapter 14
PORTFOLIO CHOICE THEORY WITH NON-GAUSSIAN
DISTRIBUTED RETURNS
SERGIO ORTOBELLI
University of Bergamo, Italy
ISABELLA HUBER
University of Karlsruhe, Germany
SVETLOZAR T. RACHEV
Department of Statistics and Applied Probability, University of California, Santa Barbara, USA
Institute of Statistics and Mathematical Economics, University of Karlsruhe, Germany
e-mail: rachev@lsoe-4.wiwi.uni-karlsruhe.de
EDUARDO S. SCHWARTZ
Anderson School of Management, University of California, Los Angeles, USA
Contents
Abstract 548
1. Introduction 549
2. Choices determined by a finite number of parameters 553
2.1. Portfolio choice with institutional restrictions 553
2.2. Portfolio choice when unlimited short sales are allowed 558
2.3. Relations with Ross' multi-parameter models 560
3. The asymptotic distributional classification of portfolio choices 561
3.1. The sub-Gaussian stable model 567
3.2. A three fund separation model in the domain of attraction of a stable law 570
3.3. A k + 1 fund separation model in the domain of attraction of a stable law 573
4. A first comparison between the normal multivariate distributional assumption
and the stable sub-Gaussian one 574
4.1. An optimal allocation problem 575
4.2. Stable versus normal optimal allocation: a first comparison 578
5. Conclusions 581
Acknowledgment 582
Appendix A: Proofs 582
Appendix B: Tables 585
References 590
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
548 S. Ortobelli et al.
Abstract
This chapter discusses the parametric distributions of asset returns and proposes portfolio
choice models consistent with the maximization of the expected utility. We analyze
multi-parameter models to select nonstochastically dominated portfolios when short sales
are allowed and when short sales are not allowed. We also concentrate our attention on the
stable distributional approach in order to derive optimal portfolios with heavy-tailed distributed
financial returns. Finally, we examine and compare optimal allocations obtained
with the multivariate normal model and the sub-Gaussian stable one.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 549
1. Introduction
The purpose of this chapter is to describe the admissible classes of parametric distribution
functions of return portfolios and to analyze their consistency with the maximization
of the expected utility, in order to choose the optimal portfolios. In particular,
we present a general theory and a unifying framework with the following aims:
(1) Understanding the distributional approach applied to the portfolio choice theory;
(2) Studying the implications of the classical market restrictions on the portfolio distributions;
(3) Considering the asymptotic behavior of return data. We conclude our analysis
comparing the normal multivariate approach with the sub-Gaussian approach here pre-
sented.
The theory of portfolio choice is based on the assumption that investors allocate their
wealth across the available assets in order to maximize their expected utility. Markowitz
(1952, 1959, 1987) and Tobin (1958, 1965) were among the first to give rigorous results
approximating to the portfolio selection problem in terms of the mean and the variance.
Their analysis was extended to an equilibrium theory by Sharpe (1964), Lintner (1965)
and Mossin's (1966) capital asset pricing model (CAPM). In a mean­variance world the
investor is concerned with only two parameters of the probability distribution of total returns
on investment: the mean of the return and the variance of the return. The simplicity
of the equilibrium theory and the intuitive appeal of the mean­variance analysis attracted
and directed the attention toward determining their generality and extensibility. The foundation
of the whole theory lays in the arbitrage pricing theory (APT) and in the stochastic
dominance analysis. As a matter of fact, both theories are strictly grounded on the equilibrium
theory [see, among others, Ross (1975), Dybvig and Ross (1987), Jarrow (1986)].
The arbitrage pricing theory and the fund separation theorems [see Ross (1976, 1978a)]
justify and extend CAPM to multi-parameter linear models. Whereas the stochastic dominance
analysis justifies the partial consistency of the mean­variance framework with the
expected utility maximization when the portfolios are elliptically distributed [see, among
others, Bawa (1975), Chamberlein (1983), Owen and Rabinovitch (1983)]. In the same
years and subsequently, the theory was further generalized to intertemporal finance and to
consumption-based model. Since the space of feasible consumption bundles is quite generally
a linear space [as Ross (1978b), Cox and Leland (1982), Rubinstein (1976) and many
others have emphasized], the original dynamic problem can be replaced with an equivalent
one-period problem, which has appropriate terminal state prices, if all consumption
takes place at the end. More generally, if preferences are time separable and if we treat
consumption at each date separately, the analysis is unchanged. For this reason, here we
propose a static approach. A first generalization to an intertemporal approach can be found
in Ortobelli, Rachev and Schwartz (2002).
The mean­variance theory has survived theoretical criticism and empirical rejection,
such as that of Samuelson (1967, 1969) and Samuelson and Merton (1975) who have underlined
the limits of the approximation given only by the mean and the variance of a
portfolio. Later, Roll (1977, 1978, 1979a, b) was the first to understand clearly the weaknesses
of the theory and the empirical deficiencies. Then, Dybvig and Ingersoll (1982)
550 S. Ortobelli et al.
verified that the mean­variance pricing and the complete market hypothesis still can lead
to arbitrage opportunities. They also proved that the standard mean­variance separation
theorem holds in a complete market only if all investors have quadratic utility. Further,
Dybvig and Ross (1982) have demonstrated that efficient sets generally are not convex
and the market portfolio could be inefficient. Bawa (1976) considered the case of a market
with no short sale opportunities and jointly normally distributed returns. Under these
assumptions, if there are some investors with increasing nonconcave utility functions [for
example Friedman and Savage type utility functions (Friedman and Savage, 1948)], the
market portfolio could be inefficient. Moreover, Dybvig and Ross (1985a) observed that
assuming symmetric information and an inefficient index, the security market line analysis
can be grossly misleading, since in general efficient and inefficient portfolios can plot
above and below the security market line. In another paper [see Dybvig and Ross (1985b)]
they also argued that differential information disrupts the validity of the security market
line analysis, since it takes us outside the domain of the mean­variance analysis. On the
other side, the fundamental work of Mandelbrot (1963a, b, 1967), Mandelbrot and Taylor
(1967) and Fama (1963, 1965a, b) has sparked considerable interest in studying the
empirical distribution of financial assets. The excess kurtosis, found in Mandelbroťs and
Fama's investigations, led them to reject the normal assumption (generally used to justify
the mean­variance approach) and to propose the stable Paretian distribution as a statistical
model for asset returns. Other relevant empirical studies on postwar US data have
shown that the slope of the mean­standard deviation frontier or of the expected returnbeta
lines is much higher than the reasonable risk aversion and consumption volatility
estimates suggest. This is the equity premium puzzle (Merha and Prescott, 1985; Hansen
and Jagannathan, 1991) that could be generated by one or more of the following conditions:
(a) the investors are much more risk averse than the academics might have thought;
(b) the stock returns of the last 50 years are due to good luck rather than an equilibrium
compensation for risk; (c) something is deeply wrong with the model [see Cochrane
(1999)].
The many lacks and contrasting results on the empirical and theoretical mean­variance
analysis represented the main justifications and reasons of the alternative mean­dispersion
models proposed in the last decades [see Markowitz' (1959) mean­semi-varianceapproach,
Yitzhaki (1982) and Shalit and Yitzhaki's (1984) mean­Gini portfolio theory, Dybvig's
(1988a, b) distributional approach for complete market, Speranza (1993) and Konno
and Jamazaki's mean­absolute deviation approach (MAD) (Konno and Yamazaki, 1991),
the Ogryczak and Ruszczynski's (1999, 2001) mean­semi-deviation models]. However,
from a conceptual point of view, the stochastic dominance theory has shown that the
variance, as any other dispersion measure, cannot always be considered as a risk measure.
As a matter of fact, given a lottery X with a given dispersion measure, we can
always find another lottery Y which has a greater, lower or equal dispersion and it
is preferred to the first one by some non-satiable or risk averse investors [see among
others Levy (1992)]. Then, the implicit problems to solve are the following: "When
can we use a dispersion measure as a risk measure? Which relations there exist between
the dispersion measure and the other parameters that characterize the para-
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 551
metric family of portfolio distribution functions?". To answer these questions, we recall
a distributional stochastic dominance analysis of the parametric families consistent
with the maximization of the expected utility, see Ortobelli (2001). The distributional
analysis proposed is formally different from Dybvig's (1988a) distributional one.
Dybvig's model for complete markets has been applied successfully in theoretical work
to value the magnitude of inefficiency of some dynamic portfolio strategies [see Dybvig
(1988b)]. However, Dybvig's model is not easily applicable from an empirical point
of view. In this chapter we classify clearly the multi-parameter optimization problems
to solve so as to obtain optimal portfolios. The proposed multi-parameter approach is
alternative to Ross' one and it is a unifying and generalizing extension of the classic
moment analysis in portfolio choice theory [see among others Jean (1971), Fishburn
(1980), Ingersoll (1987)]. As a matter of fact, when the family of return portfolios
belongs to a parametric family uniquely determined by a finite number of moments,
we show what kind of optimization problems we have to solve in order to
find frontiers of admissible choices. As a consequence, we obtain a further delimitation
of the admissible portfolio choices. Therefore, we could classify the distributions
for which Markowitz and Tobin's mean­variance rule is an optimal selection rule in
a market with and without institutional restrictions (no short sale and limited liability).
Moreover, this analysis justifies the stochastic dominance properties of the meandispersion
models in the works recalled above and of some classic mean­dispersionskewness
approaches [see, for example, Kraus and Litzenberger (1976), Simaan (1993),
Ingersoll (1987)].
In second analysis we study the asymptotic distributional behavior of data. The behavior,
generally stationary over time of returns, and the Central Limit Theorem and Central
Pre-limit Theorem for normalized sums of i.i.d. random variables [see Zolotarev (1986),
Klebanov, Rachev and Szekely (2001), Klebanov, Rachev and Safarian (2000)] theoretically
justify the stable Paretian approach proposed by Mandelbrot and Fama. As a matter
of fact, their conjecture was supported by numerous empirical investigations in the subsequent
years [see Mittnik, Rachev and Paolella (1997), Rachev and Mittnik (2000)]. The
practical and theoretical appeal of the stable non-Gaussian approach is given by its attractive
properties that are almost the same as the normal one. A relevant desirable property
of a stable distributional assumption is that stable distributions have domain of attraction.
Therefore, any distribution in the domain of attraction of a specified stable distribution
will have properties close to those of the stable distribution. Another attractive aspect of
the stable Paretian assumption is the stability property, i.e., stable distributions are stable
with respect to summation of i.i.d. random stable variables. Hence, the stability governs
the main properties of the underlying distribution [detailed accounts for theoretical aspects
of stable distributed random variables can be found in Samorodnitsky and Taqqu (1994),
Janicki and Weron (1994)]. Here, we adapt the above mentioned multi-parameter approach
to portfolio choice problems using stable laws. We find an equivalent parameterization of
the stable laws (in terms of some moments) that characterizes the stable laws generally
used. Then, we recall three admissible fund separation models where the asset returns are
in the domain for attraction of stable laws [see Ortobelli, Rachev and Schwartz (2002)].
552 S. Ortobelli et al.
We first consider the portfolio allocation among n -stable sub-Gaussian distributed risky
assets (with 1 <  < 2) and the riskless one. The joint stable sub-Gaussian family is an
elliptical family. Hence, as argued by Owen and Rabinovitch (1983), in this case, we can
use a mean­dispersion analysis. The resulting efficient frontier is formally the same as
Markowitz­Tobin's mean­variance analysis, but, instead of considering the variance as a
risk parameter, we have to consider the scale parameter of the stable distributions. All the
stable parameters can be estimated. In order to consider the possible asymmetry of asset
returns, we describe a three-fund separation model for returns in the domain of attraction
of a stable law. In case of asymmetry, the model results from a new stable version of the
Simaan's model, see Simaan (1993). In case of symmetry of returns, we obtain a version
of a model recently studied by Götzenberger, Rachev and Schwartz (1999), that can also
be viewed as a particular version of the two-fund separation of Fama's (1965b) model.
In this case too, it is possible to estimate all parameters. Finally, the last model proposed
deals with the case of optimal allocation among stable distributed portfolios with different
indexes of stability. To overcome the difficulties of the most general case of the stable law,
we introduce a k + 1 fund separation model. Then, we show how to express the modeĺs
multi-parameter admissible frontier.
Finally, we analyze an investment allocation problem. It consists of the maximization
of the mean minus a measure of portfolio risk. We propose a mean­risk analysis
that facilitates the interpretation of the results. In the allocation problem, we consider
as the risk measure the expected value of a power absolute deviation. When the power
is equal to two, we obtain the classic quadratic utility functional. We examine the optimal
allocation among a riskless return and 23 risky returns, then we compare the allocation
obtained with the Gaussian and the stable sub-Gaussian distributional assumption
for the risky returns. We choose the 6% annual rate as riskless return. The model parameters
are estimated using the methodology based on the moment method. We show
that there are significant differences in the allocation when the data fit the stable subGaussian
or the normal distributions. By comparing the joint normal distribution with
the joint stable sub-Gaussian law one, it has occurred that the results performed under
the examined optimal allocation problems are substantially different. In particular, the
stable market portfolio is generally less risky than the Gaussian market portfolio. This
intuitive result is confirmed by the comparison of the optimal allocations when different
distributional hypotheses are assumed. Therefore, the investors who fit the data with
the stable distributions are generally more risk preserving than the investors who fit the
data with the normal laws because they consider the component of risk due to the heavy
tails.
Section 2 presents a first classification of the parametric distributions consistent with
the maximization of the expected utility. Section 3 analyzes the asymptotic distributional
assumption. In Section 4 we compare the stable sub-Gaussian multivariate approach
with the normal multivariate one. In the last section, we briefly summarize the
results.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 553
2. Choices determined by a finite number of parameters
In this section we propose a distributional analysis of the optimal portfolio choice problem
among n + 1 assets: n of those assets are risky with gross returns1 Z = [Z1,...,Zn] , and
the (n + 1)-th asset has risk-free gross return Z0. When unlimited short selling is allowed,
every portfolio of gross returns is a linear combination of the constant riskless gross return
Z0, and the risky returns Zi, i.e.:
x0Z0 +
n
i=1
xiZi, (1)
where (x0,x)  Rn+1, x  Rn. Therefore, the distribution functions of all admissible investments
belong to a translation and scale invariant family2 determined by a finite number
of parameters.
Assume price taker agents have preferences depending only on the probability distribution
of terminal wealth. This assumption allows von Neumann­Morgenstern's preferences
(1953) over wealth or more generally Machina's preferences (1982) over wealth but it
precludes state dependent preferences.
Assume that the market faced by a decision maker comes from a standard model of perfect
market (no transaction costs, taxes, asymmetric information, or arbitrage opportunities
and all securities are perfectly divisible) which may not be complete.
Thus, in order to classify the parametric portfolio distribution functions consistent with
the expected utility maximization, we distinguish and analyze the differences in portfolio
allocation when:
(1) institutional restrictions (no short sales, limited liability) are allowed; or,
(2) unlimited short selling is allowed without penalty.
2.1. Portfolio choice with institutional restrictions
When limited liability and no short sales are allowed, portfolios of gross returns (i.e.,
x Z 0 where Zi > 0 and xi 0, i) are positive random variables. Thus, we assume
that the portfolios of gross returns are positive random variables belonging to a scale invariant
family, denoted with +
k (a), that admits positive translations and it has the following
characteristics:
1 Generally, we assume the standard definition of i-th gross return between time t and time t + 1, Zi =
(Pt+1,i + d[t,t+1],i)/Pt,i, where Pt,i is the price of the i-th asset at time t and d[t,t+1],i the total amount of
cash generated by the instrument between t and t + 1. We distinguish the definition of gross return (with the
capital letter) from the definition of return denoted zi = Zi - 1 (or the alternative definition of continuously
compounded return ri = logZi).
2 Recall that a parametric family of distribution functions is translation invariant if whenever the distribution
FX(x) = P (X x) belongs to , then for every t  R, FX+t  as well. Similarly, we say that a family is
scale invariant if whenever the distribution FX belongs to , then for every  > 0, FX belongs to as well.
554 S. Ortobelli et al.
(1) Every distribution FX belonging to +
k (a) is associated to a positive random variable
X and is identified from k parameters
(mX,X,a1,X,...,ak-2,X)  A  Rk
,
where mX is the mean of X, X is the positive scale parameter of X.3 We assume that
the class +
k (a) is weakly determined from its parametrization. That is, the equality
(mX,X,a1,X,...,ak-2,X) = (mY ,Y ,a1,Y ,...,ak-2,Y )
implies that FX
d
= FY , but the converse is not necessarily true.
(2) For every admissible real t 0, the distribution function FX  +
k (a) has the same
parameters as FX+t  +
k (a), except the mean and the dispersion measure. In particular,
the application f (t) = X+t is a nonincreasing continuous function.
(3) For every admissible positive , the distribution function FX  +
k (a) has, the same
parameters of the distribution FX except for the mean that is mX and the scale
parameter that is X (where mX and X are respectively the mean and the scale
parameter of the random variable X).
When portfolios belong to a +
k (a) class, we can identify stochastic dominance relations4
among portfolios and the following theorem holds.
Theorem 1. Assume all random admissible portfolios of gross returns belong to a +
k (a)
class. Let w Z and y Z be a couple of portfolios respectively determined by the parameters
(mw Z,w Z,a1,p,...,ak-2,p) and (my Z,y Z,a1,p,...,ak-2,p).
Then, the following implications hold:
(1) Suppose
mw Z
w Z
=
my Z
y Z
, then w Z FSD y Z if and only if w Z > y Z.
(2)
mw Z
w Z
my Z
y Z
and w Z y Z with at least one inequality strict, implies w Z FSD
y Z.
3 In our context we use the mean as location parameter but the analysis can be extended to translation invariant
families which do not admit finite the first moment. Moreover, we recall Pitman's seminal work (1939) on the
estimation of location and scale parameters.
4 Recall that the portfolio x Z first order stochastically deaminates (FSD) y Z if and only if for every increasing
utility functions u, E(u(x Z)) E(u(y Z)) and the inequality is strict for some u. Equivalently x Z FSD
y Z if and only if P (x Z t) P (y Z t) for every real t and strictly for some t. Analogously, we say
that x Z second order stochastically dominates (SSD) y Z, if and only if for every increasing, concave utility
function u, E(u(x Z)) E(u(y Z)) and the inequality is strict for some u. Equivalently, x Z SSD y Z, if and
only if t
- Fx Z(v)dv t
- Fy Z(v)dv for every real t and strictly for some t [see, among others, Quirk
and Saposnik (1962), Fishburn (1964), Hanoch and Levy (1969), Hadar and Russel (1969)]. We also say that
x Z Rothschild Stiglitz stochastically dominates (R­S) y Z if and only if for every concave utility functions
u, E(u(x Z)) E(u(y Z)) and the inequality is strict for some u. Equivalently x Z R­S y Z if and only if
E(x Z) = E(y Z) and x Z SSD y Z [see Rothschild and Stiglitz (1970)]. However, there exist many other
stochastic orders used in Economics and Finance, see, among others, Levy (1992), Shaked and Shanthikumar
(1994).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 555
(3) w Z R­S y Z, if and only if mw Z = my Z and w Z < y Z.
(4)
mw Z
w Z
my Z
y Z
and mw Z my Z with at least one inequality strict, implies w Z SSD
y Z.
(5) w Z y Z and w Z SSD y Z, implies w Z FSD y Z.
(6) mw Z my Z and w Z y Z with at least one inequality strict, implies w Z SSD
y Z.
The proofs of Theorem 1 and of the next results are given in the appendix.
Observe that there exist counterexamples to the converse of implications (2), (4), (5)
and (6) in Theorem 1. Thus, in order to obtain the converse of these implications, we
need additional hypotheses [see Ortobelli (2001)]. Theorem 1 stresses the limits of meanvariance
rule. In fact, suppose the portfolios of gross returns (without considering the riskless
gross return) belong to a +
2 (a) class uniquely determined by the mean and the
variance. Then, all non-satiable investors will choose portfolio solutions of the following
constrained system
max
x
x Qx subject to
E(x Z)

x Qx
= h, where x e = 1, (2)
xi 0, i = 1,...,n,
for some h, where e = [1,...,1] , Q is the variance­covariance matrix of the vector of
gross returns Z = [Z1,...,Zn] . Let  be the maximum standard deviation of all admissible
portfolios. Let us denote with  the portfolio mean of gross returns with maximum
variance. As a consequence of Theorem 1 and Bawa's results (1976), when the variancecovariance
matrix Q is not singular and h varies in the following interval:


h ZQ-1Z = max
x
x Z

x Qx
, (3)
where Z is the mean of the vector of gross returns Z, then the solutions of optimization
problem (2) describe a set that contains the efficient frontier for agents with utility functions
monotonically increasing in wealth. Moreover, under our assumptions, there exists a
nonempty neighborhood U of the global minimum variance portfolio Z Q-1e/(e Q-1e)
such that every admissible portfolio belonging to U(Z Q-1e/(e Q-1e)) is not a solution
of optimization problem (2).
With reference to the portfolio selection problem, recall that Markowitz (1952, 1987)
and Tobin (1958, 1965) proposed the following selection rule for non-satiable risk averse
investors: "From among a given set of investment alternatives (which includes the set of
securities available in the market as well as all possible linear combinations of those basic
securities), the admissible set of alternatives is obtained by discarding those investments
with a lower mean and higher variance than a member of the given set". On the basis of
556 S. Ortobelli et al.
Fig. 1. The continuous curve represents efficient portfolios for non-satiable investors considering restrictions of
nonnegative wealth; - - - dominated portfolios. The class of non-satiable investor's optimal choices (which are
contained in the arc ABC) is different from the class of risk averse investor's optimal choices (which are contained
in the arc DEAB), even if they generally have many common choices. Therefore, when we consider the risk averse
non-satiable investor's optimal choices, we obtain the feasible optimal portfolios (which are contained in the arc
AB) that are only a part of portfolios given by the classic Markowitz and Tobin's rule (arc EAB).
Theorem 1, we find that Markowitz­Tobin's selection rule is not optimal for non-satiable
risk averse investors. In this context it is necessary to underline that no short sales or limited
liability restrictions are imposed in a market where no riskless return is allowed. As a
consequence, all portfolios are random variables uniformly bounded from below.5 As a
matter of fact, Theorem 1 cannot be extended to nonpositive random variables. Markowitz,
Tobin, Bawa and many other authors left behind this observation in their considerations
using normal distributions for returns. They have considered as efficient the portfolios
on the upper neighborhood of global minimum variance (EA in Figure 1) but the same
portfolios whose domain is under this restriction are not all efficient. Therefore, we proved
that Markowitz and Tobin selection rule cannot be optimal even when portfolios belong
to a family uniquely determined from the mean and the variance. It is well known that
a lower variance does not imply a better choice for a non-satiable risk averse investor
[see, example, in Hanoch and Levy (1969)]. Moreover, in an opportune neighborhood of
global minimum variance portfolio, optimal portfolios for non-satiable investors do not
exist. However, when riskless borrowing or lending is allowed, the mean­variance rule
provides a sharper decision which permits to derive the efficient set for decision making
with increasing and concave utility functions. In fact, if riskless asset is allowed, the global
5 Recall that a random variable X is bounded from below (above) if there exists a real t such that P (X t) = 0
(P (X t) = 0). Analogously, a parametric family is uniformly bounded from below (above) if there exists a
real t such that, for every random variable X  , P (X t) = 0 (P (X t) = 0).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 557
minimum dispersion portfolio is the riskless asset itself. Thus, as shown by Levy and Kroll
(1976) and Kroll and Levy (1979), the classification of the efficient frontiers given by the
stochastic dominance analysis assumes a simpler form.
Under institutional restriction on the market (no short sales, limited liability), we can
assume that the family of all admissible portfolios of gross returns x Z belongs to a
scale invariant family which admits positive translations. If every distribution function
FX  is associated to a positive random variable X uniquely determined6 by
(mX,X,p1,X,...,pk-2,X), where mX is the mean, X is the standard deviation and
pi,X =
E((X - E(X))i+2)
i+2
X
for i = 1,...,k - 2
are the first k - 2 nontrivial fundamental ratios, then, the family is a particular +
k (a).
Note for i = 1 and i = 2 the i-th fundamental ratios are respectively Pearson's asymmetry
and kurtosis coefficients of the random variable X. Thus, all risk averse investors will
choose non-R­S stochastically dominated portfolios among the solutions of the following
constrained optimization problem
min
x
x Qx subject to
E(x Z) = m, x e = 1,
(4)
E((x Z - E(x Z))i)
(x Qx)i/2
= qi, i = 3,...,k,
xj 0, j = 1,...,n,
for some m and qi, i = 3,...,k, where e = [1,...,1] , Q is the variance­covariancematrix
of the vector of gross returns Z = [Z1,...,Zn] . Moreover, all non-satiable investors will
choose portfolio weights, solutions of the following optimization problem
max
x
x Qx subject to
E(x Z)

x Qx
h, x e = 1,
(5)
E((x Z - E(x Z))i)
(x Qx)i/2
= qi, i = 3,...,k,
xj 0, j = 1,...,n,
6 Recall that a class of distributions is uniquely determined from k parameters when the equality
(mX,X,a1,X,...,ak-2,X) = (mY ,Y ,a1,Y ,... ,ak-2,Y ) implies the equality of the respective distributions,
i.e., FX
d
= FY , and vice versa.
558 S. Ortobelli et al.
for some qi, i = 3,...,k, and h /, where  is the maximum standard deviation
of all admissible portfolios and  is the mean of that portfolio of gross returns. Similarly,
all non-satiable risk averse investors will choose portfolio weights among the solutions of
the following optimization problem
max
x
E(x Z) subject to
E(x Z)

x Qx
h, x e = 1,
(6)
E((x Z - E(x Z))i)
(x Qx)i/2
= qi, i = 3,...,k,
xj 0, j = 1,...,n,
for some qi, i = 3,...,k, and h j /j , where j is the maximum mean of all primary
gross returns and j is the standard deviation of that return. We obtain optimization
problems analogous to (4), (5) and (6) when we consider the riskless asset. In this case,
the mean is given by E(x Z) + (1 - x e)Z0 and we require that 0 x e 1 instead of
requiring x e = 1. Theorem 1 is used for positive random variables. However, the above
results can be generalized to families of random variables, which are uniformly bounded
from below. In fact, without loss of generality, we can consider a translation that makes all
random variables positive.
2.2. Portfolio choice when unlimited short sales are allowed
In the last fifty years the researchers of portfolio choice theory often used unbounded random
variables for portfolio of returns, typically: the Gaussian laws. They also used to
study continuously compounded portfolio of returns, say x r = n
i=1 xi logZi, where7
ri = logZi.
In particular, we assume that the distribution functions of portfolios belong to a translation
and scale invariant family denoted with k(a) with the following characteristics:
7 The continuously compounded portfolio of returns x r represents an approximation to the portfolio of returns
x z (i.e., n
i=1 xi logZi  x z, where zi = Zi - 1). Thus, continuously compounded portfolio of returns x r are
equivalently identified and called portfolio of returns. However, observe that
X FSD Y if and only if logX FSD logY ,
while
X SSD Y implies log X SSD logY
but the converse is not necessarily true (you can find a simple counterexample with the log-normal class). Hence,
when we study the optimal choices by considering the approximation n
i=1 xi log Zi  x z, we find a set of
choices that would be closer to the efficient set as well as the approximation would be right. (The approximation
is good enough when we consider daily ­ or weekly ­ data in the empirical analysis).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 559
(1) Every distribution FX belonging to k(a) is identified from k parameters (mX,X,
a1,X,...,ak-2,X)  A  Rk where mX is the mean of X, X is the positive scale parameter
of X. We assume that the class k(a) is weakly determined from its parameterization.
That is the equality (mX,X,a1,X,...,ak-2,X) = (mY ,Y ,a1,Y ,...,ak-2,Y )
implies that FX
d
= FY but the converse is not necessarily true.
(2) For every admissible real t, the distribution function FX  k(a) has the same parameters,
except the mean, as FX+t  k(a) (the translated of FX).
(3) For every admissible positive , the distribution function FX  k(a) has the same
parameters of the distribution FX  k(a) except for the mean that is mX and the
scale parameter that is X (where mX and X are respectively the mean and the scale
parameter of the random variable X).
The random variables associated to the distribution functions of a k(a) class are not
uniformly bounded from below because every k(a) class is translation invariant. When
portfolios belong to a k(a) class, we can identify a stochastic dominance relation among
portfolios unbounded from below and the following theorem holds.
Theorem 2. Suppose the distribution functions of all random portfolios belong to the same
class k(a). Let w r and y r be a couple of random portfolios unbounded from below
respectively determined by the parameters
(mw r,w r,a1,p,...,ak-2,p) and (my r,y r,a1,p,...,ak-2,p).
Then, the following properties are equivalent
(1) E(w r) E(y r), w r y r with at least one inequality strict.
(2) w r SSD y r and y r
d
= w r - (E(w r) - E(y r)) +  and E(/w r) = 0.
As for Theorem 2, when all portfolios are random variables unbounded from below and
their distribution functions belong to a 2(a) class, two portfolios X and Y such that
X > Y and X SSD Y cannot exist. On the contrary, when the random portfolios considered
in Theorem 2 are random variables bounded from below, we need further assumptions
to get the above equivalence [see Ortobelli (2001)].
According to Theorem 2, it follows that when all portfolios are unbounded random variables
belonging to a k(a) class, it is easier to characterize their stochastic dominance
properties. In this sense, the continuously compounded portfolios of returns x r, are natural
candidates for a simpler stochastic dominance analysis.
Samuelson (1969), Samuelson and Merton (1975) were among the first to investigate
the conditions for the mean­variance criterion to provide an approximate optimum. Chamberlein
(1983) has shown that when the riskless return is allowed, the families of elliptical
distributions with finite variance are necessary and sufficient for the expected utility of final
wealth to be a function only of the mean and the variance. Hence, when the portfolios
560 S. Ortobelli et al.
are unbounded random variables8 with distribution functions belonging to the same elliptical
distribution family having finite variance, we can use Markowitz and Tobin's rule to
individuate the optimal portfolios. Similarly, assuming that:
(a) there is no riskless asset;
(b) the portfolios of returns are unbounded random variables;
(c) the last n - 1 components of the return random vector are elliptically distributed (with
finite variance) conditional on the first component which has an arbitrary distribution
with finite variance [see Chamberlein (1983)];
then, Markowitz and Tobin's rule can be used to individuate the optimal portfolios. Thus,
Theorems 1 and 2 underline a further limitation (the above point (b)) of the previous studies
on this issue.
We can now find optimal portfolios when all returns are unbounded random variables
uniquely determined by a finite number of moments. Thus, if short sale is allowed, all risk
averse investors will choose non-R­S stochastically dominated portfolios that are solutions
of the constrained optimization problem (4) without the constrain xj 0, j = 1,...,n.
Similarly, we obtain optimal solutions for non-satiable investors maximizing the mean for
some fixed central moments.
2.3. Relations with Ross' multi-parameter models
Consider the problem of optimal allocation among n + 1 assets: n of those assets are risky
with non-redundant returns r = [r1,...,rn] , and the (n + 1)-th asset return is z0 risk-free.
Then,we are interested in the cases of portfolio distributions belonging to a k(a) family
with k < n. As argued by Ross (1978a), in order to reduce the variables of the portfolio
choice problem, we have to assume some restrictions on the vector  = (1,2,...,n) in
the following representation of the returns:
ri =
q
p=1
bi,p Yp - E(Yp) + i, i = 1,...,n, (7)
where Yi and i are random variables and bi,j are scalars. Differently from Ross, we propose
to study the case where all random variables x  belong to a s(a) family. Then, the
scale parameter x  of random variable x , has to verify the properties relatively to the
s(a) class. Thus, consider the parameterization given by:
(1) the parameters of the s(a) family, and
(2) the parameters c
j = x b,j /x , for j = 1,...,q.
This parameterization verifies the properties of a k(a) family with k = s + q. In fact,
for every positive real  the parameters of x (r - ) do not change except for the scale
8 The elliptical families with finite variance are symmetric around the mean and are not necessarily associated
to unbounded random variables, see Ingersoll (1987), Owen and Rabinovitch (1983). Then, following Theorem 2
we need to specify when elliptically distributed random variables have to be unbounded.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 561
parameter that becomes x . Then, all portfolios belong to a k(a) family with k = s+q
when the returns admit the form (7) and all admissible random variables x  belong to a
s(a) family. Typical examples of this approach are the stable Paretian models presented
in the next section.
Note that the above proposed moment analysis generalizes many of the three moment
models presented in the last decades [see, for example, Kraus and Litzenberger (1976), Ingersoll
(1987), Simaan (1993)]. Moreover, we do not need to require that portfolio returns
verify the fund separation conditions as happened in the three moment models. Therefore,
the above theorems represent a first classification of portfolio distribution functions which
is alternative to those proposed from Ross (1976, 1978a). In fact, we underline the following
differences from Ross' models:
(1) We express necessary and sufficient conditions to identify optimal portfolios. We can
derive the efficient frontiers solving a constrained optimization problem.
(2) We do not require the closure of the random law under addition.
(3) The above theorems are an unifying and generalizing extension of moment analysis
in the portfolio selection theory. In particular, the previous analysis describes further
restrictions in using Markowitz and Tobin's selection rule as optimal portfolio selection
rule.
(4) We express a portfolio choice theory dependent on a finite number of parameters consistent
with expected utility maximization. We do not specify which parameters identify
the distribution functions of asset returns. We only require very general properties
which determine the existence of a scale parameter and a shift parameter.
(5) The above results can be applied to every economic choice in uncertainty conditions
when the distribution functions are weakly determined by a finite number of
parameters and verify properties of k(a) or of +
k (a) classes. Besides, this classification
of choices under uncertainty conditions implies a first classification of
the admissible dispersion measures [see Ortobelli (2001), Giacometti and Ortobelli
(2001)].
As it follows from the previous considerations, the models introduced here can be theoretically
improved and empirically tested. However, a more general theoretical and empirical
analysis with further discussion, studies and comparison of the above models does not
enter in the objective of this chapter and it will be the subject of future research.
3. The asymptotic distributional classification of portfolio choices
In this section we study the portfolio choice problem analyzing the asymptotic behavior of
data. In particular, we consider unbounded random portfolios of stable distributed returns,
x r, that, with abuse of notation, we continue to call as portfolios of stable distributed
returns.9
9 If logZi is stable distributed, then Zi = 1 + zi is log-stable distributed.
562 S. Ortobelli et al.
The recent crashes observed in the stock market showed that the stock returns are more
volatile than those predicted by the models with finite variance of the asset returns. In the
empirical financial literature, it is well documented that the asset returns have a distribution
whose tail is heavier than that of the distributions with finite variance, i.e.,
P |ri| > x  x-i Li(x) as x  , (8)
where 0 <  < 2 and Li(x) is a slowly varying function at infinity, i.e.,
lim
x
Li(cx)
Li(x)
 1 for all c > 0,
see Rachev and Mittnik (2000) and the references therein. In particular, in the data observed
until now 1 <  < 2. The constrain 1 <  < 2 and the relation (8) imply that returns ri
admit finite mean and non-finite variance. The tail condition in (8) also implies that the
vector of returns r = [r1,...,rn] is in the domain of attraction of (1,...,n)-stable law.
That is, given T i.i.d (independent and identically distributed) observations on r, namely
r(t)
= r
(t)
1 ,...,r(t)
n , t = 1,2,...,T,
then, there exist normalizing constants
a(T )
= a(T )
1 ,...,a(T )
n  Rn
+ and b(T )
= b(T )
1 ,...,b(T )
n  Rn
,
such that
T
i=1
r
(i)
1
a(T )
1
+ b
(T )
1 ,...,
T
i=1
r
(i)
1
a(T )
n
+ b(T )
n
d
- S(1,...,n) as T  , (9)
where S(1,...,n) is (1,...,n)-stable random variable. This convergence result is
a consequence of the stationary behavior of returns and of the Central Limit Theorem
for normalized sums of i.i.d. random variables which determines the domain of attraction
of each stable law [see Zolotarev (1986)]. Therefore, any distribution in the domain of
attraction of a specified stable distribution will have properties close to those of the stable
distribution. The constants a
(T )
j in (9) have the form
a
(T )
j = T 1/j Lj (T ),
where Lj (T ) are slowly varying functions as T  .
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 563
Each component of S(1,...,n) = (s1,...,sn) has a Pareto­Lévy stable distribution,
i.e., its characteristic function is given by
j (t) =



exp -
j
j |t|j 1 - ij sgn(t)tan
j
2
+ ij t if j = 1,
exp -j |t| 1 + ij
2

sgn(t)log|t| + ij t if j = 1,
(10)
where j  (0,2) is the so-called stable (tail) index of sj ,  > 0 is the scale (or dispersion)
parameter,   [-1,1] is a skewness parameter and  is a location parameter. Moreover,
for every fixed , the Pareto­Lévy -stable law is a 3(a) class. When  > 1 the location
parameter  is the mean. However, there is a considerable debate in literature concerning
the applicability of -stable distributions as they appear in Lévy's central limit theorems.
A serious drawback of Lévy's approach is that in practice one can never know whether
the underlying distribution is heavy tailed, or just has a long but truncated tail. Limit theorems
for stable laws are not robust with respect to truncation of the tail or with respect to
any change from light to heavy tail, or conversely. Based on finite samples, one can never
justify the specification of a particular tail behavior. Hence, one cannot justify the applicability
of classical limit theorems in probability theory. Therefore, instead of relying on limit
theorems, we can use the so-called pre-limit theorem which provides an approximation for
distribution functions in case the number of observation T is "large" but not too "large"
[see Klebanov, Rachev and Szekely (2001), Klebanov, Rachev and Safarian (2000)]. In
particular the "pre-limiting" approach helps to overcome the drawback of Lévy-type central
limit theorems. As a matter of fact, we can assume that returns are bounded "far away",
say daily returns cannot be outside the interval [-0.5,0.5]. Thus, considering the empirical
observation on asset returns, we can assume that the asset returns ri are truncated
i-stable distributed with support, [-0.5,0.5]. Even if the returns will be attracted by the
CLT to the Gaussian law, pre-limit theorems show that for any reasonable T the truncated
stable laws will be attracted to the stable laws. Therefore, it is plausible assuming that
the vector of returns r = [r1,...,rn] is in the domain of attraction of a n-dimensional
(1,...,n)-stable law.
In order to express a multi-parameter choice in portfolio selection theory coherent with
the empirical evidence and consistent with the expected utility maximization, we need the
asymptotic distributional assumption consisting in:
(1) (Heavy tailedness assumption) Portfolios x r are unbounded random variables
belonging to Lp with 1 < p 2 and the return vector r = [r1,...,rn] is in the domain
of attraction of (1,...,n)-stable law (1 < i 2, i = 1,...,n). The assumption
1 < i 2 is supported by increasing empirical results as shown by Mandelbrot
(1963a, b, 1967), Fama (1963, 1965, b), Mittnik, Rachev and Paolella (1997), Rachev
and Mittnik (2000).
(2) (Consistency with the expected utility maximization) The distributions of the portfolio
returns x r belong to the same k(a) class of distribution functions.
564 S. Ortobelli et al.
Under these assumptions, as for Theorem 2, we obtain an admissible frontier for nonsatiable
and non-satiable risk averse investors.
A simpler way to express the asymptotic behavior of data consists in considering every
portfolio in the domain of attraction of a Pareto­Lévy  stable distribution with  > 1.
Given that, we implicitly assume that all optimal choices are identified by four parameters
of the underlined stable law. Therefore, every portfolio x r can be well approximated by a
stable distribution, i.e., we can assume:
x r + (1 - x e)z0
d
= S(x) (x),(x),(x) , (11)
where z0 is the riskless return, (x)  (min1 i n i,2) is the index of stability, j > 1 is
the index of stability of the j-th asset return, (x) is the scale parameter, (x) = x E(r) +
(1 - x e)z0 is the mean and (x) is the skewness parameter. Properties of 4(a) class are
verified with this parameterization, so according to Theorem 2 every risk averse investor
will choose a portfolio weight, solution of the following constrained problem
min
x
(x) subject to
x E(r) + (1 - x e)z0 = m,
(12)
(x) = 
,
(x) = 
for some m, , . In this case, we are not able to find a closed form of the efficient
frontier because we do not know a priori the joint distribution of the asset returns. In order
to overcome this problem, we could consider another admissible parameterization of
stable distribution for problem (11). For example, we can prove that the mean (x) =
x E(r) + (1 - x e)z0, the scale parameter s(x) = E(|x r - x E(r)|) and the fundamental
ratios
1(x) =
E(|x r - x E(r)|q1)
(s(x))q1
and 2(x) =
E((x r - x E(r)) q2 )
(s(x))q2
,
where q1,q2  (1,min1 i n i); represent a parameterization which verifies the properties
of 4(a) class.10 In fact, first observe that 1(x) and 2(x) do not depend on (x) and
(x) because
x r - x E(r)
q1 d
= (x)q1 S(x)(1,(x),0)
q1
,
10 The symbology x t stands for sgn(x)|x|t .
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 565
and also
x r - x E(r)
q2 d
= (x)q2 S(x)(1,(x),0)
q2
.
Thus, as a consequence of Property 1.2.17 in Samorodnisky and Taqqu (1994)
1(x) =
E(|x r - x E(r)|q1)
(s(x))q1
= K
(1 - q1/(x))cos(arctan((x)tan((x)/2))q1/(x))
( (1 - 1/(x))cos(arctan((x)tan((x)/2))1/(x)))q1
,
where K is a constant that depends only on q1. Hence, for every q1  (1,min1 i n i)
and for every fixed (x), 1(x) is a decreasing function of (x) on the existence interval.
Moreover, 1(x) is an even function of (x) and it decreases in |(x)| for fixed
(x)  (min1 i n i,2). Instead, 2(x) is an increasing odd function of  for every
q2  (1,min1 i n i) and for every fixed (x)  (min1 i n i,2). These relations imply
that 1(x) and 2(x) uniquely determinate (x) and (x). Then, under the assumption
(11), every risk averse investor will choose a portfolio weight, solution of the following
constrained problem
min
x
E x r - x E(r) subject to
x E(r) + (1 - x e)z0 = m,
(13)
E(|x r - x E(r)|q1)
(s(x))q1
= 1,
E((x r - x E(r)) q2 )
(s(x))q2
= 2
for some m, 1, 2. Differently from problem (12), problem (13) does not require the
knowledge of the joint distribution of asset returns but it is still computationally too complex.
Generally, in order to identify the efficient frontier and reduce the number of parameters,
we assume that i =  for all i = 1,...,n. Observe that stable distributions are
stable with respect to summation of i.i.d. random stable variables and the vector of returns
r = [r1,...,rn] is -stable distributed with  > 1 if and only if all linear combinations
are stable [see Samordinsky and Taqqu (1994, Theorems 2.1.2 and 2.1.5)]. In this case the
joint characteristic function of returns is given by
r(t) = exp -
Sn
|t s|
1 - isgn(t s)tan

2
 (ds) + it  ,
where  is the index of stability,  (ds) is the spectral measure concentrated on Sn =
{s  Rn | s = 1}.
566 S. Ortobelli et al.
Thus, when the vector of returns is  stable distributed (with  > 1), every portfolio
x r + (1 - x e)z0 (except the riskless return, i.e., x = 0) is distributed as
x r + (1 - x e)z0
d
= S (x),(x),(x) ,
where
(x) = x E(r) + (1 - x e)z0,
(x) =
Sn
|x s|
 (ds)
1/
and (x) =
Sn
|x s| sgn(x s) (ds)
((x))
are respectively the mean, the scale parameter and the skewness parameter of the portfolio
x r + (1 - x e)z0. Under this distributional assumption, every risk averse investor will
choose a portfolio weight, solution of the following constrained problem
min
x
(x) subject to
x E(r) + (1 - x e)z0 = m, (14)
(x) = 
for some m and . In order to determine estimates of the scale parameter and of the
skewness parameter, we can consider the tail estimator for the index of stability  and the
estimator for the spectral measure  (ds) proposed by Rachev and Xin (1993) and Cheng
and Rachev (1995). However, even if the estimates of the scale parameter and the skewness
parameter are computationally feasible, they require numerical calculations. Thus,
model (14) does not present an easy applicability from an empirical point of view. Similarly
to problem (13), we can fix q <  and propose a different representation based on
the moments type constrains. Therefore, instead of model (14), we obtain the following
constrained problem
min
x
E x r - x E(r) subject to
x E(r) + (1 - x e)z0 = m, (15)
E((x r - x E(r)) q2 )
(E(|x r - x E(r)|))q2
= 2
for some m and 2. Optimization problems (15) and (13) can be used in a more general
setting than optimization problems (12), (14). In fact, a priori other classes of distribution
functions (not only stable distributions) for returns uniquely determined by the parameters
m(x), s(x), 1(x) and 2(x) could exist. Next, in order to overcome the intrinsic difficulties
of the problems (12)­(14) and (15), we analyze different fund separation models that
consider the asymptotic distributional assumption.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 567
3.1. The sub-Gaussian stable model
Assume the vector of returns r = [r1,...,rn] is sub-Gaussian -stable distributed with
1 <  < 2. Then, the characteristic function of r has the following form
r(t) = E exp(it r) = exp -(t Qt)/2
+ it  , (16)
where Q = [Ri,j /2] is a positive definite (n × n)-matrix,  = E(r) is the mean vector, and
 (ds) is the spectral measure with support concentrated on Sn = {s  Rn | s = 1}. The
term Ri,j is defined by
Ri,j
2
= [~ri, ~rj ] ~rj
2-
 , (17)
where ~rj = rj - j are the centralized return, the covariation [~ri, ~rj ] between two jointly
symmetric stable random variables ~ri and ~rj is given by
[~ri, ~rj ] =
S2
si|sj |-1
sgn(sj ) (ds),
in particular, ~rj  = ( S2
|sj | (ds))1/ = ([~rj , ~rj ])1/. Here the spectral measure
 (ds) has support on the unit circle S2.
This model can be considered as a special case of Owen­Rabinovitch's elliptical model
[see Owen and Rabinovitch (1983)]. However, no estimation procedure of the model parameters
is given in the elliptical models with non-finite variance. In our approach we use
(16) and (17) to provide a statistical estimator of the stable efficient frontier. To estimate
the efficient frontier for returns given by (16), we need to consider an estimator for the
mean vector  and an estimator for the dispersion matrix Q. The estimator of  is given
by the vector ^ of sample averages. Using Lemma 2.7.16 in Samorodnitsky and Taqqu
(1994) we can write for every p such that 1 < p < 
[~ri, ~rj ]
~rj


=
E(~ri ~r
p-1
j )
E(|~rj |p)
, (18)
where the scale parameter j can be written ~rj  = j . It can be approximated by the
moment method suggested by Samorodnitsky and Taqqu (1994) (Property 1.2.17) in the
case  = 0

p
j = ~rj
p
 =
E(|~rj |p)p
+
0 u-p-1 sin2
udu
2p-1 (1 - p/)
. (19)
It follows
Ri,j
2
= 2
j
E(~zi ~z
p-1
j )
E(|~zj |p)
= 
2-p
j
p
+
0 u-p-1 sin2
udu
2p-1 (1 - p/)
E ~zi ~z
p-1
j .
568 S. Ortobelli et al.
The above suggests the following estimator Q = [Ri,j /2] for the entries of the unknown
covariation matrix Q
Ri,j
2
= ^
2-p
j
p
+
0 u-p-1 sin2
udu
2p-1 (1 - p/)
1
N
N
k=1
~z
(k)
i ~z
(k)
j
p-1
, (20)
where the 2
j is estimated as follows
^2
j =
Rj,j
2
=
1
N
N
k=1 |~r(k)
j |pp
+
0 u-p-1 sin2
udu
2p-1 (1 - p/)
2/p
. (21)
The moment estimator makes most sense for each fixed p  (1,). The rate of convergence
of the empirical matrix Q = [Ri,j /2] to the unknown matrix Q (to be estimated),
will be faster, if p is as large as possible, see Rachev (1991).
Now, let us recall that our portfolio satisfies the relation
x r
d
= S(x r,x r,mx r)
and furthermore, W = z0 when x = 0, otherwise
W = x r + (1 - x e)z0
d
= S(x r,x r,mW ),
where  is the index of stability, x r =

x Qx is the scale (dispersion) parameter, x r = 0
is the skewness parameter and mW = x E(r) + (1 - x e)z0 is the mean of W. In particular,
every sub-Gaussian -stable family is a particular 2(m,) class.
In view of what stated before, when the returns r = [r1,...,rn] are jointly sub-Gaussian
-stable distributed, every risk averse investor will choose an optimal portfolio among all
portfolio solutions of the following optimization problem:
min
x
x Qx subject to x  + (1 - x e)z0 = mW (22)
for some given mean mW , where W = x r + (1 - x e)z0. Thus, every optimal portfolio
that maximizes a given concave utility function u,
max
x
E u x r + (1 - x e)z0
belongs to the mean­dispersion frontier
 =



m - z0
( - ez0) Q-1( - ez0)
if m z0,
z0 - m
( - ez0) Q-1( - ez0)
if m < z0,
(23)
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 569
where  = E(r); m = x  + (1 - x e)z0; e = [1,...,1] ; and 2 = x Qx. Besides, the
optimal portfolio weights x satisfy the following relation:
x = Q-1
( - z0e)
m - z0
( - ez0) Q-1( - ez0)
. (24)
Note that (23) and (24) have the same forms as the mean­variance frontier. In particular,
they assume a more general form for nonnecessarily symmetric dispersion matrix Q.
As a matter of fact, even if Q is a symmetric matrix (it is definite positive) the estimator
proposed in the sub-Gaussian cases (21) and (22) generally is not necessarily symmetric.
Therefore, in some extreme cases we could obtain the inconsistent situation of
stable distribution associated to a portfolio x with square scale parameter equal or lower
than zero.11 Moreover, (24) exhibits the two fund separation property for both the stable
and the normal case but the matrix Q and the parameter have different meaning. In
the normal case, Q is the variance­covariance matrix and  is the standard deviation,
while in the stable case Q is a dispersion matrix and  is the scale (dispersion) parameter,
 =

x Qx. According to the two-fund separation property of the sub-Gaussian -stable
approach, we can assume that the market portfolio is equal to the risky tangent portfolio
under the equilibrium conditions (as in the classical mean­variance Capital Asset Pricing
Model (CAPM)). Therefore, every optimal portfolio can be seen as the linear combination
between the market portfolio
x r =
r Q-1( - z0e)
e Q-1 - e Q-1ez0
, (25)
and the riskless asset return z0. Following the same arguments as in Sharpe, Lintner,
Mossin's mean­variance equilibrium model, the return of asset i is given by:
E(ri) = z0 + i,m E(x r) - z0 , (26)
11 Observe that for every x  Rn, we get x Qx > 0 if and only if (Q + (Q) )/2 is a definite positive matrix.
Thus, we can verify that (Q + (Q) )/2 is definite positive in order to avoid stable portfolios x z with negative
scale parameter estimators. Moreover, we observe that the symmetric matrix (Q + (Q) )/2 is an alternative
estimator of the dispersion matrix Q whose statistical properties have to be proved. In particular, if we want to
simulate the vector ~z = [~z1,... , ~zn] of the centred  stable sub-Gaussian return distributions, we generally use
the dispersion matrix  = (Q + (Q) )/2. As a matter of fact, we first generate the vector G = [G1,...,Gn]
of the joint Gaussian distribution G = N(0,) using the Cholesky decomposition matrix. Then, the vector of
returns [see Samorodnitsky and Taqqu (1994)] is given by:
~z =

AG,
where A
d
= S/2(2(cos(/4))2/,1,0) is an /2 stable random variable independent of the Gaussian vector G.
570 S. Ortobelli et al.
where i,m = x Qei/x Qx, with ei the vector with 1 in the i-th component and zero in all
the other components. As a consequence of Ross' necessary and sufficient conditions of
two-fund separation [see Ross (1978a)], the above model admits the form
ri = i + biY + i, i = 1,...,n,
where i = E(ri), E(/Y) = 0,  = (1,2,...,n) , b = [b1,...,bn] and the vector
bY +  is sub-Gaussian -stable distributed with zero mean.
Hence, our sub-Gaussian -stable version of CAPM is not much different from
Gamrowski­Rachev's (1999) version of the two-fund separation -stable model. As a matter
of fact, Gamrowski and Rachev (1999) propose a generalization of Fama's -stable
model (1965b) assuming ri = i + biY + i, for every i = 1,...,n, where i and Y are
-stable distributed and E(/Y) = 0. In view of their assumptions,
E(ri) = z0 + ~i,m E(x r) - z0 ,
where
~i,m =
1
 x ~r 

 x ~r 

 xi
=
[~ri, x ~r]
x ~r 

.
Furthermore, the coefficient [~ri, x ~r]/ x ~r 
 can be estimated using the above formula
(18).
Now, we see that in the above sub-Gaussian symmetric -stable model x Qx = x ~r 2

and x Qei = 1
2 x ~r 2
/ xi. Thus, we get the equivalence between the coefficient i,m of
model (26) and ~i,m of Gamrowski­Rachev's model, i.e.:
i,m =
x Qei
x Qx
=
1
x r
x r
 xi
=
[~ri, x ~r]
x ~r 

= ~i,m,
where x r is the scale parameter of market portfolio.
3.2. A three fund separation model in the domain of attraction of a stable law
Let us assume that the vector r = [r1,...,rn] describes the following three-fund separating
stable model of security returns:
ri = i + biY + i, i = 1,...,n, (27)
where the random vector  = (1,2,...,n) is independent from Y and follows a joint
sub-Gaussian 1-stable distribution (1 < 1 < 2 ), with zero mean and characteristic func-
tion
(t) = exp -|t Qt|1/2
,
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 571
where Q is the definite positive dispersion matrix. On the other hand, Y
d
= S2 (Y ,Y ,0)
is 2-stable distributed random variable independent from , with 1 < 2 < 2 and zero
mean. Under these assumptions, the portfolios are in the domain of attraction of a  stable
law with  = min(1,2) and belong to a 3(a) family. A testable case in which Y
is 2-stable symmetric distributed (i.e., Y = 0), was recently studied by Götzenberger,
Rachev and Schwartz (1999). When Y = 0 and 1 = 2, our model can lead to the twofund
separation Fama's model. The characteristic function of the vector of returns r =
[r1,r2,...,rn] is given by:
r(t) = (t)Y (t b)eit 
= exp -|t Qt|1/2
- |t bY |2 1 - iY sgn(t b)tan
2
2
+ it  , (28)
where b = [b1,...,bn] is the coefficient vector and  = [1,...,n] is the mean vector.
Next we shall estimate the parameter in model (27), (28). First, the estimator of  is
given by the vector ^ of sample average. Then, we consider as factor Y a centralized index
return (for example the market portfolio (25) given by the above sub-Gaussian model).
Therefore, given the sequence of observations Y(k), we can estimate its stable parameters.
Observe that the random vector  admits a representation as a product of random variable
V and Gaussian vector G:
 = V G.
V =

A, where A is an 1/2-stable subordinator, that is
A
d
= S1/2 cos
1
4
2/1
,1,0 ;
G is a (n × 1)-Gaussian vector with null mean and variance­covariance matrix Q and
it is independent from A. We can generate values Ak, k = 1,...,N, of A independent
from G. We address to Paulauskas and Rachev's work (1999) the problem of generating
such values Ak. Using the centralizing returns ~rj = rj - j on Y we write the following
OLS estimators12 for b = [b1,...,bn] and Q:
^bi =
N
k=1 Y(k) ~r(k)
i /Ak
N
k=1 (Y(k))2/Ak
, i = 1,...,n,
and
Q =
1
N
N
k=1
(~r(k) - ^bY(k))(~r(k) - ^bY(k))
Ak
.
12 For a discussion see Tokat, Rachev and Schwartz (2002).
572 S. Ortobelli et al.
The selection of 1 is a separate problem. A possible way to estimate 1 is to consider the
OLS estimator ~bi = N
k=1 Y(k) ~r
(k)
i / N
k=1(Y(k))2 and then to evaluate the sample residuals
~(k) = ~r(k) - ~bY(k). If these residuals are heavy tailed, one can take the tail exponent
as an estimator for 1. The asymptotic properties of the above estimator can be derived
arguing similarly with Paulauskas and Rachev (1999) and Götzenberger, Rachev and
Schwartz (1999).
In order to determine portfolios R­S non-dominated when unlimited short selling is
allowed, we have to minimize the scale parameter W =

x Qx for some fixed mean
mW = x  + (1 - x e)z0 and ~b = x b/

x Qx. Alternatively, as shown by Ortobelli,
Rachev and Schwartz (2002), we can obtain these portfolios from the solution of the following
quadratic programming problem:
min
x
x Qx subject to
x  + (1 - x e)z0 = mW ,
x b = b
(29)
for some mW and b. Thus, under our assumptions, every portfolio that maximizes the
expected value of a given concave utility function u,
max
x
E u(x r)
belongs to the following frontier
(1 - 2 - 3)z0 + 2
r Q-1( - z0e)
e Q-1( - z0e)
+ 3
r Q-1b
e Q-1b
(30)
spanned by the riskless return z0, and the two risky portfolios
u(1)
=
r Q-1( - z0e)
e Q-1( - z0e)
and u(2)
=
r Q-1b
e Q-1b
.
Observe in (28) that when  = 1 = 2 > 1, every portfolio x r is an -stable distribution
and satisfies the relation
W = (1 - x e)z0 + x r
d
= S x r,x r,(1 - x e)z0 + mx r
and W = z0 when x = 0, where

x r = (x Qx)/2
+ |x bY |
, x r =
|x bY | sgn(x b)Y

x r
, mx r = x E(r).
Hence, this jointly -stable model is a fund separation model whose solutions are given
by the optimization problem (14) and these solutions satisfy the quadratic programming
problem (29).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 573
3.3. A k + 1 fund separation model in the domain of attraction of a stable law
As empirical studies show in the stable case one of the most severe restrictions of performance
measurement and asset pricing is the assumption of a common index of stability
for all assets ­ individual securities and portfolio alike. It is well understood that asset
returns are not normally distributed. We also know that the return distributions do not
have the same index of stability. However, under the assumption that returns have different
indexes of stability, it is not generally possible to find a closed form to the efficient frontier.
Generalizing the above model instead, we get the following k + 1 fund separation model
[for details on k fund separation models see Ross (1978a)]:
ri = i + bi,1Y1 +  + bi,k-1Yk-1 + i, i = 1,...,n. (31)
Here, n k 2, the vector  = (1,2,...,n) is independent from Y1,...,Yk-1 and
follows a joint sub-Gaussian symmetric k-stable distribution with 1 < k < 2, zero
mean and characteristic function (t) = exp(-|t Qt|k/2), and the random variables
Yj
d
= Sj (Yj ,Yj ,0), j = 1,...,k - 1, are mutually independent13 j -stable distributed
with 1 < j < 2 and zero mean. Under these assumptions, the portfolios belong to a
k+1(a) class. If we need to insure the separation obtained in situations where the above
model degenerates into a p-fund separation model with p < k + 1, we require the rank
condition [see Ross (1978a)]. In order to determine portfolios R­S non-dominated when
unlimited short selling is allowed, we have to minimize the scale parameter W =

x Qx
for some fixed mean mW = x  + (1 - x e)z0 and ~bj = x b,j /

x Qx, j = 1,...,k - 1.
Alternatively, as shown by Ortobelli, Rachev and Schwartz (2002), we can obtain these
portfolios from the solution of the following quadratic programming problem:
min
x
x Qx subject to
x  + (1 - x e)z0 = mW ,
x b,j = cj , j = 1,...,k - 1.
(32)
13 In order to estimate the parameters, we need to know the joint law of the vector (Y1,...,Yk-1). Therefore,
we assume independent random variables Yj , j = 1,... ,k - 1. Then the characteristic function of the vector of
returns r = [r1,...,rn] is given by
r(t) = (t)
k-1
j=1
Yj
(t b,j )eit .
Under this additional assumption, we can approximate all parameters of any optimal portfolio using a similar
procedure of the previous three fund separation model. However, if we assume a given joint (1,... ,k-1)
stable law for the vector (Y1,...,Yk-1), we can generally determine estimators of the parameters studying the
characteristics of the multivariate stable law.
574 S. Ortobelli et al.
By solving the optimization problem (32), we obtain that the riskless portfolio and other
k risky portfolios span the efficient frontier for the risk averse investors given by
1 -
k
j=1
j z0 + 1
r Q-1( - z0e)
e Q-1( - z0e)
+
k-1
j=1
j+1
r Q-1b,j
e Q-1b,j
.
The above multivariate models are motivated by arbitrage considerations as in the Arbitrage
Pricing Theory (APT) [see Ross (1976)]. Without going into details, it should be
noted that there are two versions of the APT for -stable distributed returns, a so-called
equilibrium [see Chen and Ingersoll (1983), Dybvig (1983), Grinblatt and Titman (1983)]
and an asymptotic version [see Huberman (1982)]. Connor (1984) and Milne (1988) introduced
a general theory which encompassed the equilibrium APT as well as the mutual fund
separation theory for returns belonging to any normed vector space (hence also symmetric
-stable distributed returns). While Gamrowski and Rachev (1999) provide the proof
for the asymptotic version of -stable distributed returns. Hence, it follows from Connor
and Milne's theory that the above model in the domain of attraction of a stable law of the
return is coherent with the classic arbitrage pricing theory and the mean returns can be
approximated by the linear pricing relation
i  z0 + bi,11 +  + bi,k-1k-1,
where p, for p = 1,...,k - 1, are the risk premiums relative to the different factors. The
above k + 1 fund separation model concludes the examples of models in the domain of attraction
of stable laws. In the next section we compare the Gaussian multivariate approach
with the sub-Gaussian stable one.
4. A first comparison between the normal multivariate distributional assumption
and the stable sub-Gaussian one
In this section we examine and compare the stable sub-Gaussian assumption with the normal
distributional one. Thus, we implicitly assume that returns belong to a 2(m,) class
where m is the mean and  is either the scale parameter of stable distributions or the standard
deviation of normal distributions.
In a recent work Ortobelli, Rachev and Schwartz (2002) compare the stable nonGaussian
assumption and the normal one by analyzing optimal allocations between a riskless
return and a benchmark index. Three different indexes have been taken into consideration:
CAC40, DAX30 and S&P 500. Their analysis has indicated that either the heavy
tails of data or a greater centralization of data around the mean can have a significant
impact on the approximation of the investors' choices. However, the stable non-Gaussian
allocation is generally more risk preserving than the normal one. Precisely, the stable approach
considers a further component of risk which is due to the fat tails of the return
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 575
distributions. This fact does not surprise us excessively. As a matter of fact, also Mehra
and Prescotťs empirical analysis (1985) underlines that asset pricing puzzles can be justified
thinking of people much more risk averse. Clearly, we do not believe that the equity
premium puzzle can be explained only considering the sub-Gaussian stable distribution
instead of the Gaussian one. However, we believe that the distributional differences between
the data and the classic model used in finance can help to understand asset pricing
puzzles. This conjecture is partly confirmed by assuming the stable distributions in place
of the Gaussian one [see, for example, Kocherlakota's (1997) test on CCAPM with heavytailed
pricing errors].
Next, we extend Ortobelli, Rachev and Schwartz's comparison to the multivariate case.
This comparison is formally and theoretically different from the previous one because
here the benchmark index is given by the market portfolio which generally will change,
if the distributional assumptions change too. Thus, as a consequence of Roll (1977, 1978,
1979a, b), Dybvig and Ross' (1985a, b) analysis, we observe that:
(a) an investor, who fits the return distributions with a joint 1-stable sub-Gaussian distribution,
will consider as inefficient the choice of another investor who fits the return
distributions with a joint 2-stable sub-Gaussian distribution with 1 = 2; and
(b) the stable CAPM is still subject of some of the criticism already addressed to the
classical one.
Nevertheless, it seems that the stable case explains better the empirical data. This is the
main reason why here we interpret and analyze the different behavior between the investor
who fits the data with joint stable sub-Gaussian distribution and the investor who fits the
data with the joint normal distribution.
4.1. An optimal allocation problem
First, we consider the optimal allocation among 24 assets: 23 of those assets are risky assets
with returns r = [r1,r2,...,r23] and the 24th is riskfree with annual rate 6%. We analyze
the portfolio choice problems when short sales are allowed and when short sales are not
allowed. In view of this comparison, we discuss and study the differences in portfolio
choice problems without examining them so as to choose one of the two assumptions
(Gaussian or sub-Gaussian).14
In our comparison we use daily data taken from 23 international risky indexes valued
in USD and quoted from January 1995 to January 1998. In the analysis proposed we first
consider the maximum likelihood estimation of the stable parameters and of the Gaussian
ones for every risky asset. Thus, Tables 1 and 2 assembles the approximating parameters
obtained from using the program STABLE.15
In order to compare the different stable sub-Gaussian joint distributions and the joint
normal distributions for the asset returns, we assume that the vector r is sub-Gaussian
14 On this topic, recent studies [see Ortobelli et al. (2001), Ortobelli, Huber and Schwartz (2002)] have shown
that sub-Gaussian multivariate models present a superior performance with respect to the mean­variance model.
15 See Nolan (1997) and the web site www.ca.american.edu/jpnolan.
576 S. Ortobelli et al.
-stable distributed, with  = k, k = 1,2 where 1 = 1.7488 represents the average of
the indexes of stability and 2 = 1.8856 represents the maximum of the indexes of stability
(see Table 2).16 Moreover, when in the following tables we consider the index of stability
 = 2, we implicitly assume that the returns are jointly normal distributed. Thus, every
portfolio of risky assets is stable distributed in the following way:
x r
d
= Sk (x r,x r,mx r),
where k is one of the considered index of stability k = 1,2, x r = (x Qkx)1/2 is the
respective scale parameter, Qk = [Rij /2]k is the dispersion matrix, with k = 1,2, x r = 0
is the skewness parameter, and mx r represents the mean of x r. Observe that the matrix
Qk is estimated with the method defined in the previous section and thus it depends on the
index of stability k for k = 1,2. As observed previously, the rate of convergence of the
empirical matrix Qk to the unknown matrix Qk will be faster, if p is as large as possible.
In our estimations we use p1 = 1.7 (relative to 1 = 1.7488) and p2 = 1.8 (relative to
2 = 1.8856).
We assume the investors wish to maximize the following utility functional:
U(W) = E(W) - cE W - E(W)
q
, (33)
where c and q are positive real numbers, W = z0 + (1 - )x r is the return on the portfolio,
z0 is the risk-free asset return, and
x r =
r Q-1
k ( - z0e)
e Q-1
k  - e Q-1
k ez0
is the tangent portfolio of returns (25). With reference to the allocation problem (33), we
observe:
(1) Problem (33) is equivalent to the following maximization of the utility functional
aE(W) - bE W - E(W)
q
, (34)
assuming c = b/a in (33) for every a,b > 0. Thus, E(|W - E(W)|q) represents a
particular risk measure of portfolio loss, which satisfies (under the opportune standardization)
the main characteristics of the typical dispersion measures. Solving the optimal
allocation problem (33), the investor implicitly maximizes the expected mean of
the increment wealth aW as well as minimizes the individual risk bE(|W - E(W)|q).
(2) Furthermore, when q = 2, the maximization of utility functional (33) motivates the
mean­variance approach in terms of preference relations.
16 We consider different indexes of stability, in order to value the effects of heavy-tailedness on the portfolio
selection problems.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 577
Suppose X dominates Y in the sense of R­S. Since E(X) = E(Y) and f (x) = c|x E(X)|q
is a concave utility function, for every q  [1,), it follows that:
U(X) = E(X) - cE X - E(X)
q
U(Y), q  [1,).
The above inequality implies that every risk averse investor with utility functional (34)
should choose a portfolio W = z0 + (1 - )x r that maximizes the utility functional (33)
for some real  and some q  [1,).
We know that for  = 1, all the portfolio returns W = z0 + (1 - )x r admits stable
distribution
Sk |1 - |x r,0,z0 + (1 - )mx r , k = 1,2,
and W = z0 when  = 1. Now, in order to solve the asset allocation problem
max

E(W) - cE W - E(W)
q
,
notice first that, for all q  [1,) and 1 <  < 2, we get
U(W) = E(W) - cE W - E(W)
q
= z0 + (1 - )mx r - c H(,0,q)
q
|1 - |q

q
x r,
where
H(,0,q)
q
=
2q-1 (1 - q/)
q

0 u-q-1 sin2
udu
[see Samorodnitsky and Taqqu (1994), Hardin (1984)]. The above relation analyzes the
stable non-Gaussian case. When the vector r admits a joint normal distribution (i.e.,  = 2),
then for all q > 0,
U(W) = E(W) - cE W - E(W)
q
= z0 + (1 - )mx r - c
2q/2 ((q + 1)/2)


|1 - |q

q
x r.
Hence, the real optimal solution of the problem in the important case q  (1,), is given
by
 = 1 - sgn(1 - )
sgn(1 - )(mx r - z0)
qc
q
x rV (,0,q)
1/(q-1)
(35)
578 S. Ortobelli et al.
and
x = (1 - )x, (36)
where x is given by (25) and
V (,0,q) =



H(,0,q)
q
in the stable case (1 <  < 2),
2q/2 ((q + 1)/2)


in the normal case ( = 2).
Again, one would expect that the optimal allocation was different because the constant
V (,0,q) and the matrix Q are different in the stable sub-Gaussian and in the normal case.
4.2. Stable versus normal optimal allocation: a first comparison
We analyze the differences in optimal allocations with reference to problem (33) when the
investor chooses:
(1) joint normal distribution, or,
(2) joint k stable sub-Gaussian distribution (k = 1,2) where 1 = 1.7488, 2 = 1.8856
as a model for the asset returns in his/her portfolio. Under these distinctive assumptions,
the investors with utility functional (33) have different information about the distributional
behavior of data. In particular, we examine the different market portfolio composition and
the different investor's wealth allocation in the riskless asset.
First, when short sales are allowed and when short sales are not allowed, we examine
optimal allocation among the riskless return and 23 index-daily returns: DAX 30, DAX 100
Performance, CAC 40, FTSE all share, FTSE 100, FTSE actuaries 350, Reuters Commodities,
Nikkei 225 simple average,Nikkei 300 weighted stock average,Nikkei 300 simple
stock average, Nikkei 500, Nikkei 225 stock average, Nikkei 300, Brent Crude Physical,
Brent current month, Corn No 2 Yellow cents, Coffee Brazilian, Dow Jones Futures 1, Dow
Jones Commodities, Dow Jones Industrials, Fuel Oil No 2, Goldman Sachs Commodity,
S&P 500. We use the riskless return 6% p.a.
Using the estimated daily index parameters, we can compute the dispersion matrixes
and the approximating "market" portfolios. The dispersion matrix Q is given by either
the variance­covariance matrix (in the normal case) or the matrix Qk (in the stable cases)
which depends on the index of stability k for k = 1,2 (1 = 1.7488 and 2 = 1.8856).
Therefore, as shown by Tables 3, 4, the market portfolio weights
x =
Q-1( - z0e)
e Q-1 - e Q-1ez0
change under the different distributional assumptions. We observe that the market portfolio
composition does not change excessively when we use either the asymmetric estimator
(20) and (21) of matrix Qk or the symmetric one (Qk + (Qk) )/2. However, using daily
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 579
data the elements of the dispersion matrixes are of orders 10-6. Thus the approximation in
using data could be determinant to express elements of the matrixes. In particular, Table 3
presents the market portfolio weights when we consider all 23 asset returns and short sales
are allowed. Table 4 gives the market portfolio weights when no short sales are allowed.
Under this constraint, we value the market portfolio weights in terms of the risky portfolio
compositions which maximize the extended Sharpe ratio, i.e., the market portfolio weights
are the solution of the following optimization problem
max
x
E(x r) - z0
x r
,
x e = 1,
xi 0, i = 1,...,n.
In this case the optimal allocation is reduced only among the four risky assets: DAX 100
Performance, FTSE all share, Nikkei 300 weighted stock average, Dow Jones Industrials
and the riskless one. As argued by Roll (1977, 1978), Dybvig and Ross (1985a), different
market portfolios imply a completely different security market line analysis. Thus, the
approach which takes into account short sales presents more opportunities of earning than
the approach with no short sales constraint. Therefore, it dominates the other approaches.
Besides, if the returns are jointly k stable sub-Gaussian distributed (for some determined
k = 1,2), then the Gaussian approach is inefficient. Since, in general, efficient and inefficient
portfolios can plot above and below the "real" security market line.
The analysis of Tables 3 and 4 points out that the composition of the market portfolio
is strictly linked to the index of stability. In fact, we see that the allocation of the market
portfolio in each asset component is generally monotone with respect of the stability index.
Then the intuition suggests that the stable sub-Gaussian approaches take more into consideration
the component of risk because of the fat tails. Recall that the tail behavior of every
stable non-Gaussian distribution X
d
= S(,,), with 1 <  < 2, is given by
lim
+

P(X > ) = C
1  
2

, (37)
where C = (1 - )/( (2 - )cos 
2 ). Therefore, the fat tails of smaller stability indexes
underline the risk of the loss component of every portfolio. In particular, under the diverse
distributional assumption, we distinguish the different perception of risk in the market
portfolio components. This issue can be easily analyzed in the market portfolio weights
with reference to the 23 returns when no short sales are allowed. In fact, Table 2 shows
that the index of stability of FTSE all share is greater than the other indexes of stability
(of the assets DAX 100 Performance, Nikkei 300 weighted stock average, Dow Jones
Industrials). Observe that in Table 4 the component of the FTSE all share in the market
portfolio increases with the index of stability k of the sub-Gaussian approach and the
component of the other assets (DAX 100 Performance, Nikkei 300 weighted stock average,
580 S. Ortobelli et al.
Dow Jones Industrials) decreases with the index of stability. Thus, the market portfolios
obtained under Gaussian and sub-Gaussian distributional hypotheses consider the risks due
to heavy tails differently. On the other hand the mean of market portfolios decreases with
the index of stability. However, if we accept the idea that the market portfolios represent
in some sense the market behavior, then according to the classic mean­risk interpretation,
an optimal portfolio that has a greater mean, it has also a greater risk. This fact appears
clear enough when we consider and compare the dispersion measures xkQj xk in every
mean­risk plane for every market portfolio weights
xk =
Q-1
k ( - z0e)
e Q-1
k  - e Q-1
k ez0
,
for every k and j. Observe that ~j,k = xkQj xk is the dispersion measure of market portfolio
xkr considering the j stable Paretian approach. Therefore, for every fixed mean­risk
plane (i.e., for every fixed j stable distributional approach) we can compare the market
portfolio risk positions considering their risk position ~j,k (varying k). According to a
mean­risk interpretation, we could observe that market portfolio with greater mean admits
also a greater dispersion measure ~j,k in any mean­risk plane (see Tables 5 and 6).
As a consequence of relation (37) it follows that every stable non-Gaussian distribution
X
d
= S(,,), with 1 <  < 2, admits
E X - E(X)
q
<  for q <  and (38)
E X - E(X)
q
=  for q . (39)
Hence, the weight of the risk measure E(|X - E(X)|q) in optimization problem (33) is
generally greater for the investors who use the stable laws for asset returns when q is quite
close to the index of stability .
In Tables 7, 8 we listed the optimal allocation  for the normal and the stable fit. Recall
that  is the optimal proportion of funds invested in the risk free asset which maximizes
E(W) - cE(|W - E(W)|q), where W = z0 + (1 - )x r. We have chosen q = 1.45
in Table 7 and q = 1.55 in Table 8, so that q is strictly less than all indexes of stability
in the data set. On the other hand, we want to value and compare the different effects of
q distant or closer to the stability parameters k. For any given allocation problem, we
remark in bold character and in italics respectively the greatest and the smallest allocation
in the riskless asset. Both tables show the greatest diversity among the optimal allocations
considering small risk aversion coefficients c. Instead, the very risk averse investors assume
a less risky position with every distributional hypothesis and the allocations in the riskless
asset do not change very much.
As we see from these tables, when q = 1.45 and q = 1.55 the investors who fit the data
with the Gaussian approach generally assume a less risky position than the investors who fit
the data with the sub-Gaussian approach. Thus, if the stable sub-Gaussian approximation
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 581
presents greater performances than Gaussian one (as observed by many empirical analysis)
the "stable investors" have more opportunities of earning than the "Gaussian investors". In
particular, the investors with 1 = 1.7488 stable sub-Gaussian approach invest less in the
riskless asset than the investors who fit the data with the other approaches. However, if
we consider q very closer to 1 in optimization problem (33), then, as a consequence of
(38) and (39), the investors who fit the data with 1 stable sub-Gaussian approach assume
a less risky position than the investors who fit the data with the Gaussian approach. In
this case, the "stable investor" has a very risk preserving behavior because he prefers not
allocating too much wealth in the risky asset. In this sense, intuition suggests that the
stable approaches with lower indexes of stability generally are more risk preserving than
those with greater indexes of stability because they consider the component of risk due
to the fat tails of asset returns. Therefore, the stability index plays a strategic role in the
stable optimal portfolio selection. Conversely, q in the above optimization problem can be
an opportune measure of the magnitude to be given to the component of risk due to the
heavy-tailedness of the asset returns. The importance given to q is intuitively linked to the
conditions of the market in which the investor operates.
5. Conclusions
Firstly, we study, analyze and discuss portfolio choice models depending on a finite number
of parameters. The distributional analysis presented permits to classify the admissible
parametric families of returns. Moreover, by the interrelation between the parameters of
each parametric family, we can order the portfolio choices using the basic principles of
the stochastic dominance analysis. Thus, we can identify a dispersion measure which has
some basic characteristics and represents the implicit measure of the return portfolio risk.
In view of the classification of parametric portfolio choices, that is alternative to Ross'
multiparameter one, we can distinguish the different efficient frontiers for investors who
are non-satiable, risk averse or both (non-satiable and risk averse). In particular, we distinguish
further restrictions to the classic Markowitz­Tobin's efficient frontier when no short
sales are allowed. Besides, we can identify the optimization problems we have to solve
in order to determine more accurate estimations of the investor's optimal allocations. In
this sense, the analysis presented represents a general theory and a unifying framework to
understand the parametric distributional approach to the portfolio choice theory.
Secondly, we show a simple classification of the portfolio choices considering the asymptotic
behavior of returns with heavy tailed distributions. As a matter of fact, when
returns have a stationary behavior they are in the domain of attraction of a stable law.
Therefore, we present some examples of models in the domain of attraction of stable laws.
The first distributional model considered is the case of the sub-Gaussian stable distributed
returns. It permits a mean risk analysis pretty similar to Markowitz­Tobin's mean­variance
one. In fact, this model admits the same analytical form for the efficient frontier but the parameters
have a different meaning in the two models. Thus, the most important difference
is given by the way of estimating the parameters. In order to present heavy tailed models
582 S. Ortobelli et al.
that consider the asymmetry of returns, we study a three fund separation model where
the portfolios are in the domain of attraction of an (1,2) stable law. Next, we analyze
the case of k + 1 fund separation model with portfolios in the domain of attraction of an
(1,...,k) stable law. In all models we explicate the efficient frontier for the risk averse
investors. In this context, we have shown that if the stable optimal portfolio analysis is
stable, our approach is theoretically and empirically possible. Indeed, this work should be
viewed only as a starting point for new empirical and theoretical studies on the topic of
optimal allocation.
Finally, the comparison made between the stable sub-Gaussian and the normal approach
in terms of the allocation problems has indicated that the stable sub-Gaussian allocation
is more risk preserving than the normal one and can give more opportunities of earning.
Precisely, the stable approach, differently from the normal one, considers the component
of risk due to the fat tails. Therefore, we find that the tail behavior of sub-Gaussian and
Gaussian approaches could imply substantial differences in the asset allocation. Taken into
account that the stable approach is more adherent to the reality of the market, then, as
argued by Götzenberger, Rachev and Schwartz (1999), we can obtain models that improve
the performance measurements with the stable distributional assumption.
Acknowledgment
We are grateful to Stoyan Stoyanov and Boryana Racheva-Jotova (Sofia University and
Bravo Risk Management Group, Santa Barbara) for the computational analysis and helpful
comments.
Appendix A: Proofs
In order to prove the following results, we use some Hanoch and Levy's results [in particular
see Theorems 3 and 4 in Hanoch and Levy (1969)].
Proof of Theorem 1: Implication 1. According to definition of +
k (a) family, it follows
w Z
w Z
d
=
y Z
y Z
because the two random variables have the same parameters. If w Z > y Z, then for every
t 0
P(w Z t) = P
w Z
w Z
t
w Z
P
w Z
w Z
t
y Z
= P(y Z t)
and the above inequality is strict for some t. Conversely, if w Z FSD y Z, then E(w Z) >
E(y Z).
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 583
Implication 2. As a consequence of the assumptions, it follows
q := E(w Z) - E(y Z) 0 and w Z y Z y Z+t,
for every t 0. Moreover, for every t 0 the function g(t) = (E(y Z) + t)/y Z+t is an
increasing continuous positive function that tends to infinity as t  . As a consequence
of definition of +
k (a) family there exists t q such that the random variable (w Z)/w Z
has the same parameters of (y Z + t)/y Z+t and hence (w Z)/w Z
d
= (y Z + t)/y Z+t.
Then, for every  0:
P(w Z ) P
y Z + t
y Z+t

y Z+t
P(y Z ). (40)
Observe that at least one of the two inequalities w Z y Z and q 0 is strict by hypothesis.
Then, at least one of the previous inequalities (40) is strict for some real  0.
Therefore, w Z FSD y Z.
Implication 3. First assume mw Z = my Z and w Z < y Z. Then, mw Z/w Z >
my Z/y Z and there exists t > 0 such that mw Z/w Z = (my Z + t)/y Z+t . Therefore,
mw Z < my Z + t, 1 <
y Z+t
w Z
=:  and w Z
d
=
y Z + t

.
Hence, for every  M := t/( - 1);
P(w Z ) = P(y Z  - t) P(y Z )
and for every  M,
P(w Z ) P(y Z ).
By hypothesis mw Z = my Z, thus we cannot have w Z FSD y Z or y Z FSD w Z. Therefore,
from Hanoch and Levy Theorem 3 (1969) w Z SSD y Z and mw Z = my Z; that is
w Z R­S stochastically dominates y Z. Conversely, if w Z R­S stochastically dominates
y Z, then mw Z = my Z and w Z = y Z. Thus, by previous demonstration it follows that
w Z < y Z because the converse is absurd.
Implication 4. From Implication 2, w Z y Z, implies w Z FSD y Z which implies
w Z SSD y Z. Next, assume w Z < y Z. Therefore, there exists t 0 such that
mw Z/w Z = (my Z + t)/y Z+t and w Z
d
= (y Z + t)/ where  = y Z+t/w Z. Thus,
we can distinguish two cases:
(1) mw Z my Z + t and w Z y Z+t. As a consequence of Implication 1, w Z FSD
y Z.
(2) my Z mw Z < my Z + t and w Z < y Z+t. Then, as proved in Implication 3, w Z
SSD y Z.
584 S. Ortobelli et al.
Implication 5. First, assume w Z = y Z. As a consequence of stochastic dominance
w Z SSD y Z, the inequality E(w Z) > E(y Z) holds. Thus, as a consequence of Implication
2, w Z FSD y Z. Secondly, assume w Z > y Z. Therefore, we can distinguish two
cases:
(1) mw Z/w Z my Z/y Z. Thus, as a consequence of Implication 2, w Z FSD y Z.
(2) mw Z/w Z < my Z/y Z. Then, there exists t > 0 such that (mw Z + t)/w Z+t =
my Z/y Z and (w Z + t)/w Z+t
d
= y Z/y Z. Observe that mw Z + t > my Z. Therefore,
w Z+t > y Z and for every  M := t/( - 1) where  = (w Z+t)/y Z, it
follows Fw Z() Fy Z(). Similarly, for every  > M, Fw Z() Fy Z(). However,
a  M such that Fw Z() > Fy Z() cannot exist, because distribution functions are
right continuous and
M
- Fw Z(u)du >
M
- Fy Z(u)du, against the assumption w Z
SSD y Z.
Then, w Z FSD y Z.
Implication 6. The assumptions w Z y Z and mw Z my Z with at least one inequality
strict imply mw Z/w Z > my Z/y Z. As a consequence of Implication 4, it follows
w Z SSD y Z.
Proof of Theorem 2: From the assumptions of the theorem it follows
w r - E(w r)
w r
d
=
y r - E(y r)
y r
.
First suppose w r SSD y r. Therefore, E(w r) E(y r). If E(w r) = E(y r) and w r =
y r the equality in distribution w r
d
= y r holds, against the hypothesis. Suppose for absurd
that w r > y r . Then, for every t < M := (mw ry r - my rw r)/(y r - w r) it follows
(t - E(w r))/w r > (t - E(y r))/y r and Fw r(t) Fy r(t). The inequality is strict for
some t because w r is a random variable unbounded from below. This is a contradiction
because by hypothesis w r SSD y r. Therefore, w r y r and E(w r) E(y r) with at
least one inequality strict.
Conversely, suppose E(w r) > E(y r) and w r = y r we obtain w r FSD y r from
the properties of k(a) class [see Ortobelli (2001)]. Then, assume E(w r) E(y r) and
w r < y r. The distributions of the random variables w r and X
d
= w r -E(w r)+E(y r)
belong to the same k(a) family and E(X) = E(y r). Moreover, from the properties of
k(a) family, the random variables X and w r have the same scale parameter w r and
the same other k - 2 parameters (a1,p,...,ak-2,p). Therefore,Y := (X - E(y r))/w r
d
=
(y r - E(y r))/y r and for every real u:
u
P(X
) - P(y r ) d
=
u
P
Y
 - E(y r)
w r
- P Y
 - E(y r)
y r
d 0.
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 585
The above inequality is strict for some real u because random variable Y is unbounded
from below and w r < y r . Thus, X SSD y r. If E(w r) = E(y r) we have X
d
= w r.
Instead, if E(w r) > E(y r), from the properties of k(a) class the following stochastic
relation holds
w r FSD X SSD y r.
Therefore w r SSD y r. However, considering the stochastic dominance equivalences for
"SSD" [see Levy (1992)] the following equality in distribution holds y r
d
= w r -E(w r)+
E(y r) +  with E(/w r) = 0.
Appendix B: Tables
Table 1
Maximum likelihood estimations of the normal asset return parameters considering
daily data from 1/3/95 to 1/30/98
Assets Gaussian parameters
Mean  Standard deviation 
DAX 30 0.0007 0.0113
DAX 100 Performance 0.0007 0.0106
CAC 40 0.0005 0.011
FTSE all share 0.0007 0.007
FTSE 100 0.0008 0.0078
FTSE actuaries 350 0.0007 0.0072
Reuters Commodities -0.0002 0.0072
Nikkei 225 simple average 0.0005 0.0157
Nikkei 300 weighted stock average 0.0006 0.0137
Nikkei 300 simple stock average 0.0004 0.0129
Nikkei 500 0.0003 0.0128
Nikkei 225 stock average -0.0005 0.0158
Nikkei 300 -0.0005 0.0138
Brent Crude Physical 0.0000 0.0185
Brent current month 0.0000 0.0186
Corn No 2 Yellow cents 0.0002 0.0152
Coffee Brazilian 0.0002 0.0270
Dow Jones Futures 1 -0.0001 0.0055
Dow Jones Commodities -0.0001 0.0079
Dow Jones Industrials 0.0013 0.0086
Fuel Oil No 2 -0.0001 0.0201
Goldman Sachs Commodity 0.0000 0.0092
S&P 500 0.0009 0.0083
586 S. Ortobelli et al.
Table 2
Maximum likelihood estimators of the stable asset return parameters considering daily data from 1/3/95 to
1/30/98
Assets Stable parameters
Index of Stable Stable Stable
stability  skewness  mean  scale 
DAX 30 1.8148 -0.6682 0.0005 0.0069
DAX 100 performance 1.7996 -0.6389 0.0004 0.0064
CAC 40 1.8381 -0.1852 0.0004 0.0071
FTSE all share 1.8418 -0.5726 0.0006 0.0045
FTSE 100 1.8856 -0.5192 0.0007 0.0052
FTSE actuaries 350 1.8521 -0.5666 0.0006 0.0047
Reuters Commodities 1.7959 -0.2075 -0.0003 0.0045
Nikkei 225 simple average 1.663 -0.0483 0.0004 0.009
Nikkei 300 weighted stock average 1.6962 0.0869 0.0006 0.0079
Nikkei 300 simple stock average 1.7064 0.085 0.0004 0.0075
Nikkei 500 1.7253 0.0334 0.0003 0.0076
Nikkei 225 stock average 1.6798 -0.0721 -0.0006 0.0091
Nikkei 300 1.6994 0.0303 -0.0005 0.008
Brent Crude Physical 1.7423 -0.229 -0.0003 0.0112
Brent current month 1.7405 -0.2039 -0.0001 0.0112
Corn No 2 Yellow cents 1.6869 -0.1565 0.0002 0.0083
Coffee Brazilian 1.5876 -0.0153 0.0007 0.0144
Dow Jones Futures 1 1.8063 -0.4641 -0.0002 0.0035
Dow Jones Commodities 1.6806 -0.1389 -0.0001 0.0037
Dow Jones Industrials 1.7368 -0.2886 0.0012 0.0049
Fuel Oil No 2 1.7338 -0.1961 -0.0002 0.0117
Goldman Sachs Commodity 1.8036 -0.2663 -0.0002 0.0058
S&P 500 1.7052 -0.0881 0.0010 0.0047
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 587
Table 3
Stable sub-Gaussian and Gaussian market portfolio weights when short sales are allowed
Assets Weights for Weights for Gaussian weights
 = 1.7488  = 1.8856 ( = 2)
DAX 30 0.3929 0.7182 1.2784
DAX 100 Performance -0.0704 -0.4594 -1.1205
CAC 40 -0.0694 -0.1217 -0.2022
FTSE all share 9.1802 10.6690 14.0729
FTSE 100 -3.0513 -2.5989 -1.6515
FTSE actuaries 350 -4.5968 -6.3873 -10.4307
Reuters Commodities -1.2833 -1.3451 -1.4679
Nikkei 225 Simple average 0.9302 0.0114 -1.8360
Nikkei 300 weighted stock average 19.7594 19.4478 19.7910
Nikkei 300 simple stock average -12.7675 -12.0352 -11.2504
Nikkei 500 15.8399 15.5310 15.5214
Nikkei 225 stock average -2.1322 -1.2088 0.6395
Nikkei 300 -21.2774 -21.3744 -22.4705
Brent Crude Physical 0.2125 0.1850 0.1653
Brent current month 0.0044 0.0260 0.0460
Corn No 2 Yellow cents 0.0295 0.0032 -0.0306
Coffee Brazilian 0.0175 0.0170 0.0201
Dow Jones Futures 1 -0.2376 -0.2771 -0.4155
Dow Jones Commodities -0.7123 -0.6422 -0.5327
Dow Jones Industrials 3.7278 3.7832 4.0654
Fuel Oil No 2 -0.2806 -0.2740 -0.2714
Goldman Sachs Commodity 0.2473 0.2356 0.2143
S&P 500 -2.8630 -2.9032 -3.1345
588 S. Ortobelli et al.
Table 4
Stable sub-Gaussian and Gaussian market portfolio weights when no short sales are allowed
Assets Weights for Weights for Gaussian weights
 = 1.7488  = 1.8856 ( = 2)
DAX 100 performance 0.0746 0.0742 0.0732
FTSE all share 0.2329 0.2379 0.2496
Nikkei 300 weighted stock average 0.0509 0.0502 0.0485
Dow Jones Industrials 0.6416 0.6377 0.6287
Other assets 0 0 0
Table 5
Stable sub-Gaussian and Gaussian market portfolio parameters for every mean­dispersion plane when short sales
are allowed
Parameters Market portfolio Market portfolio Gaussian market
 = 1.7488  = 1.8856 portfolio
Mean 0.0243 0.0236 0.0231
Dispersion if  = 1.7488 0.0104 0.0101 0.0101
Dispersion if  = 1.8856 0.0191 0.0185 0.0183
Standard deviation 0.0451 0.0433 0.0418
Table 6
Stable sub-Gaussian and Gaussian market portfolio parameters for every mean­dispersion plane when no short
sales are allowed
Parameters Market portfolio Market portfolio Gaussian market
 = 1.7488  = 1.8856 portfolio
Mean 0.001079 0.001077 0.001072
Dispersion if  = 1.7488 0.001553 0.001549 0.001541
Dispersion if  = 1.8856 0.002807 0.002800 0.002783
Standard deviation 0.006393 0.006376 0.006339
Ch. 14: Portfolio Choice Theory with Non-Gaussian Distributed Returns 589
Table 7
Optimal allocation for the optimization problem max E(W) - cE(|W - E(W)|1.45) when different distributional
assumptions are considered
Allocation  in the riskless asset considering the market portfolio
on 23 assets when unlimited short sales are allowed
Coefficient c of the Optimal allocation Optimal allocation Optimal allocation
optimization  when  when  when
problem  = 1.7488  = 1.8856  = 2
1.8 -10.2855 -2.3803 -0.0715
2.2 -6.2252 -1.1642 0.3140
3 -2.6268 -0.0863 0.6557
4.2 -0.7171 0.4857 0.8370
5 -0.1655 0.6509 0.8893
6 0.2227 0.7672 0.9262
7 0.4482 0.8347 0.9476
10 0.7502 0.9252 0.9763
15 0.8985 0.9696 0.9904
21 0.9520 0.9856 0.9954
Allocation  in the riskless asset considering the market portfolio
on 23 assets when short sales are not allowed
Coefficient c of the Optimal allocation Optimal allocation Optimal allocation
optimization  when  when  when
problem  = 1.7488  = 1.8856  = 2
1.4 0 0 0.3788
1.5 0 0 0.4671
1.8 0 0 0.6446
2.2 0 0.3024 0.7725
3 0 0.6498 0.8858
4.2 0.4559 0.8342 0.9459
5 0.6306 0.8875 0.9633
6 0.7537 0.9250 0.9755
7 0.8251 0.9467 0.9826
10 0.9208 0.9759 0.9921
This table computes the optimal allocation  in the riskless return 6% annual rate (daily z0 = 0.00016) for
different risk aversion coefficient c of the optimization problem max E(W) - cE(|W - E(W)|1.45) where
W = z0 + (1 - ) x r and x r is either the Gaussian market portfolio (for  = 2) or the sub-Gaussian market
portfolio (for  = 1.7488 or  = 1.8856).
590 S. Ortobelli et al.
Table 8
Optimal allocation for the optimization problem max E(W) - cE(|W - E(W)|1.55) when different distributional
assumptions are considered
Allocation  in the riskless asset considering the market portfolio
on 23 assets when unlimited short sales are allowed
Coefficient c of the Optimal allocation Optimal allocation Optimal allocation
optimization  when  when  when
problem  = 1.7488  = 1.8856  = 2
1.5 -11.4475 -4.5346 -1.2292
1.8 -7.9354 -2.9730 -0.6003
2.2 -5.2038 -1.7584 -0.1111
3 -2.5298 -0.5695 0.3678
4.2 -0.9146 0.1487 0.6571
5 -0.3944 0.3800 0.7503
6 -0.0010 0.5549 0.8207
7 0.2437 0.6637 0.8646
10 0.6046 0.8242 0.9292
15 0.8108 0.9159 0.9661
Allocation  in the riskless asset considering the market portfolio
on 23 assets when short sales are not allowed
Coefficient c of the Optimal allocation Optimal allocation Optimal allocation
optimization  when  when  when
problem  = 1.7488  = 1.8856  = 2
1.3 0 0 0
1.5 0 0 0
1.8 0 0 0.0859
2.2 0 0 0.3654
3 0 0.1239 0.6389
4.2 0 0.5248 0.8042
5 0.2305 0.6539 0.8574
6 0.4476 0.7515 0.8976
7 0.5826 0.8123 0.9226
10 0.7818 0.9019 0.9595
This table computes the optimal allocation  in the riskless return 6% annual rate (daily z0 = 0.00016 ) for
different risk aversion coefficient c of the optimization problem max E(W) - cE(|W - E(W)|1.55)where W =
z0 + (1- ) x r and x r is either the Gaussian market portfolio (for  = 2) or the sub-Gaussian market portfolio
(for  = 1.7488 or  = 1.8856).
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Chapter 15
PORTFOLIO MODELING WITH HEAVY TAILED RANDOM
VECTORS
MARK M. MEERSCHAERT
Department of Mathematics, University of Nevada, Reno, USA
e-mail: mcubed@unr.edu
HANS-PETER SCHEFFLER
Fachbereich Mathematik, University of Dortmund, 44221 Dortmund, Germany
e-mail: hps@math.uni-dortmund.de
Contents
Abstract 596
Keywords 596
1. Introduction 597
2. Heavy tails 597
3. Central limit theorems 600
4. Matrix scaling 607
5. The spectral decomposition 610
6. Sample covariance matrix 613
7. Dependent random vectors 616
8. Tail estimation 619
9. Tail estimator proof for dependent random vectors 626
10. Conclusions 636
References 637
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
596 M.M. Meerschaert and H.-P. Scheffler
Abstract
Since the work of Mandelbrot in the 1960s there has accumulated a great deal of empirical
evidence for heavy tailed models in finance. In these models, the probability of a large
fluctuation falls off like a power law. The generalized central limit theorem shows that
these heavy-tailed fluctuations accumulate to a stable probability distribution. If the tails
are not too heavy then the variance is finite and we find the familiar normal limit, a special
case of stable distributions. Otherwise the limit is a nonnormal stable distribution, whose
bell-shaped density may be skewed, and whose probability tails fall off like a power law.
The most important model parameter for such distributions is the tail thickness , which
governs the rate at which the probability of large fluctuations diminishes. A smaller value
of  means that the probability tails are fatter, implying more volatility. In fact, when
 < 2 the theoretical variance is infinite. A portfolio can be modeled using random vectors,
where each entry of the vector represents a different asset. The tail parameter  usually
depends on the coordinate. The wrong coordinate system can mask variations in , since
the heaviest tail tends to dominate. A judicious choice of coordinate system is given by the
eigenvectors of the sample covariance matrix. This isolates the heaviest tails, associated
with the largest eigenvalues, and allows a more faithful representation of the dependence
between assets.
Keywords
multivariable regular variation, moment estimates, moving averages, generalized domains
of semistable attraction, R­O varying measures
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 597
1. Introduction
In order to construct a useful probability model for an investment portfolio, we must consider
the dependence between assets. If we accept the premise that price changes are heavy
tailed, then we are lead to consider random vectors with heavy tails. In this chapter, we
survey those portions of the theory of heavy tailed random vectors that seem relevant to
portfolio analysis. The most flexible models recognize the possibility that the thickness of
probability tails varies in different directions, implying the need for matrix scaling. A judicious
change of coordinates often simplifies the model, and may uncover features masked
by the original coordinates. The original coordinates are the price changes (or returns) for
each asset. The new coordinates can be interpreted as market indices, chosen to capture
certain features of the market. In some popular heavy-tailed finance models, the tails are
so heavy that the theoretical variance of price changes is undefined. For these models, the
theoretical covariance matrix is also undefined. Of course the sample variance and the sample
covariance matrix can always be computed for any data set, but these statistics are not
estimating the usual model parameters. One of the most interesting discoveries in heavy
tailed modeling is that, in the infinite variance case, the sample covariance matrix actually
contains quite a bit of important information about the underlying distribution. In fact, the
eigenvectors of this matrix provide a very useful coordinate system. We illustrate the application
of this principle, and we also include a previously unpublished proof, extending
the method to more general heavy tailed vector models with time dependence.
2. Heavy tails
A probability distribution has heavy tails if some of its moments fail to exist. Suppose that
X is a random variable with density f (x) so that
P(a X b) =
b
a
f (x)dx.
The k-th moment of the random variable X is defined by an improper integral
k = E Xk
=

-
xk
f (x)dx.
The mean  = 1, variance 2 = 2 - 2
1, skewness and kurtosis depend on these moments.
Because k is an improper integral, it may not exist. If f (x) is a normal density,
a lognormal density, or any other density whose tails fall off exponentially then all of the
moments k exist. But if f (x) has heavy tails that fall off like a power law, then some
of the moments k will not exist. The simplest example of a heavy tailed distribution is a
Pareto, invented to model the distribution of incomes. A Pareto random variable satisfies
598 M.M. Meerschaert and H.-P. Scheffler
P(X > x) = Cx- so that the probability of large outcomes falls off like a power law. The
Pareto density is defined by
f (x) = Cx--1 for x > C1/,
0 otherwise
so that
k =

C1/
Cxk--1
dx = Ck/

1
yk--1
dy = Ck/ ykk
- 

y=1
using the substitution x = C1/y. If k <  then the limit at infinity is zero and k =
Ck//( - k), but if k  then this improper integral diverges, so that the k-th moment
does not exist.
Pareto distributions are closely related to some other familiar distributions. If U has a
uniform distribution on (0,1), then X = U-1/ has a Pareto distribution with tail parameter
. To check this, write
P(X > x) = P U-1/
> x = P U < x=
x-
.
If X is Pareto with P(X > x) = x-, then Y = lnX has an exponential distribution with
rate . To see this, note that
P(Y > y) = P(ln X > y) = P X > ey
= ey =
e-y
.
Some other familiar distributions have Pareto-like power law tails, causing some moments
to diverge. If Y has a Student-t distribution with  degrees of freedom, then P(|Y| > y) 
Cy- where  = .1 Then E(Yk) exists only for k < . If Y has a Gamma distribution with
density proportional to yp-1 e-qy then the log-Gamma random variable X defined by Y =
lnX satisfies P(X > x)  Cx- for x large, where  = q. Some other distributions with
Pareto-like tails are the stable and operator stable distributions, which will be discussed
later in this chapter.
Heavy tailed random variables with P(|X| > x)  Cx- are observed in many real
world applications. Estimation of the tail parameter  is important, because it determines
which moments exist. Anderson and Meerschaert (1998) find heavy tails in a river flow
with   3, so that the variance is finite but the fourth moment is infinite. Tessier et
al. (1996) find heavy tails with 2 <  < 4 for a variety of river flows and rainfall accumulations.
Hosking and Wallis (1987) find evidence of heavy tails with   5 for
annual flood levels of a river in England. Benson, Wheatcraft and Meerschaert (2000),
Benson et al. (2001) model concentration profiles for tracer plumes in groundwater using
1 Here f (x)  g(x) means that f (x)/g(x)  1 as x  .
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 599
stochastic models whose heavy tails have 1 <  < 2, so that the mean is finite but the
variance is infinite. Heavy tail distributions with 1 <  < 2 are used in physics to model
anomalous diffusion, where a cloud of particles spreads faster than classical Brownian motion
predicts (Blumen, Zumofen and Klafter, 1989; Klafter, Blumen and Shlesinger, 1987;
Shlesinger, Zaslavsky and Frisch, 1994). More applications to physics with 0 <  < 2
are cataloged in Uchaikin and Zolotarev (1999). Resnick and St˘aric˘a (1995) examine the
quiet periods between transmissions for a networked computer terminal, and find heavy
tails with 0 <  < 1, so that the mean and variance are both infinite. Several additional
applications to computer science, finance, and signal processing appear in Adler, Feldman
and Taqqu (1998). More applications to signal processing can be found in Nikias
and Shao (1995).
Mandelbrot (1963) and Fama (1965a) pioneered the use of heavy tail distributions in
finance. Mandelbrot (1963) presents graphical evidence that historical daily price changes
in cotton have heavy tails with   1.7, so that the mean exists but the variance is infinite.
Jansen and de Vries (1991) argue that daily returns for many stocks and stock indices have
heavy tails with 3 <  < 5, and discuss the possibility that the October 1987 stock market
plunge might be just a heavy tailed random fluctuation. Loretan and Phillips (1994) use
similar methods to estimate heavy tails with 2 <  < 4 for returns from numerous stock
market indices and exchange rates. This indicates that the variance is finite but the fourth
moment is infinite. Both daily and monthly returns show heavy tails with similar values of
 in this study. Rachev and Mittnik (2000) use different methods to find heavy tails with
1 <  < 2 for a variety of stocks, stock indices, and exchange rates. McCulloch (1996) uses
similar methods to re-analyze the data in Jansen and de Vries (1991), Loretan and Phillips
(1994), and obtains estimates of 1.5 <  < 2. This is important because the variance of
price returns is finite if  > 2 and infinite if  < 2. While there is disagreement about
the true value of , depending on which model is employed, all of these studies agree
that financial data is typically heavy tailed, and that the tail parameter  varies between
different assets.
Portfolio analysis involves the joint probability distribution of several prices or returns
X1,...,Xd, where d is the number of assets in the portfolio. It is natural to model this
set of numbers as a d-dimensional random vector X = (X1,...,Xd) . We say that X has
heavy tails if E( X k) is undefined for some k = 1,2,3,.... Let us consider the practical
problem of portfolio modeling. We choose d assets and research historical performance
to obtain data of the form Xi(t) where i = 1,...,d is the asset and t = 0,...,n is the
time variable. Typically the distribution of values Xi(0),...,Xi(n) has a heavy tail whose
parameter i can be estimated from this data. The research of Jansen and de Vries (1991),
Loretan and Phillips (1994), and Rachev and Mittnik (2000) indicates, not surprisingly, that
i will vary depending on the asset. Then the random vectors Xt = (X1(t),...,Xd(t))
will have heavier tails in some directions than in others. Despite this well known fact, most
existing research on heavy tailed portfolio modeling has assumed that the probability tails
are the same in every direction. Nolan, Panorska and McCulloch (2001) consider such a
model, based on the multivariable stable distribution, for a vector of two exchange rates.
600 M.M. Meerschaert and H.-P. Scheffler
They argue that  is the same for both.2 Rachev and Mittnik (2000) use a multivariable
stable model for portfolio analysis, so that  is the same for every asset. The same approach
was also applied to portfolio analysis by Bawa, Elton and Gruber (1979), Belkacem, Véhel
and Walter (2000), Chamberlain, Cheung and Kwan (1990), Fama (1965b), Gamba (1999),
Press (1982), Rachev and Han (2000), and Ziemba (1974). If this modeling approach can
be enhanced to allow i to vary with the asset, a more realistic and flexible representation
of financial portfolios can be achieved. The goal of this chapter is to show how this can be
accomplished, using modern central limit theory.
3. Central limit theorems
Normal and log-normal models are popular in finance because of their simplicity and familiarity.
Their use can also be justified by the central limit theorem. If X,X1,X2,X3,...
are independent and identically distributed (IID) random variables with mean m = E(X)
and finite variance 2 = E[(X - m)2] then the central limit theorem says that
(X1 +  + Xn) - nm
n1/2
 Y, (3.1)
where Y is a normal random variable with mean zero and variance 2, and  means
convergence of probability distributions. Essentially, (3.1) means that X1 +  + Xn is
approximately normal (with mean nm and variance n2) for n large. If the summands
Xi represent independent price shocks, then their sum is the price change over a period of
time. If price changes are accumulations of many IID shocks, then they should be normally
distributed. If price changes accumulate multiplicatively, taking logs changes the product
into a sum, leading to a log-normal model.
For portfolio analysis, we need to consider a vector of prices. Suppose that X,X1,
X2,X3,... are IID random vectors on a d-dimensional Euclidean space Rd . If X =
(X1,...,Xd) then the mean m = E(X) is a vector with i-th entry mi = E(Xi), the covariance
matrix C is a d × d matrix with ij entry
cij = Cov(Xi,Xj ) = E (Xi - mi)(Xj - mj ) ,
and the central limit theorem says that
(X1 +  + Xn) - nm

n
 Y, (3.2)
2 Example 8.1 gives an alternative operator stable model for the same data set.
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 601
where Y is a normal random vector with mean zero and covariance matrix C = E[YY ].
In this case, it simplifies the analysis to change coordinates. If the matrix P defines the
change of coordinates then it follows from (3.2) that
(PX1 +  + PXn) - nPm

n
 PY , (3.3)
where PY is multivariate normal with mean zero and covariance matrix PCP =
E[(PY)(PY) ]. If we take the new coordinate system defined by the eigenvectors of
the covariance matrix C, then the limit PY has independent normal marginals. The
eigenvalues of C determine the variance of each marginal, so their square roots measure
volatility. The corresponding marginals of PX are all linear combinations of the
original assets, chosen to be asymptotically independent. This coordinate system is one
of the cornerstones of Markowitz's theory of optimal portfolios, see for example Elton
and Gruber (1995).
For heavy tailed random variables, the central limit theorem may not hold, because the
second moment might not exist. An extended central limit theorem applies in this case.
If X,X1,X2,X3,... are IID random variables we say that X belongs to the domain of
attraction of some random variable Y, and we write X  DOA(Y), if
(X1 +  + Xn) - bn
an
 Y. (3.4)
For mathematical reasons we exclude the degenerate case where Y = c with probability
one. The limits in (3.4) are called stable. If E(X2) exists then the classical central limit
theorem shows that Y is normal, a special case of stable. In this case, we can take an = n1/2
and bn = nE(X). If X has heavy tails with P(|X| > r)  Cr- then the situation depends
on the tail thickness . If  > 2 then E(X2) exists and sums are asymptotically normal. But
if 0 <  2 then E(X2) =  and (3.4) holds with an = n1/ as long as a tail balancing
condition holds:
P(X > r)
P(|X| > r)
 p and
P(X < -r)
P(|X| > r)
 q as r   (3.5)
for some 0 p,q 1 with p + q = 1.
A proof of the extended central limit theorem can be found in Gnedenko and Kolmogorov
(1968), see also Feller (1971) and Meerschaert and Scheffler (2001a). The condition
for X  DOA(Y) is stated in terms of regular variation. A function f (r) varies
regularly if
lim
r
f (r)
f (r)
= 
for all  > 0. (3.6)
For Y stable with index 0 <  < 2, so that Y is not normal, a necessary and sufficient
condition for X  DOA(Y) is that P(|X| > r) varies regularly with index - and (3.5)
602 M.M. Meerschaert and H.-P. Scheffler
holds for some 0 p,q 1 with p + q = 1. If we have P(|X| > r)  Cr- then it
is easy to see that P(|X| > r) varies regularly with index -, but the definition also
allows a slightly more general tail behavior. For example, if P(|X| > r)  Cr- logr
then P(|X| > r) still varies regularly with index -. The norming constants an in
(3.4) can always be chosen according to the formula nP(|X| > an)  C. If we have
P(|X| > r)  Cr- this leads to an = n1/. In practical applications, it is common to
assume that P(|X| > r)  Cr- because a practical procedure exists for estimating the
parameters C, for a given heavy tailed data set.3
Stable distributions are typically specified in terms of their characteristic functions
(Fourier transforms). If Y is stable with density f (y) its characteristic function
E eikY
=

-
eiky
f (y)dy
is of the form e(k) where
(k) =



ibk - |k| 1 - i sign(k)tan

2
for  = 1,
ibk - |k| 1 + i
2

sign(k)ln|k| for  = 1.
(3.7)
The entire class of nondegenerate stable laws on R1 is given by these formulas with index
  (0,2], scale   (0,), skewness   [-1,+1], and center b  (-,). The
stable distribution with these parameters will be written as S(,,b) using the notation
of Samorodnitsky and Taqqu (1994). The skewness  = p - q governs the deviations of
the distribution from symmetry, so that f (y) is symmetric if  = 0. The scale  and the
center b have the usual meaning that if Y has a S(1,,0) distribution then Y + b has
a S(,,b) distribution, except that for  = 1 and  = 0 multiplication by  introduces
a nonlinear change in the shift. The stable index  governs the tails of Y, and in fact
P(|Y| > r)  Cr- where

=



C
(2 - )
1 - 
cos

2
for  = 1,
C

2
for  = 1
(3.8)
in the nonnormal case 0 <  < 2. The tails are balanced so that
P(Y > r)
P(|Y| > r)
 p and
P(Y < -r)
P(|Y| > r)
 q as r  . (3.9)
3 See Section 8.
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 603
Stable laws belong to their own domain of attraction, but more is true. In fact, if Yn are IID
with Y then
(Y1 +  + Yn) - bn
n1/
d
= Y (3.10)
for some bn, where
d
= indicates that both sides have the same probability distribution.
Sums of IID stable laws are again stable with the same ,. Although there is no closed
analytical formula for stable densities, the efficient computational method of Nolan (1997,
2002; Resnick and St˘aric˘a, 1995) can be used to plot density curves. Nolan (2001) uses
these methods to compute maximum likelihood estimators for the stable parameters, see
also Mittnik et al. (1999), Mittnik, Doganoglu and Chenyao (1999).
If Xn is the price change on day n then the accumulation of these changes will be approximately
stable, assuming that Xn are IID with X and P(|X| > x)  Cx-. If  < 2,
as in the cotton prices considered in Mandelbrot (1963), then the price obtained by adding
these changes will be approximately stable with a power law tail. The balancing parameters
p and q describe the probability that a large change in price will be positive or negative,
respectively. The scale  (or equivalently, the dispersion C) depends on the price units
(e.g., US dollars). If 2 <  < 4 then the sum of these price changes will be asymptotically
normal. However, the rule of thumb that sums look normal for n 30 is no longer reliable.
The heavy tails slow the rate of convergence in the central limit theorem. To illustrate the
point, we simulated Pareto random variables with  = 3, using the fact that if U is uniform
on (0,1) then U-1/ is Pareto with tail parameter . We summed n = 50 of these
random variables, and repeated the simulation 100 times to get an idea of the distribution
of these sums. The boxplot in Figure 1 indicates that the distribution of the resulting sums
is skewed to the right, with some outliers. The normal probability plot in Figure 2 indicates
a significant deviation from normality. The moral of this story is that for heavy tailed
random variables with  > 2, sums eventually converge to a normal limit, but slower than
usual.
For heavy tailed random vectors, a generalized central limit theorem applies. If
X,X1,X2,X3,... are IID random vectors on Rd we say that X belongs to the generalized
domain of attraction of some full dimensional random vector Y on Rd, and we write
X  GDOA(Y), if
An(X1 +  + Xn - bn)  Y (3.11)
for some d × d matrices An and vectors bn  Rd. The limits in (3.11) are called operator
stable (Jurek and Mason, 1993; Sharpe, 1969). If E( X 2) exists then the classical
central limit theorem shows that Y is multivariable normal, a special case of operator stable.
In this case, we can take An = n-1/2I and bn = nE(X). If X has heavy tails with
P( X > r)  Cr- then the situation depends on the tail thickness . If  > 2 then
604 M.M. Meerschaert and H.-P. Scheffler
Fig. 1. Sums of 50 Pareto variables with  = 3. Their distribution is skewed to the right with several outliers.
Fig. 2. Sums of 50 Pareto variables with  = 3. Upper tail shows systematic deviation from normal distribution.
E( X 2) exists and sums are asymptotically normal. But if 0 <  < 2 then E( X 2) = 
and (3.11) holds with An = n-1/I as long as a tail balancing condition holds:
P( X > r, X/ X  B)
P( X > r)
 M(B) as r   (3.12)
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 605
for all Borel subsets4 B of the unit sphere S = {  Rd:  = 1} whose boundary has
M-measure zero, where M is a probability measure on the unit sphere which is not supported
on any d - 1 dimensional subspace of Rd. A proof of the generalized central limit
theorem can be found in Rvaˇceva (1962) or Meerschaert and Scheffler (2001a). In this case,
where the tails of X fall off at the same rate in every direction, the limit Y is multivariable
stable (Samorodnitsky and Taqqu, 1994), a special case of operator stable.
If Y is multivariable stable with density f (y) its characteristic function
E eikY
= eiky
f (y)dy
is of the form e(k) where
(k) = ib  k - 
 =1
|  k|
1 - isign(  k)tan

2
M(d)
for  = 1 and
(k) = ib  k - 
 =1
|  k| 1 + i
2

sign(  k)ln|  k| M(d)
for  = 1. The entire class of multivariable stable laws on Rd is given by these formulas
with index   (0,2], scale  > 0, mixing measure M and center b  Rd . We say that Y has
distribution S(,M,b) in this case. The mixing measure M is a probability distribution
on the unit sphere in Rd that governs the tails of Y, so that f (y) is symmetric if M is
symmetric. The center b and scale  have the usual meaning that if Y has a S(1,M,0)
distribution then Y + b has a S(,M,b) distribution, except when  = 1. The stable
index  governs the tails of Y in the nonnormal case (0 <  < 2). In fact, P( Y > r) 
Cr- where C is given by (3.8). The mixing measure M is a multivariable analogue of the
skewness . If d = 1 then M{+1} = p and M{-1} = q, since the unit sphere on R1 is the
two point set {-1,+1}. In this case, Y is stable with skewness  = p - q. The tails of a
multivariable stable random vector are balanced so that
P( Y > r, Y/ Y  B)
P( Y > r)
 M(B) as r  . (3.13)
If d = 1 this reduces to the tail balancing condition (3.9) for stable random variables.
Multivariable stable laws belong to their own domain of attraction, and if Yn are IID with
Y then
(Y1 +  + Yn) - bn
n1/
d
= Y (3.14)
4 The class of Borel subsets is the smallest class that include open sets and is closed under complements and
countable unions.
606 M.M. Meerschaert and H.-P. Scheffler
for some bn, so that sums of IID multivariable stable laws are again multivariable stable
with the same . When Y is nonnormal multivariable stable with distribution S(,M,b)
for some 0 <  < 2, the necessary and sufficient condition for X  DOA(Y) is that
P( X > r) varies regularly with index - and the balanced tails condition (3.12)
holds.
Example 3.1. The mixing measure governs the radial direction of large price jumps. Take
Ri IID Pareto random variables with P(R > r) = Cr-. Take i to be IID random unit
vectors with distribution M, independent of (Ri). Then Xi = Rii are IID random vectors
with P( Xi > r) = Cr- and
P( Xi > r, Xi/ Xi  B)
P( Xi > r)
= P(i  B) = M(B)
for any Borel subset B of the unit sphere, and so Xi  DOA(Y) where Y is multivariable
stable with distribution S(,M,b) for any b  Rd. We can take An = n-1/I in (3.11),
and b depends on the choice of centering bn. We call these heavy tailed random vectors
multivariable Pareto. If we use a multivariable Pareto model for large jumps in the vector
of prices for a portfolio, the parameter  governs the radius and the mixing measure M
governs the angle of large jumps. Sums of these IID jumps are asymptotically multivariable
stable with the same index  and mixing measure M. The radius R = Y satisfies
P(R > r)  Cr- and the distribution of the radial component  = Y/ Y conditional
on P( Y > r) tends to M as r   in view of the tail balancing condition (3.13). In
other words, multivariable stable random vectors are asymptotically multivariable Pareto
on their tails. In a multivariable stable model for price jumps, the mixing measure determines
the direction of large jumps. If M is discrete with M(i) = pi, then it follows
from the characteristic function formulas that Y can be represented as the sum of independent
stable components laid out along the i directions, and the methods of Nolan (1997,
2002) can be used to plot multivariable stable densities, see Byczkowski, Nolan and Rajput
(1993). The same idea is used by Modarres and Nolan (1994) to simulate stable random
vectors with discrete mixing measures. For an arbitrary mixing measure, multivariable stable
laws can be simulated using sums of independent, identically distributed multivariable
Pareto laws. If 0 <  < 1 then the random vector n-1/(X1 +  + Xn) is approximately
S(,M,0) where C is given by (3.8). If 1 <  < 2 then n-1/(X1 +  + Xn - nEX1)
is approximately S(,M,0) where C is given by (3.8) and
E(X1) = E(R1)E(1) =
C1/
 - 1  =1
M(d).
Remark 3.2. Previously a different type of multivariable Pareto distribution was considered
by Arnold (1990), see also Kotz, Balakrishnan and Johnson (2000).
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 607
4. Matrix scaling
The multivariable stable model is the basis for the work of Nolan, Panorska and McCulloch
(2001) on exchange rates, and the portfolio models in Rachev and Mittnik (2000).
Under the assumptions of this model, the probability tail of the random vector Xt is
assumed to fall off at the same power law rate in every radial direction. Suppose that
Xt = (X1(t),...,Xd(t)) where Xi(t) is the price change of the i-th asset on day t.
If Xt belongs to the domain of attraction of some multivariable stable random vector
Y = (Y1,...,Yd) with index , and that (3.11) holds with An = n-1/I. Projecting onto
the i-th coordinate axis shows that
Xi(1) +  + Xi(n) - bi(n)
n1/
 Yi, (4.1)
where bn = (b1(n),...,bd(n)) , so that Yi is stable with index  and Xi(t) belongs to the
domain of attraction of Yi . According to Jansen and de Vries (1991), Loretan and Phillips
(1994), and Rachev and Mittnik (2000), the stable index i should vary depending on the
asset. Then (4.1) is replaced by
Xi(1) +  + Xi(n) - bi(n)
n1/i
 Yi for each i = 1,...,d (4.2)
so that Yi is stable with index i. Mittnik and Rachev (1993) seem to have been the first
to apply such models to a problem in finance, see also Section 8.6 in Rachev and Mittnik
(2000). Assuming the joint convergence
An








X1(1)
X2(1)
...
Xd(1)



 +  +




X1(n)
X2(n)
...
Xd(n)



 -




b1(n)
b2(n)
...
bd(n)







 




Y1
Y2
...
Yd



 (4.3)
and changing to vector-matrix notation we get (3.11) with diagonal norming matrices
An =



n-1/1 0  0
0 n-1/2 0
...
...
...
0 0  n-1/d


 (4.4)
which we will also write as An = diag(n-1/1 ,...,n-1/d ). The matrix scaling is natural
since we are dealing with random vectors, and it allows a more realistic portfolio model.
The i-th marginal Yi of the operator stable limit vector Y is stable with index i , so the
tail behavior of Y varies with angle. The convergence (3.11) with An diagonal was first
considered in Resnick and Greenwood (1979), see also Meerschaert (1991).
608 M.M. Meerschaert and H.-P. Scheffler
Matrix notation also leads to a natural analogue of the stable index . Let exp(A) =
I + A + A2/2! + A3/3! +  be the usual exponential operator for d × d matrices.
This operator occurs, for example, in the theory of linear differential equations. If
A = diag(a1,...,ad) then an easy matrix computation using the Taylor series formula
ex = 1 + x + x2/2! + x3/3! +  shows that exp(A) = diag(ea1,...,ead ). See Hirsch and
Smale (1974) or Section 2.2 of Meerschaert and Scheffler (2001a) for details and additional
information. Now define E = diag(1/1,...,1/d). Then the norming matrices An
in (4.4) can also be written in the more compact form An = n-E = exp(-E lnn), since
-E lnn = diag(-(1/1)lnn,...,-(1/d)lnn) and e-(1/i)lnn = n-1/i . The matrix E,
called an exponent of the operator stable random vector Y, plays the role of the stable index
. This matrix E need not be diagonal. Diagonalizable exponents involve a change of
coordinates, degenerate eigenvalues thicken probability tails by a logarithmic factor, and
complex eigenvalues introduce rotational scaling, see Meerschaert (1990). The case of a
diagonalizable exponent plays an important role in Example 8.1.
The generalized central limit theorem for matrix scaling can be found in Meerschaert
and Scheffler (2001a). Matrix scaling allows for a limit with both normal and nonnormal
components. Since Y is infinitely divisible, the Lévy representation [Theorem 3.1.11 in
Meerschaert and Scheffler (2001a)] shows that the characteristic function E[eikY ] is of the
form e(k) where
(k) = ib  k -
1
2
k  Ck +
x=0
eikx
- 1 ik
 x
1 + x 2
(dx)
for some b  Rd, some nonnegative definite symmetric d × d matrix C and some Lévy
measure . The Lévy measure satisfies {x: x > 1} <  and
0< x <1
x 2
(dx) < .
For a multivariable stable law,
 x: x > r,
x
x
 B = Cr-
M(B)
and the characteristic function formulas for multivariable stable laws follow by a lengthy
computation, see Section 7.3 in Meerschaert and Scheffler (2001a) for complete details. If
 = 0 then Y is normal with mean b and covariance matrix C. If C = 0 then a necessary
and sufficient condition for (3.11) to hold is that
nP(AnX  B)  (B) as n   (4.5)
for Borel subsets B of Rd \{0} whose boundary have -measure zero, where  is the Lévy
measure of the limit Y. Proposition 6.1.10 in Meerschaert and Scheffler (2001a) shows
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 609
that the convergence (4.5) is equivalent to regular variation of the probability distribution
(B) = P(X  B). If (4.5) holds then Proposition 6.1.2 in Meerschaert and Scheffler
(2001a) shows that the Lévy measure satisfies
t(dx) =  t-E
dx for all t > 0 (4.6)
for some d × d matrix E. Then it follows from the characteristic function formula that Y
is operator stable with exponent E, and that for Yn IID with Y we have
n-E
(Y1 +  + Yn - bn)
d
= Y (4.7)
for some bn, see Theorem 7.2.1 in Meerschaert and Scheffler (2001a). Hence operator
stable laws belong to their own GDOA, so that the probability distribution of Y also varies
regularly, and sums of IID operator stable random vectors are again operator stable with
the same exponent E. If E = aI then Y is multivariable stable with index  = 1/a, and
(4.5) is equivalent to the balanced tails condition (3.12).
Example 4.1. Multivariable Pareto random vectors with matrix scaling extend the model
in Example 3.1. Suppose Y is operator stable with exponent E and Lévy measure . Define
Fr,B = sE
: s > r,   B
and let (B) = (F1,B) for any Borel subset B of the unit sphere S whose boundary
has -measure zero.5 Let C = (S) and define the probability measure M(B) = (B)/C.
Take Ri IID standard Pareto random variables with P(R > r) = Cr-1, i IID random
unit vectors with distribution M and independent of (Ri), and finally let Xi = RE
i i .
Since tEF1,B = Ft,B we have (Ft,B) = (tEF1,B) = t-1(F1,B) = Ct-1M(B) in view
of (4.6). Then
nP n-E
Xi  Ft,B = nP RE
i i  Fnt,B
= nP(Ri > nt,i  B)
= nC(nt)-1
M(B) = (Ft,B)
for n > 1/t, so that (4.5) holds for the sets Ft,B with An = n-E. Then Xi  GDOA(Y).
Operator stable laws can be simulated using sums of these IID random vectors. If every
eigenvalue of E has real part greater than one, then n-E(X1 +  + Xn) is approximately
operator stable with exponent E and Lévy measure . If every eigenvalue of E has real
5 The measure  is called the spectral measure of Y.
610 M.M. Meerschaert and H.-P. Scheffler
part less than one, then n-E(X1 +  + Xn - nm) is approximately operator stable with
exponent E and Lévy measure  where
m = C
 =1

C
rE

dr
r2
M(d)
is the mean of X1.
5. The spectral decomposition
The tail behavior of an operator stable random vector Y is determined by the eigenvalues
of its exponent E. If E = (1/)I then Y is multivariable stable and P(|Y  | > r) 
C r- for any  = 0. If E = diag(a1,...,ad) then Y = (Y1,...,Yd) where Yi is a stable
random variable with index i = 1/ai. This requires 0 < i 2 so that ai 1/2. For any
d × d matrix E there is a unique spectral decomposition based on the real parts of the
eigenvalues, see for example Theorem 2.1.14 in Meerschaert and Scheffler (2001a). This
decomposition allows us to write E = PBP-1 where P is a change of coordinates matrix
and B is block-diagonal with
B =




B1 0  0
0 B2 0
...
...
...
0 0  Bp



 (5.1)
where Bi is a di ×di matrix, every eigenvalue of Bi has real part equal to ai, a1 <  < ap,
and d1 ++dp = d. Let e1 = (1,0,...,0) , e2 = (0,1,0,...,0) , ..., ed = (0,...,0,1)
be the standard coordinates for Rd and define pik = Pej when j = d1 + + di-1 + k for
some k = 1,...,di. Then
Vi = span{pi1,...,pidi
} =
di
k=1
tkpik: t1,...,tdi real
is a di-dimensional subspace of Rd. Any vector y  Rd can be written uniquely in the
form y = y1 +  + yp with yi  Vi for each i = 1,...,p. This is called the spectral
decomposition of Rd with respect to E. Since B is block-diagonal and E = PBP-1, every
Epik is a linear combination of pi1,...,pidi
and therefore Eyi  Vi for every yi  Vi.
This means that Vi is an E-invariant subspace of Rd . Given a nonzero vector   Rd, write
 = 1 +  + p with i  Vi for each i = 1,...,p and define
() = min
1
ai
: i = 0 . (5.2)
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 611
Since the probability distribution of Y varies regularly with exponent E, Theorem 6.4.15
in Meerschaert and Scheffler (2001a) shows that for any small  > 0 we have
r-()<
P |Y  | > r < r-()+
for all r > 0 sufficiently large. In other words, the tail behavior of Y is dominated by the
component with the heaviest tail. This also means that E(|Y  |) exists for 0 <  < ()
and diverges for  > (). If we write Y = Y1 +  + Yp with Yi  Vi for each i =
1,...,p, then projecting (4.7) onto Vi shows that Yi is an operator stable random vector
on Vi with some exponent Ei. We call this the spectral decomposition of Y with respect
to E. Since every eigenvalue of Ei has the same real part ai we say that Yi is spectrally
simple, with index i = 1/ai. Although Yi might not be multivariable stable, it has similar
tail behavior. For any small  > 0 we have
r-i<
P Yi > r < r-i+
for all r > 0 sufficiently large, so E( Yi
) exists for 0 <  < i and diverges for  > i .
If X  GDOA(Y) then Theorem 8.3.24 in Meerschaert and Scheffler (2001a) shows that
the limit Y and norming matrices An in (3.11) can be chosen so that every Vi in the spectral
decomposition of Rd with respect to the exponent E of Y is An-invariant for every n, and
V1,...,Vp are mutually perpendicular. Then the probability distribution of X is regularly
varying with exponent E and X has the same tail behavior as Y. In particular, for any small
 > 0 we have
r-()<
P |X  | > r < r-()+
for all r > 0 sufficiently large. In this case, we say that Y is spectrally compatible with X,
and we write X  GDOAc(Y).
Example 5.1. If Y is operator stable with exponent E = aI then (4.7) shows that Y is
multivariable stable with index  = 1/a. Then p = 1, P = I, and B = E. There is only
one spectral component, since the tail behavior is the same in every radial direction. If
asset price change vectors are IID with X = (X1,...,Xd)  GDOA(Y), then every asset
has the same tail behavior. If j measures the amount of the j-th asset in a portfolio, price
changes for this portfolio are IID with the random variable X   = X11 +  + Xdd.
Since the probability tails of X are uniform in every direction, the probability of a large
jump in price falls off like r- for any portfolio.
Example 5.2. If Y is operator stable with exponent E = diag(a1,...,ad) where a1 <
 < ad then p = d, P = I, B = E, Bi = ai and Vi is the i-th coordinate axis. The spectral
decomposition of Y = (Y1,...,Yd) with respect to E is Y = Y1 +  + Yd with
Yi = Yiei , the i-th marginal laid out along the i-th coordinate axis. Projecting (4.7) onto
the i-th coordinate axis shows that Yi is stable with index i = 1/ai, so that P(|Yi | > r) 
612 M.M. Meerschaert and H.-P. Scheffler
Cir-i . If  = 0 then P(|Y  | > r) falls off like r-() where () = min{i: i = 0}.
In other words, the heaviest tail dominates. If asset price change vectors are IID with
X  GDOAc(Y), then the assets are arranged in order of increasing tail thickness. If i
measures the amount of the i-th asset in a portfolio, the probability of a large jump in price
falls off like r-().
Example 5.3. If Y is operator stable with exponent E = diag(1,...,d) then Bi = aiI
for some ai 1/2 and di counts the number of diagonal entries j for which j = 1/i.
The matrix P sorts 1,...,d in increasing order, and the vectors pik are the coordinates
ej for which j = ai. The vectors Yi are multivariable stable with index i = 1/ai, so that
P( Y i > r)  Cir-i . For nonzero vectors   Vi we have
P |Y  | > r = P |Yi  | > r  C r-i
by the balanced tails condition for multivariable stable laws. For any other nonzero vector
, P(|Y  | > r)  C r-() where () = min{1/j : j = 0}. Again, the heaviest
tail dominates. If asset price change vectors are IID with X  GDOAc(Y), then X has
essentially the same tail behavior as Y, and P sorts the assets in order of increasing tail
thickness.
Example 5.4. Take B = diag(a1,...,ad) where a1 <  < ad and P orthogonal, so that
P-1 = P . If Y = (Y1,...,Yd) is operator stable with exponent E = PBP-1 then p = d,
Bi = ai and V1,...,Vd are the coordinate axes in the new coordinate system defined by
the vectors pi = Pei for i = 1,...,d. The spectral component Yi is the stable random
variable Y pi with index i = 1/ai, laid out along the Vi axis. Since Yj = Y ej is a linear
combination of stable laws of different indices, it is not stable. The change of coordinates
P rotates the coordinate axes to make the marginals stable. Since n-PBP-1
= Pn-B P-1
it follows from (4.7) that
Pn-B
P-1
(Y1 +  + Yn - bn)
d
= Y,
n-B
P-1
Y1 +  + P-1
Yn - P-1
bn
d
= P-1
Y
so that Y0 = P-1Y is operator stable with exponent B. Then the tail behavior of Y = PY0
follows from Example 5.2 and the change of coordinates. If we write  = 1p1 ++dpd
in these coordinates then P(|Y | > r)  C r-() where () = min{i: i = 0}. If asset
price change vectors are IID with X  GDOAc(Y), then the tail behavior of X is essentially
the same as Y. In particular, taking  = p1 gives a portfolio with the lightest probability
tails.
Example 5.5. Suppose that Y is operator stable with exponent E = PBP-1 where P is
orthogonal and B is given by (5.1), with di ×di blocks Bi = aiI for some 1/2 a1 <  <
ap. Let D0 = 0 and Di = d1 + + di for 1 i p. Then pik = Pej when j = Di-1 + k
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 613
for some k = 1,...,di and Vi = span{pik: k = 1,...,di}. To avoid double subscripts we
will also write qj = Pej , so that qj = pik when j = Di-1 +k for some k = 1,...,di. The
j-th column of the matrix P is the vector qj , and
Eqj = PBP-1
qj = PBej = Paiej = aiPej = aiqj
when qj  Vi, so that qj is a unit eigenvector of the matrix E with corresponding eigenvalue
ai. The spectral component
Yi =
di
k=1
(Y  pik)pik
is the orthogonal projection of Y onto the di-dimensional subspace Vi. The random vector
Yi is multivariable stable with index i = 1/ai, so that P( Yi > r)  Cir-i , and every
marginal Yik = Y  pik is stable with the same index i. The change of coordinates P
rotates the coordinate axes to find a set of orthogonal unit eigenvectors for E, so that the
marginals of Y in the new coordinate system are all stable random variables. The matrix
P also sorts the corresponding eigenvalues in increasing order. For any nonzero vector
  Rd ,
P |Y  | > r  C r-()
,
where () = i for the largest i such that the orthogonal projection of  onto the subspace
Vi is not equal to zero. If asset price change vectors are IID with X  GDOAc(Y), then the
tail behavior of X is essentially the same as Y . If  = 1e1 + ded so that i measures
the amount of the i-th asset in a portfolio, price changes for this portfolio are IID with
X   = X11 +  + Xdd. In particular, any   V1 gives a portfolio with the lightest
probability tails.
6. Sample covariance matrix
Given a data set of price changes (or log returns) X1,X2,...,Xn for a given asset, the k-th
sample moment
^k =
1
n
n
t=1
Xk
t
estimates the k-th moment k = E(Xk). These sample moments are used to estimate the
mean, variance, skewness and kurtosis of the data. If Xt are IID with P(|Xt | > r)  Cr-,
then Xk
t are also IID and heavy tailed with
P Xk
t > r = P |Xt| > r1/k
 Cr-/k
614 M.M. Meerschaert and H.-P. Scheffler
so the extended central limit theorem applies. Recall from Section 2 that k exists for
k <  and diverges for k . If  > 4 then Var(X2
t ) = 4 - 2
2 exists and the central limit
theorem (3.1) implies that
n1/2
( ^2 - 2) = n-1/2
n
t=1
X2
t - 2  Y, (6.1)
where Y is normal. When 2 <  < 4, the mean 2 = E(X2
t ) of these summands exists but
Var(X2
t ) is infinite, and the extended central limit theorem (3.4) implies that
n1-2/
( ^2 - 2) = n-2/
n
t=1
X2
t - 2  Y,
where Y is stable with index /2. When 0 <  < 2 the mean 2 = E(X2
t ) of the squared
price change diverges, and the extended central limit theorem implies that
n1-2/
^2 = n-2/
n
t=1
X2
t  Y,
where again Y is stable with index /2. In this case, the sample second moment ^2 exists
but the second moment 2 does not. When 0 <  < 2, or when 2 <  < 4 and 1 = 0, the
sample variance
^2
=
1
n
n
t=1
(Xt - ^1)2
= ^2 - ^2
1 (6.2)
is asymptotically equivalent to the sample second moment, see for example Anderson and
Meerschaert (1997). Since we can always center to zero expectation when 2 <  < 4,
both have the same asymptotics. If  > 4 the sample variance is asymptotically normal,
and when 0 <  < 4 the sample variance is asymptotically stable. Since the variance is a
measure of price volatility, the sample variance estimates volatility. Confidence intervals
for the variance are based on normal asymptotics when  > 4 and stable asymptotics when
2 <  < 4. When  < 2 the variance is undefined, but the sample variance still captures
some important features of the data, see Section 8.
Suppose that Xt = (X1(t),...,Xd(t)) where Xi(t) is the price change of the i-th asset
on day t. The covariance matrix characterizes dependence between price changes of different
assets over the same day, and the sample covariance matrix estimates the covariance
matrix. As before, it is simpler to begin with the uncentered estimate
Mn =
1
n
n
t=1
Xt Xt, (6.3)
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 615
where X denotes the transpose of the vector X = (X1,...,Xd) and hence
XX =


X1
...
Xd

(X1,...,Xd) =




X1X1  X1Xd
X2X1  X2Xd
...
...
...
XdX1  XdXd




is an element of the vector space Md
s of symmetric d × d matrices. The ij entry of Mn is
Mn(i,j) =
1
n
n
t=1
Xi(t)Xj (t)
which estimates E(XiXj ). If Xt are IID with X, then XtXt are IID random matrices and
we can apply the central limit theorems from Section 3 [see Section 10.2 in Meerschaert
and Scheffler (2001a) for complete proofs]. If the probability distribution of X is regularly
varying with exponent E and (4.5) holds with t{dx} = {t-E dx} for all t > 0, then the
distribution of XX is also regularly varying with
nP(AnXX An  B)  (B) as n   (6.4)
for Borel subsets B of Md
s that are bounded away from zero and whose boundary has
-measure zero. The exponent  of the limit measure {d(xx )} = {dx} is defined by
M = EM + ME for M  Md
s . Using the matrix norm
M =
d
i=1
d
j=1
M(i,j)2
1/2
we get
XX 2
=
d
i=1
d
j=1
(XiXj )2
=
d
i=1
X2
i
d
j=1
X2
j = X 4
so that XX = X 2. If every eigenvalue of E has real part ai < 1/4, then E( XX 2) =
E( X 4) <  and the multivariable central limit theorem (3.2) shows that
n1/2
(Mn - C) = n-1/2
n
t=1
(XtXt - C)  W, (6.5)
where W is a Gaussian random matrix and C is the (uncentered) covariance matrix
C = E(XX ). The estimates of Jansen and de Vries (1991) and Loretan and Phillips (1994)
616 M.M. Meerschaert and H.-P. Scheffler
indicate tail estimates in the range 2 <  < 4. In this case, every eigenvalue of E has real
part 1/4 < ai < 1/2. Then E( XX 2) = E( X 4) = , but E( XX ) = E( X 2) <
 so the covariance matrix C = E(XX ) exists. Now the generalized central limit theorem
(3.11) gives
nAn(Mn - C)An = An
n
t=1
(XtXt - C) An  W, (6.6)
where the limit W is a nonnormal operator stable random matrix. The estimates in Rachev
and Mittnik (2000) give tail estimates in the range 1 <  < 2, so that every eigenvalue of
E has real part ai > 1/2. Then E( XX ) = E( X 2) =  and the covariance matrix
C = E(XX ) diverges. In this case,
nAnMnAn  W (6.7)
holds with W operator stable. Since the covariance matrix is undefined, there is no reason
to believe that the sample covariance matrix contains useful information. However, we will
see in Section 8 that even in this case the sample covariance matrix characterizes the most
important distributional features of the random vector X.
The centered sample covariance matrix is defined by
n =
1
n
n
i=1
Xi - Xn Xi - Xn ,
where Xn = n-1(X1 +  + Xn) is the sample mean. In the heavy tailed case ai > 1/4,
Theorem 10.6.15 in Meerschaert and Scheffler (2001a) shows that n and Mn have the
same asymptotics, similar to the one dimensional case. In practice, it is common to meancenter
the data, so it does not matter which form we choose.
7. Dependent random vectors
Suppose that Xt = (X1(t),...,Xd(t)) where Xi(t) represents the price change (or log
return) of the i-th asset on day t. A model where Xt are IID with X  GDOA(Y) allows
dependence between the price changes Xi(t) and Xj (t) on the same day t, which
is commonly observed in practice. If we also want to model dependence between days,
we need to relax the IID assumption. A wide variety of time series models can be mathematically
reduced to a linear moving average. This reduction may involve integer or
fractional differencing, detrending and deseasoning, and nonlinear mappings. Asymptotics
for the underlying moving average are established in Section 10.6 of Meerschaert and
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 617
Scheffler (2001a). Assume that Z,Z1,Z2,Z3,... are IID random vectors on Rd whose
probability distribution is regularly varying with exponent E, so that
nP(AnZ  B)  (B) as n   (7.1)
for Borel subsets B of Rd \ {0} whose boundary have -measure zero, and t(dx) =
(t-E dx) for all t > 0. If every eigenvalue of E has real part ai > 1/2 then Z  GDOA(Y)
and
An(Z1 +  + Zn - nbn)  Y, (7.2)
where Y is operator stable with exponent E and Lévy measure . Define the moving
average process
Xt =

j=0
Cj Zt-j , (7.3)
where Cj are d × d real matrices. If every eigenvalue of E has real part ai < ap then the
moving average (7.3) is well defined as long as

j=0
Cj

<  (7.4)
for some  < 1/ap with  1. If every eigenvalue of E has real part ai < 1/2, then
E( Xt
2) exists and the asymptotics are normal, see Brockwell and Davis (1991). If every
eigenvalue of E has real part ai > 1/2, and if for each j either Cj = 0, or else C-1
j exists
and AnCj = Cj An for all n, then Theorem 10.6.2 in Meerschaert and Scheffler (2001a)
shows that
An X1 +  + Xn - n

j=0
Cj bn 

j=0
Cj Y. (7.5)
The limit in (7.5) is operator stable with no normal component and Lévy measure j Cj ,
where Cj  = 0 if Cj = 0 and otherwise Cj (dx) = (C-1
j dx).
If every eigenvalue of E has real part ai < 1/2, then both the mean m = E(Xt) and the
lag h covariance matrix
 (h) = E (Xt - m)(Xt+h - m)
exist. The matrix  (h) tells us when price changes on day t are correlated with price
changes (of the same asset or some other asset) h days later. These correlations are useful
618 M.M. Meerschaert and H.-P. Scheffler
to identify leading indicators, and they are the basic tools of time series modeling. The
sample covariance matrix at lag h 0 for the moving average Xt is defined by
n(h) =
1
n - h
n-h
t=1
Xt - X Xt+h - X , (7.6)
where X = (X1 +  + Xn)/n. If every eigenvalue of E has real part ai < 1/4, then
E( Xt
4) <  and n(h) is asymptotically normal, see Brockwell and Davis (1991). If
every eigenvalue of E has real part 1/4 < ai < 1/2, the estimates of Jansen and de Vries
(1991) and Loretan and Phillips (1994), then
An
n
t=1
(ZtZt - D) An  U (7.7)
as in Section 6, where U is a nonnormal operator stable random matrix and D = E(ZZ ).
Then Theorem 10.6.15 in Meerschaert and Scheffler (2001a) shows that
nAn n(h) -  (h) An 

j=0
Cj UCj+h (7.8)
for any h 0. The asymptotics (7.8) determine which elements of the sample covariance
matrix n(h) are statistically significantly different from zero.
If every eigenvalue of E has real part ai > 1/2, as in the estimates of Rachev and Mittnik
(2000), then
An
n
t=1
Zt Zt An  U (7.9)
and Theorem 10.6.15 in Meerschaert and Scheffler (2001a) shows that
nAnn(h)An 

j=0
Cj UCj+h (7.10)
for any h 0. In this case the covariance matrix  (h) does not exist, but the sample
covariance matrix n(h) still contains useful information about the time series Xt of price
changes. In the next section, we will explain this apparent paradox.
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 619
8. Tail estimation
Given a set of price changes (or log-returns) X1,...,Xn for some asset, it is important
to estimate the tail behavior. If the price changes Xt are identically distributed6 with X
and P(X > r)  Cr-, then the dispersion C and the tail index  determine the central
limit behavior, as well as the extreme value behavior, of the price change distribution.
Mandelbrot (1963) pioneered a graphical estimation method for C and . If y = P(X > r)
 Cr- then logy  logC -  logr. Ordering the data so that X(1) X(2)  X(n)
we should have approximately that r = X(i) when y = i/n. Then a plot of logX(i) versus
log(i/n) should be approximately linear with slope - and logC can be estimated from
the vertical axis intercept. If P(X > r)  Cr- for r large, then the upper tail should be
approximately linear. We call this a Mandelbrot plot. Several Mandelbrot plots for stock
market and exchange rate returns appear in Loretan and Phillips (1994) as evidence of
heavy tails with 2.5 <  < 3. Replacing X by -X gives information about the left tail.
Least squares estimators for  based on the Mandelbrot plot were proposed by Schultze
and Steinebach (1996), see also Csörg˝o and Viharos (1997).
The most popular numerical estimator for C and  is due to Hill (1975), see also Hall
(1982). Sort the data in decreasing order to obtain the order statistics X(1) X(2) 
X(n). Assuming that P(X > r) = Cr- for large values of r > 0, the maximum likelihood
estimates for  and C based on the m + 1 largest observations are
^ =
1
m
m
i=1
lnX(i) - lnX(m+1)
-1
,
(8.1)
C =
m
n
X^
(m+1),
where m is to be taken as large as possible, but small enough so that the tail condition
P(X > r) = Cr- remains valid. Replacing X by -X gives estimates for the left tail.
Replacing X by |X| gives estimates for the combined tail. This is often advantageous,
because it allows us to combine the data from both tails, and increase the number m of order
statistics used. Finding the best value of m is a challenge, and creates a certain amount of
controversy. Jansen and de Vries (1991) use Hilĺs estimator with a fixed value of m = 100
for several different assets. Loretan and P hillips (1994) tabulate several different values of
m for each asset. Hilĺs estimator ^ is consistent and asymptotically normal with variance
2/m, so confidence intervals are easy to construct. These intervals clearly demonstrate
that the tail parameters in Jansen and de Vries (1991) and Loretan and Phillips (1994) vary
depending on the asset.
Aban and Meerschaert (2001) develop a more general Hilĺs estimator to account for a
possible shift in the data. If P(X > r) = C(r - s)- for r large, the maximum likelihood
6 Note that we are not assuming IID here.
620 M.M. Meerschaert and H.-P. Scheffler
estimates for  and C based on the m + 1 largest observations are
^ =
1
m
m
i=1
ln(X(i) - ^s) - ln(X(m+1) - ^s)
-1
,
(8.2)
C =
m
n
(X(m+1) - ^s)^
,
where ^s is obtained by numerically solving the equation
^(X(m+1) - ^s)-1
= (^ + 1)
1
m
m
i=1
(X(i) - ^s)-1
(8.3)
over ^s < X(m+1). Once the optimal shift is computed, ^ and C come from Hilĺs estimator
applied to the shifted data. One practical implication is that, since the Pareto model is not
shift-invariant, it is a good idea to try shifting the data to get a linear Mandelbrot plot.
If Xt is the sum of many IID price shocks, then it can be argued that the distribution of
Xt must be (at least approximately) stable with distribution S(,,b). Maximum likelihood
estimation for the stable parameters is now practical, using the efficient method
of Nolan (1997) for computing stable densities, see also Mittnik et al. (1999), Mittnik,
Doganoglu and Chenyao (1999). Since the stable index 0 <  2, the stable MLE for 
cannot possibly agree with the estimates found in Jansen and de Vries (1991) and Loretan
and Phillips (1994). Rachev and Mittnik (2000) use a stable model for price changes, and
their estimates yield 1 <  < 2 for a variety of assets. McCulloch (1997) argues that the
 > 2 estimates found in Jansen and de Vries (1991) and Loretan and Phillips (1994) are
inflated due to a distributional misspecification. The Pareto tail of a stable random variable
X disappears as   2, so that it may be impossible to take m large enough for a reliable
estimate, see Fofack and Nolan (1999) for a more detailed discussion. The estimator in
Aban and Meerschaert (2001) corrects for the fact that Hilĺs ^ is not shift-invariant, and
may go some distance towards correcting the problem identified by McCulloch (1997).
Maximum likelihood estimation is quite sensitive to deviations from the proscribed distribution,
and it is no surprise that the MLE computations of Jansen and de Vries (1991)
and Loretan and Phillips (1994), based on the Pareto model, differ significantly from the
estimates of Rachev and Mittnik (2000), based on a stable model. Part of the controversy
stems from the fact that the range of  is limited to (0,2] for the stable model. Akgiray and
Booth (1988) interpret the results of Hilĺs estimator for stock returns as evidence against
the stable model. Actual finance data does not exactly fit either the stable or Pareto-tail
models, and in our opinion, parameter estimates are only valid with respect to the model
used to obtain them, so that Pareto-based estimates of  > 2 in no way invalidate the stable
model.
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 621
Meerschaert and Scheffler (1998) propose a robust estimator
^ =
2 lnn
lnn + ln ^2
(8.4)
based on the sample variance (6.2). This estimator can be applied whenever X  DOA(Y)
and Y is stable with index 0 <  < 2. Then X can be stable or Pareto, or any distribution
with balanced power-law tails. The estimator is also applicable to dependent data, since
it also applies when Xt = j cj Zt-j , Zt is IID with Z  DOA(Y), and Y is stable with
index 0 <  < 2. The estimator is based on the simple idea that
n1-2/
^2
 Y,
ln n^2
-
2

lnn  lnY,
2 lnn
ln(n^2)
2 lnn
-
1

 lnY
so that ln(n^2)/(2 lnn) estimates 1/. If X has heavy tails with  2 then ^  2. In this
case, we can apply the estimator to Xk, which also has heavy tails with tail parameter /k.
It is interesting, and even somewhat ironic, that the sample variance can be used to estimate
tail behavior, and hence tells us something about the spread of typical values, even in this
case 0 <  < 2 where the variance is undefined.
Portfolio modeling requires a vector model to incorporate dependence between price
changes for different assets. In these vector models, the sample variance is replaced by
the sample covariance matrix. For heavy tailed price changes with infinite variance, the
covariance matrix does not exist. Even so, we will see that the sample covariance matrix
is a very useful tool for portfolio modeling. Suppose that Xt = (X1(t),...,Xd(t)) where
Xi(t) is the price change of the i-th asset on day t. If Xt are identically distributed with X
and if X has heavy tails with P( X > r)  Cr- then the vector norms X1 ,..., Xn
can be used to estimate the tail parameter . Alternatively, we can apply one variable tail
estimators to the i-th marginal to get an estimate ^i of the tail parameter. If the probability
tails of X fall off at the same rate r- in every radial direction, then these estimates should
all be reasonably close. In that case, we might assume that X is multivariable stable with
distribution S(,M,b). The mean b can be estimated using the sample mean in the usual
case 1 <  < 2. Several estimators now exist for the scale  and the mixing measure M, or
equivalently, for the spectral measure (d) = M(d). Those estimators are surveyed
in another chapter in this Handbook (Kozubowski, Panorska and Rachev, 2003), so we
will not dwell on them here. If  > 2, one might consider the multivariable Pareto laws
introduced in Example 3.1. If P( X > r)  Cr- and the balanced tails condition (3.13)
holds for some mixing measure M, then the tail behavior of X is multivariable Pareto.
Multivariable stable random vectors have this property with 0 <  < 2. If  > 2 then
multivariable Pareto could offer a reasonable alternative, which to our knowledge has not
been pursued in the finance literature.
622 M.M. Meerschaert and H.-P. Scheffler
While experts disagree on the range of  for typical assets, there seems to be general
agreement that the tail index depends on the asset. Then it is appropriate to assume that the
probability distribution of X varies regularly with some exponent E. For IID random vectors,
a method for estimating the exponent E can be found in Section 10.4 of Meerschaert
and Scheffler (2001a). In Section 9 we show that the same methods also apply to dependent
random vectors which are identically distributed. The method is applicable when the
eigenvalues of E all have real part ai > 1/2, the infinite variance case. To be concrete, we
adopt the model of Example 5.5, which is the simplest model flexible enough for realism.
This model assumes that E has a set of d mutually orthogonal unit eigenvectors. Note
that if the eigenvalues of E are all distinct then these unit eigenvectors are unique up to a
factor of 1. On the other hand, if E = aI for some a > 1/2 then any set of d mutually
orthogonal unit vectors can be used.
Recall the spectral decomposition E = PBP-1 from Example 5.5, where P is orthogonal
and B is given by (5.1), with di × di blocks Bi = aiI for some 1/2 a1 <  < ap.
Let D0 = 0 and Di = d1 +  + di for 1 i p. Then qj = Pej is a unit eigenvector
of the matrix E and the di dimensional subspace Vi = span{qj : Di-1 < j Di} contains
every eigenvector of E with associated eigenvalue ai. Our estimator for E is based on the
sample covariance matrix Mn defined in (6.3). Since Mn is symmetric and nonnegative
definite, there exists an orthonormal basis of eigenvectors for Mn with nonnegative eigenvalues.
Eigenvalues and eigenvectors of Mn are easily computed using standard numerical
routines, see for example Press et al. (1987). Sort the eigenvalues
1  d
and the associated unit eigenvectors
1,...,d
so that Mnj = j j for each j = 1,...,d. Now Theorem 10.4.5 in Meerschaert and
Scheffler (2001a) shows that
logn + logj
2 logn
 ai as n  
in probability for any Di-1 < j Di. This is a multivariable analogue for the one variable
tail estimator (8.4). Furthermore, Theorem 10.4.8 in Meerschaert and Scheffler (2001a)
shows that the eigenvectors j converge in probability to V1 when j D1, and to Vp when
j > Dp-1. This shows that the eigenvectors estimate the coordinate vectors in the spectral
decomposition, at least for the lightest and heaviest tails.
Now we illustrate the practical application of the multivariable tail estimator. Recall
that Xt = (X1(t),...,Xd(t)) where Xi(t) is the price change of the i-th asset on day t.
Compute the (uncentered) sample covariance matrix Mn using the formula (6.3) and then
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 623
compute the eigenvalues 1  d and the associated eigenvectors
1 = 1(1),...,d(1)
... (8.5)
d = 1(d),...,d(d)
of the matrix Mn. A change of coordinates is essential to the method. Write
Zj (t) = Xt  j = X1(t)1(j) +  + Xd(t)d(j)
for each j = 1,...,d. Our portfolio model is based on these new coordinates. Let
^j =
2 logn
logn + logj
for each j = 1,...,n. Since the eigenvalues are sorted in increasing order we will have
^1  ^d . Our model assumes that Zj (t) are identically distributed with Zj , and the
tail parameter ^j governs the j-th coordinate Zj . If ^j < 2 then P(|Zj | > r) falls off like
r-^j and if ^j 2 then a finite variance model for Zj is adequate. We can also use any
other one variable tail estimator to get j for each of the new coordinates Zj (t). The new
coordinates unmask variations in  that would go undetected in the original coordinates.
Example 8.1. We look at a data set of n = 2853 daily exchange rate log-returns X1(t)
for the German Deutsch Mark and X2(t) for the Japanese Yen, both taken against the US
Dollar. We divide each entry by 0.004 which is the approximate median for both |X1(t)|
and |X2(t)|. This has no effect on the eigenvectors but helps to obtain good estimates of
the tail thickness. Then we compute
Mn =
1
n
n
t=1
X1(t)2 X1(t)X2(t)
X1(t)X2(t) X2(t)2 =
3.204 2.100
2.100 3.011
which has eigenvalues 1 = 1.006, 2 = 5.209 and associated unit eigenvectors 1 =
(0.69,-0.72) , 2 = (0.72,0.69) . Next we compute
^1 =
2 ln2853
ln 2853 + ln1.006
= 1.998,
(8.6)
^2 =
2 ln2853
ln 2853 + ln5.209
= 1.656
indicating that Z1(t) = 0.69X1(t) - 0.72X2(t) fits a finite variance model but Z2(t) =
0.72X1(t) + 0.69X2(t) fits a heavy tailed model with  = 1.656. Then we can model
624 M.M. Meerschaert and H.-P. Scheffler
Fig. 3. Exchange rates against the US dollar. The new coordinates uncover variations in the tail parameter .
Zt = (Z1(t),Z2(t)) as being identically distributed with the random vector Z = (Z1,Z2)
where P(|Z2| > r)  C1r-1.656 and Var(Z1) < . The simplest model with these properties
is to take Z1(t) normal and Z2(t) stable with index  = 1.656 and independent of
Z1(t).
Next we explain the operator stable model based on these estimates. The random vectors
Zt are operator stable with exponent
B =
0.50 0
0 0.60
since 0.50 = 1/1.998 and 0.60 = 1/1.656. The change of coordinates matrix
P =
0.69 -0.72
0.72 0.69
so that
Zt =
Z1(t)
Z2(t)
=
0.69 -0.72
0.72 0.69
X1(t)
X2(t)
= PXt .
Since
P-1
=
0.69 0.72
-0.72 0.69
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 625
(rounded off to two decimal places) we also have
Xt = P-1
Zt =
X1(t)
X2(t)
=
0.69 0.72
-0.72 0.69
Z1(t)
Z2(t)
so that
X1(t) = 0.69Z1(t) + 0.72Z2(t),
(8.7)
X2(t) = -0.72Z1(t) + 0.69Z2(t).
Both exchange rates have a common heavy-tailed stable factor Z2(t) and so both exchange
rates have heavy tails with the same tail index  = 1.656. It is tempting to interpret Z2(t)
as the common influence of fluctuations in the US dollar, and the remaining light-tailed
factor Z1(t) as the accumulation of other price shocks independent of the US dollar.
We also take the opportunity to fill in the details of Example 5.4 in this simple case. The
original data Xt = P-1Zt is modeled as operator stable with exponent
E = PBP-1
=
0.55 0.05
0.05 0.55
.
In this case, Z1(t) and Z2(t) are independent so the density of Zt is the product of the
two marginal densities, and then the density of Xt can be obtained by a simple change of
variables. The columns of the change of variables matrix P are the eigenvectors j of the
sample covariance matrix, which estimate the theoretical coordinate system vectors pj in
the spectral decomposition.
Remark 8.2. This exchange rate data in Example 8.1 was also analyzed by Nolan,
Panorska and McCulloch (2001) using a multivariable stable model. Since both marginals
X1(t) and X2(t) have heavy tails with the same , there is no obvious reason to employ
a more complicated model. However, the change of coordinates in Example 8.1 uncovers
variations in the tail parameter , an important modeling insight.
Remark 8.3. Kotz, Kozubowski and Podgórski (2001) employ a very different model for
the data in Example 8.1, based on the Laplace distribution. This distribution, and its multivariable
analogues, assume exponential probability tails for the data. These models have
heavier tails than the Gaussian, but they have moments of all orders.
Remark 8.4. The simplistic model in Example 8.1 assumes that the two factors Z1 and
Z2 are independent. If we assume that Z is operator stable with Z1 normal and Z2 stable
then these components must be independent, in view of the general characteristic function
formula for operator stable laws. Another alternative is to assume that Z1 is stable with
index  = 1.998, very close to a normal distribution. In this case, the two components can
626 M.M. Meerschaert and H.-P. Scheffler
be dependent. The dependence is captured by the mixing measure or spectral measure, see
Example 4.1. Scheffler (1999) provides a method for estimating the spectral measure from
data for an operator stable random vector with a known exponent. This provides a more
flexible model including dependence between the two factors.
9. Tail estimator proof for dependent random vectors
In this section, we provide a proof that the multivariable tail estimator of Section 8 is
still valid for certain sequences of dependent heavy tailed random vectors. We say that a
sequence (Bn) of invertible linear operators is regularly varying with index -E if for any
 > 0 we have
B[n]B-1
n  -E
as n  .
For further information about regular variation of linear operators see Meerschaert and
Scheffler (2001a, Chapter 4).
In view of Theorem 2.1.14 of Meerschaert and Scheffler (2001a) we can write Rd =
V1    Vp and E = E1    Ep for some 1 p d where each Vi is E invariant,
Ei :Vi  Vi and Re() = ai for all real parts of the eigenvalues of Ei and some a1 <  <
ap. By Definition 2.1.15 of Meerschaert and Scheffler (2001a) this is called the spectral
decomposition of Rd with respect to E. By Definition 4.3.13 of Meerschaert and Scheffler
(2001a) we say that (Bn) is spectrally compatible with -E if every Vi is Bn-invariant for
all n. Note that in this case we can write Bn = B1n    Bpn and each Bin :Vi  Vi
is regularly varying with index -Ei. [See Proposition 4.3.14 of Meerschaert and Scheffler
(2001a).] For the proofs in this section we will always assume that the subspaces Vi in
the spectral decomposition of Rd with respect to E are mutually orthogonal. We will also
assume that (Bn) is spectrally compatible with -E. Let i denote the orthogonalprojection
operator onto Vi. If we let Pi = i + + p and Li = Vi   Vp then Pi :Rd  Li is
a orthogonal projection. Furthermore, Pi = 1 +  + i is the orthogonal projection onto
Li = V1    Vi.
Now assume 0 < a1 <  < ap. Since (Bn) is spectrally compatible with -E, Proposition
4.3.14 of Meerschaert and Scheffler (2001a) shows that the conclusions of Theorem
4.3.1 of Meerschaert and Scheffler (2001a) hold with Li = Vi    Vp for each
i = 1,...,p. Then for any  > 0 and any x  Li \ Li+1 we have
n-ai Bnx
n-ai+
(9.1)
for all large n. Then
log Bnx
logn
 -ai as n   (9.2)
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 627
and since this convergence is uniform on compact subsets of Li \ Li+1 we also have
log iBn
logn
 -ai as n  . (9.3)
It follows that
log Bn
logn
 -a1 as n  . (9.4)
Since (Bn)-1 is regularly varying with index E , a similar argument shows that for any
x  Li \ Li-1 we have
nai-
(Bn)-1
x nai+
(9.5)
for all large n. Then
log (Bn)-1x
logn
 ai as n   (9.6)
and since this convergence is uniform on compact subsets of Li \ Li-1 we also have
log i(Bn)-1
logn
 ai as n  . (9.7)
Hence
log (Bn)-1
logn
 ap as n  . (9.8)
Suppose that Xt , t = 1,2,..., are Rd-valued random vectors and let Mn be the sample
covariance matrix of (Xt ) defined by (6.3). Note that Mn is symmetric and positive semidefinite.
Let 0 1n  dn denote the eigenvalues of Mn and let 1n,...,dn be the
corresponding orthonormal basis of eigenvectors.
Basic Assumptions. Assume that for some exponent E with real spectrum 1/2 < a1 <
 < ap the subspaces Vi in the spectral decomposition of Rd with respect to E are mutually
orthogonal, and there exists a sequence (Bn) regularly varying with index -E and
spectrally compatible with -E such that:
(A1) The set {n(BnMnBn): n 1} is weakly relatively compact.
(A2) For any limit point M of this set we have:
(a) M is almost surely positive definite.
(b) For all unit vectors  the random variable  M has no atom at zero.
628 M.M. Meerschaert and H.-P. Scheffler
Now let Rd = V1    Vp be the spectral decomposition of Rd with respect to E. Put
di = dimVi and for i = 1,...,p let bi = di +  + dp and bi = d1 +  + di. Our goal is
now to estimate the real spectrum a1 <  < ap of E as well as the spectral decomposition
V1,...,Vp. In various situation, these quantities completely describe the moment behavior
of the Xt .
Theorem 9.1. Under our basic assumptions, for i = 1,...,p and bi-1 < j bi we have
log(njn)
2 logn
 ai in probability as n  .
The proof of Theorem 9.1 is in parts quite similar to the Theorem 2 in Meerschaert
and Scheffler (1999b). See also Section 10.4 in Meerschaert and Scheffler (2001a), and
Scheffler (1998). We include it here for sake of completeness.
Proposition 9.2. Under our basic assumptions we have
log(ndn)
2 logn
 ap in probability.
Proof: For  > 0 arbitrary we have
P
log(ndn)
2 logn
- ap >  P dn > n2(ap+)-1
+ P dn < n2(ap-)-1
.
Now choose 0 <  <  and note that by (9.8) we have (Bn)-1 nap+ for all large n.
Using assumption (A1) we obtain for all large n
P dn > n2(ap+)-1
= P Mn > n2(ap+)-1
P (Bn)-1 2
nBnMnBn > n2(ap+)
P nBnMnBn > n2(-)
and the last probability tends to zero as n  .
Now fix any 0  Lp \ Lp-1 and write (Bn)-10 = rnn for some unit vector n and
rn > 0. Theorem 4.3.14 of Meerschaert and Scheffler (2001a) shows that every limit point
of (n) lies in the unit sphere in Vp. Then since (9.5) holds uniformly on compact sets we
have for any 0 <  <  that nap- rn nap+ for all large n. Then for all large n we get
P dn < n2(ap-)-1
= P max
 =1
Mn   < n2(ap-)-1
P Mn0  0 < n2(ap-)-1
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 629
= P nBnMnBnn  n < r-2
n n2(ap-)-1
P nBnMnBnn  n < n2(-)
.
Given any subsequence (n ) there exists a further subsequence (n )  (n ) along which
n  . Furthermore, by assumption (A1) there exists another subsequence (n )  (n )
such that nBnMnBn  M along (n ). Hence by continuous mapping [see Theorem 1.2.8
in Meerschaert and Scheffler (2001a)] we have
nBnMnBnn  n  M   along (n ).
Now, given any 1 > 0 by assumption (A2)(b) there exists a  > 0 such that P{M   < }
< 1/2. Hence for all large n = n we have
P nBnMnBnn  n < n2(-)
P{nBnMnBnn  n < }
P{M   < } +
1
2
< 1.
Since for any subsequence there exists a further subsequence along which
P nBnMnBnnn < n2(-)
 0,
this convergence holds along the entire sequence which concludes the proof.
Proposition 9.3. Under the basic assumptions we have
log(n1n)
2 logn
 a1 in probability.
Proof: Since the set GL(Rd) of invertible matrices is an open subset of the vector space
of d × d real matrices, it follows from (A1) and (A2)(a) together with the Portmanteau
Theorem [cf., Theorem 1.2.2 in Meerschaert and Scheffler (2001a)] that limn P{Mn 
GL(Rd)} = 1 holds. Hence we can assume without loss of generality that Mn is invertible
for all large n.
Given any  > 0 write
P
log(n1n)
2 logn
- a1 >  P 1n > n2(a1+)-1
+ P 1n < n2(a1-)-1
.
To estimate the first probability on the right-hand side of the inequality above choose a
unit vector 0  L1 and write (Bn)-10 = rnn as above. Then, since (9.5) holds uniformly
630 M.M. Meerschaert and H.-P. Scheffler
on the unit sphere in L1 = V1, for 0 <  <  we have na1- rn na1+ for all large n.
Therefore for all large n
P 1n > n2(a1+)-1
P min
 =1
Mn   > n2(a1+)-1
P Mn0  0 > n2(a1+)-1
P nBnMnBnn  n > n2(-)
.
It follows from assumption (A1) together with the compactness of the unit sphere in Rd
and continuous mapping that the sequence (nBnMnBnn  n) is weakly relatively compact
and hence by Prohorov's Theorem this sequence is uniformly tight. Since  >  it follows
that P{1n > n2(a1+)-1}  0 as n  .
Since the smallest eigenvalue of Mn is the reciprocal of the largest eigenvalue of M-1
n
we have
P 1n < n2(a1-)-1
= P
1
1n
> n2(-a1)+1
= P max
 =1
M-1
n    > n2(-a1)+1
= P M-1
n > n2(-a1)+1
P
1
n
(Bn)-1
M-1
n B-1
n > Bn
-2
n2(-a1)
.
It follows from (9.4) that for any 0 <  <  there exists a constant C > 0 such that Bn
Cn-a1+ for all n and hence for some constant K > 0 we get Bn
-2 Kn2(a1-) for
all n. Note that by assumptions (A1) and (A2)(a) together with continuous mapping the
sequence
1
n
(Bn)-1
M-1
n B-1
n
is weakly relatively compact and hence by Prohorov's theorem this sequence is uniformly
tight. Hence
P
1
n
(Bn)-1
M-1
n B-1
n > Bn
-2
n2(-a1)
P
1
n
(Bn)-1
M-1
n B-1
n > Kn2(-)
 0
as n  . This concludes the proof.
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 631
Proof of Theorem 9.1: Let Cj denote the collection of all orthogonal projections onto
subspaces of Rd with dimension j. The Courant­Fischer Max­Min Theorem [see, Rao
(1965, p. 51)] implies that
jn = min
PCj
max
 =1
PMnP  
= max
PCd-j+1
min
 =1
PMnP  . (9.9)
Note that P2
i = Pi and that Bn and Pi commute for all n,i. Furthermore (PiBn) is regularly
varying with index Ei    Ep. Since
n(PiBn)PiMnPi(BnPi) = nPi(BnMnBn)Pi
it follows by projection from our basic assumptions that the sample covariance matrix
formed from the Li valued random variables PiXt satisfies again those basic assumptions
with E = Ei    Ep on Li. Hence if n denotes the smallest eigenvalue of the matrix
PiMnPi it follows from Proposition 9.3 that
log(nn)
2 logn
 ai in probability.
Similarly, the sample covariance matrix formed in terms of the Li-valued random vectors
LiXt again satisfies the basic assumptions with E = E1    Ei as above. Then,
if n denotes the largest eigenvalue of the matrix PiMnPi it follows from Proposition 9.2
above that
log(nn)
2 logn
 ai in probability.
Now apply (9.9) to see that
n jn n
whenever bi-1 < j bi. The result now follows easily.
After dealing with the asymptotics of the eigenvalues of the sample covariance in Theorem
9.1 above we now investigate the convergence of the unit eigenvectors of Mn. Recall
that i :Rd  Vi denotes the orthogonal projection onto Vi for i = 1,...,p. Define the
random projection
in(x) =
bi
j=bi-1+1
(x  jn)jn.
632 M.M. Meerschaert and H.-P. Scheffler
Theorem 9.4. Under the basic assumptions we have 1n  1 and pn  p in probability
as n  .
Again the proof is quite similar to the proof of Theorem 3 in Meerschaert and Scheffler
(1999b) and Theorem 10.4.8 in Meerschaert and Scheffler (2001a). See also Scheffler
(1998). We include here a sketch of the arguments.
Proposition 9.5. Under our basic assumptions we have: If j > bp-1 and r < p then
rjn  0 in probability.
Proof: Since rjn = (rMn/jn)jn we get
rjn
rMn
jn
rB-1
n nBnMnBn (Bn)-1
njn
.
By assumption (A1) together with continuous mapping it follows from Prohorov's theorem
that (n BnMnBn ) is uniformly tight. Also, by (9.7), (9.8) and Theorem 9.1 we get
log( rB-1
n nBnMnBn (Bn)-1 )/(njn)
logn
=
log rB-1
n
logn
+
log (Bn)-1
logn
-
log(njn)
logn
 ar + ap - 2ap < 0 in probability.
Hence the assertion follows.
Proposition 9.6. Under our basic assumptions we have: If j b1 and r > 1 then
rjn  0 in probability.
Proof: Since rjn = (rM-1
n jn)jn we get
rjn rM-1
n jn rBn
1
n
(Bn)-1
M-1
n B-1
n Bn (njn).
As in the proof of Proposition 9.3 the sequence (1
n (Bn)-1M-1
n B-1
n ) is uniformly tight
and now the assertion follows as in the proof of Proposition 9.5.
Proof of Theorem 9.4: The proof is almost identical to the proof of Theorem 3 in Meerschaert
and Scheffler (1999b) or Theorem 10.4.8 in Meerschaert and Scheffler (2001a) and
therefore omitted.
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 633
Corollary 9.7. Under our basic assumptions, if p 3 then in  i in probability for
i = 1,...,p.
Proof: Obvious.
Example 9.8. Suppose that Z,Z1,Z2,... is a sequence of independent and identically
distributed (IID) random vectors with common distribution . We assume that  is regularly
varying with exponent E. That means that there exists a regularly varying sequence
(An) of linear operators with index -E such that
n(An)   as n  . (9.10)
For more information on regularly varying measures see Meerschaert and Scheffler
(2001a, Chapter 6).
Regularly varying measures are closely related to the generalized central limit theorem
discussed in Section 3. Recall that if
An(Z1 +  + Zn - nbn)  Y as n   (9.11)
for some nonrandom bn  Rd, we say that Z belongs to the generalized domain of attraction
of Y and we write Z  GDOA(Y). Corollary 8.2.12 in Meerschaert and Scheffler
(2001a) shows that Z  GDOA(Y) and (9.11) holds if and only if  varies regularly
with exponent E and (9.10) holds, where the real parts of the eigenvalues of E are
greater than 1/2. In this case, Y has an operator stable distribution and the measure 
in (9.10) is the Lévy measure of the distribution of Y. Operator stable distributions and
Lévy measures were discussed in Section 4, where (9.10) is written in the equivalent form
nP(AnZ  dx)  (dx). The spectral decomposition was discussed in Section 5. Theorem
8.3.24 in Meerschaert and Scheffler (2001a) shows that we can always choose norming
operators An and limit Y in (9.11) so that Y is spectrally compatible with Z, meaning
that An varies regularly with some exponent -E, the subspaces Vi in the spectral decomposition
of Rd with respect to E are mutually orthogonal, and these subspaces are also
An-invariant for every n. In this case, we write Z  GDOAc(Y).
Recall from Section 6 that, since the real parts of the eigenvalues of E are greater
than 1/2,
nAnMnAn  W as n  , (9.12)
where Mn is the uncentered sample covariance matrix
Mn =
1
n
n
i=1
ZiZi
and W is a random d × d matrix whose distribution is operator stable. Theorem 10.2.9
in Meerschaert and Scheffler (2001a) shows that W is invertible with probability one, and
634 M.M. Meerschaert and H.-P. Scheffler
Theorem 10.4.2 in Meerschaert and Scheffler (2001a) shows that for all unit vectors   Rd
the random variable   W has a Lebesgue density. Then the basic assumptions of this
section hold, and hence the results of this section apply.
The tail estimator proven in this section approximates the spectral index function (x)
defined in (5.2). This index function provides sharp bounds on the tails and radial projection
moments of Z. Given a d-dimensional data set Z1,...,Zn with uncentered covariance
matrix Mn, let 0 1n  dn denote the eigenvalues of Mn and 1n,...,dn the corresponding
orthonormal basis of eigenvectors. Writing xj = x  j we can estimate the
spectral index (x) by
^(x) = min{^j : xj = 0}, where ^j =
2 logn
log(njn)
using the results of this section. Hence the eigenvalues are used to approximate the tail
behavior, and the eigenvectors determine the coordinate system to which these estimates
pertain. A practical application of this tail estimator appears in Example 8.1.
Example 9.9. The same tail estimation methods used in the previous example also apply
to the moving averages considered in Section 7. This result is apparently new. Given a
sequence of IID random vectors Z,Zj whose common distribution  varies regularly with
exponent E, so that (9.10) holds, we define the moving average process
Xt =

j=Cj
Zt-j , (9.13)
where we assume that the d × d matrices Cj fulfill for each j either Cj = 0 or Cj is
invertible and AnCj = Cj An for all n. Moreover if ap denotes the largest real part of the
eigenvalues of E we assume further

j=-
Cj

<  (9.14)
for some  < 1/ap with  1. Recall from Section 7 that under those conditions Xt is
almost surely well defined, and that if the real parts of the eigenvalues of E are greater
than 1/2 we have that
nAnn(0)An  M =

j=Cj
WCj as n  , (9.15)
where the sample covariance matrix n(h) is defined by (7.6) and W is a random
d × d matrix whose probability distribution is operator stable. Suppose that the norming
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 635
operators An are chosen so that (9.11) holds and Z  GDOAc(Y). Then in view of our
basic assumptions (A1) and (A2) it remains to show:
Lemma 9.10. Under the assumptions of the paragraph above the limiting matrix M in
(9.15) is a.s. positive definite and for any unit vector  the random variable M   has no
atom at zero.
Proof: Since W in (9.12) is a.s. positive definite we have for any  = 0 that Cj WCj    =
WCj   Cj  0 for all j and strictly greater that zero for those j with Cj = 0. Hence
M   =

j=Cj
WCj    > 0
for any  = 0 so M is positive definite.
Moreover if for a given unit vector  we set zj = Cj  then zj0 = 0 for at least one j0.
Since W is almost surely positive definite we have
P{M   < t} = P

j=Wzj
 zj < t P{Wzj0  zj0 < t}  0
as t  0 using the fact that Wzj0  zj0 has a Lebesgue density as above. Hence M   has
no atom at zero.
It follows from (9.15) together with Lemma 9.10 that the Xt defined above fulfill the
basic assumptions of this section. Hence it follows from Theorems 9.1 and 9.4 that the tail
estimator used in Example 9.8 also applies to time-dependent data that can be modeled
as a multivariate moving average. We can also utilize the uncentered sample covariance
matrix (6.3), which has the same asymptotics as long as EZ = 0 [cf. Theorem 10.6.7 and
Corollary 10.2.6 in Meerschaert and Scheffler (2001a)]. In either case, the eigenvalues can
be used to approximate the tail behavior, and the eigenvectors determine the coordinate
system in which these estimates apply.
Example 9.11. Suppose now that Z1,Z2,... are IID Rd-valued random vectors with common
distribution . We assume that  is ROV(E,c), meaning that there exist (An) regularly
varying with index -E, a sequence (kn) of natural numbers tending to infinity with
kn+1/kn  c > 1 such that
kn(Akn )   as n  . (9.16)
See Meerschaert and Scheffler (2001a, Section 6.2) for more information on R­O varying
measures.
636 M.M. Meerschaert and H.-P. Scheffler
R­O varying measures are closely related to a generalized central limit theorem. In fact,
if  is ROV(E,c) and the real parts of the eigenvalues of E are greater than 1/2 then
(9.16) is equivalent to
Akn (Z1 +  + Zkn - knbn)  Y as n  ,
where Y has a so called (cE,c) operator semistable distribution. See Meerschaert and
Scheffler (2001a), Sections 7.1 and 8.2 for details. Once again, a judicious choice of norming
operators and limits guarantees that Y is spectrally compatible with Z, so that An varies
regularly with some exponent -E, the subspaces Vi in the spectral decomposition of Rd
with respect to E are mutually orthogonal, and these subspaces are also An-invariant for
every n. It follows from Theorem 8.2.5 of Meerschaert and Scheffler (2001a) that Z has
the same moment and tail behavior as for the generalized domain of attraction case considered
in Section 5. In particular, there is a spectral index function (x) taking values in the
set {a-1
1 ,...,a-1
p } where a1 <  < ap are the real parts of the eigenvalues of E. Given
x = 0, for any small  > 0 we have
r-(x)<
P |Z  x| > r < r-(x)+
for all r > 0 sufficiently large. Then E(|Z  x|) exists for 0 <  < (x) and diverges for
 > (x).
Now let
Mn =
1
n
n
i=1
ZiZi
denote the sample covariance matrix of (Zi). Then it follows from Theorem 10.2.3, Corollaries
10.2.4 and 10.2.6, Theorem 10.2.9, and Lemma 10.4.2 in Meerschaert and Scheffler
(2001a) that Mn fulfills the basic assumptions (A1) and (A2) of this section. Hence,
by Theorems 9.1 and 9.4 we rediscover Theorems 10.4.5 and 10.4.8 of Meerschaert and
Scheffler (2001a). See also Scheffler (1998). In other words, the approximation ^(x) from
Example 9.8 still functions in this more general case, which represents the most general
setting in which sums of IID random vectors can approximated in distribution via a central
limit theorem.
10. Conclusions
If one believes that asset price changes (or log-returns) have heavy tails, then there is
ample reason to seek a model where the tail thickness parameter  varies with the asset.
Operator stable random vectors provide such a model, and are justified by a central
limit theorem. Matrix-scaled sums of independent, identically distributed random vectors
can only converge (in a distributional sense) to an operator stable limit. Such ran-
Ch. 15: Portfolio Modeling with Heavy Tailed Random Vectors 637
dom vectors have regularly varying probability distributions whose tails are governed by
a matrix exponent. Time dependent models can be constructed by taking moving averages
of these random vectors. If Xi is the price change in the i-th asset then the vector of price
changes X = (X1,...,Xd) can be described by such models. If i measures the amount
of the i-th asset in a portfolio, price changes for this portfolio are of the form X   =
X11 +  + Xdd. The probability of large jumps in price depends on the mix according
to a tail index function (). If 2 < () < 4 we have a finite variance model with infinite
fourth moments. Then the sample covariance matrix plays the usual role as a descriptor of
dependence between assets, but its asymptotics are operator stable. If () < 2 indicating
heavy tails with infinite variance, the sample covariance matrix still provides some useful
information.In particular, the coordinate system that diagonalizes this matrix also identifies
the portfolios with the best or worst tail behavior.
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Chapter 16
LONG RANGE DEPENDENCE IN HEAVY TAILED STOCHASTIC
PROCESSES
BORJANA RACHEVA-IOTOVA
Faculty of Economics and Business Administration, University of Sofia, Bulgaria
Bravo Risk Management Group, 1101-C Eugenia Place, Carpinteria, Santa Barbara, CA 93013, USA
e-mail: borjana.racheva@bravo-group.com
GENNADY SAMORODNITSKY
School of Operations Research and Industrial Engineering, and Department of Statistical Science,
Cornell University, Ithaca, NY 14853, USA
e-mail: gennady@orie.cornell.edu
Contents
Abstract 642
Keywords 642
1. Introduction 643
2. What is long range dependence? 644
3. Tails and rare events 646
4. Some classes of heavy tailed processes 649
4.1. Linear processes 649
4.2. Infinitely divisible processes 650
5. Rare events, associated functionals and long range dependence 652
5.1. Unusual sample mean and long strange segments for heavy tailed linear processes 653
5.2. Ruin probability for heavy tailed linear processes 656
5.3. Rare events for stationary stable processes 657
5.4. High dimensional joint tails for a linear process with stable innovations 659
References 661
 G. Samorodnitsky's research was partially supported by NSF grants DMS-0071073 and DMI-9713549 at
Cornell University.
Handbook of Heavy Tailed Distributions in Finance, Edited by S.T. Rachev
 2003 Elsevier Science B.V. All rights reserved
642 B. Racheva-Iotova and G. Samorodnitsky
Abstract
The notion of long range dependence has traditionally been defined through a slow decay
of correlations. This approach may be completely inappropriate in the case of a stochastic
process with heavy tails. Yet long memory has been reported to be found in various fields
where heavy tails are a standard feature of the commonly used stochastic models. Financial
and communications networks data are among those often believed to exhibit long memory.
We discuss alternative points of view on long range dependence that are applicable in the
heavy tailed case. Such alternative approaches may be tailored for a particular applications
at hand.
Keywords
heavy tails, long range dependence, rare events, large deviations
Ch. 16: Long Range Dependence and Heavy Tails 643
1. Introduction
A glance at the plot on Figure 1 describing the annual minima of the water level in the Nile
river suggests that the process plotted there has at least 4 distinct time periods when the
"level" and "drift" of the process change. This needs not, however, be necessarily taken
as an indication that a nonstationary model should be used for the annual minima of water
level process. Even though using a nonstationary model is possible in such situations,
sometimes a more parsimonious model is a long memory stationary model (or a stationary
model with long range dependence). In fact, commonly used long memory models exhibit
what Benoit Mandelbrot termed "persistence", or "Joseph effect" (referring to the long
stretches of plenty and famine in Egypt of the Bible). Here is what one sees when looking
at the increments of a long memory Fractional Brownian motion (also called Fractional
Gaussian noise): "Nearly every sample looks like a "random noise" superimposed upon a
background that performs several cycles. However, there cycles are not periodic, that is,
cannot be extrapolated as the sample lengthens. In addition, one often sees an underlying
trend that need not continue in the extrapolate." (Mandelbrot, 1983, p. 251.)
The Nile river data set is a famous one; arguably, it is the data set that forced us to think
about long range dependence in the first place. It was, of course, the same Mandelbrot who
with co-workers (Mandelbrot, 1965; Mandelbrot and Van Ness, 1968; Mandelbrot and
Wallis 1968, 1969) first realized that a long memory stationary Gaussian process may explain
the behaviour a particular statistic (the so-called R/S statistic) suggested and applied
to the Nile river data by Hurst (1951, 1955).
Today long memory models are still being used in hydrology and related areas. However,
new applications have arisen, significantly in finance and communication networks.
Often observations from these latter areas feature heavy tails, and such data sets sometimes
provide extreme illustrations to the Mandelbrot remark on "spurious cycles". For example,
Figure 2 describing the load offered by a network server, suggests that the process plotted
there has at least 10 distinct time periods when the "nature" of the process changes. Once
Fig. 1. Annual minima of the water level in the Nile river for the years 622 to 1281, measured at the Roda gauge
near Cairo.
644 B. Racheva-Iotova and G. Samorodnitsky
Fig. 2. Amount of information (in bytes) sent by a server from a major telecommunication company in the middle
of a workday. Time is measured in seconds.
again, one should not automatically decide to use a nonstationary model because there are
perfectly reasonable stationary models that have a similar behavior.
Stationary models are attractive not only because of parsimony but also because it is
important to have a reasonably small class of well studied and well understood models
that have wide applicability. Hence it is important to study stationary processes that can
account for features we saw above; stationary processes with long range dependence.
This chapter is an attempt to survey stationary models with long range dependence and
heavy tails. These two features are believed to be present in various data sets of financial
and communication networks origin and, hence, attracted recently much attention. Describing
long range dependence in the heavy tailed case is especially challenging and most
of the work is still ahead of us. Nevertheless, it is an exciting task, and we argue that the insights
we hope one will obtain are likely to be useful in other areas of stochastic modeling.
2. What is long range dependence?
The obvious way to measure the length of memory in a stochastic process is by looking at
the rate at which its correlations decay with lag. Annoyingly, this requires correlations to
make sense, hence finite variance needs to be assumed.
Let, therefore, Xn, n = 0,1,2,..., be a stationary stochastic process with mean  =
EX0 and 0 < EX2
0 <  (we discuss discrete time processes, but parallel formulations
for stationary processes with finite variance in continuous time are entirely clear). Let
n = Corr(X0,Xn), n = 0,1,..., be the correlation function. For most "usual" stochastic
models: ARMA processes, GARCH processes, many Markov and Markov modulated
processes the correlations decay exponentially fast with n; this has a number of important
consequences, one of which is 
n=0 |n| < . This, in turn, guarantees that the variance
of the partial sums Sn = X1 +  + Xn, n 0, cannot grow more than linearly fast, which
says, heuristically, that we do not expect to see Sn to be more than about

n away from its
Ch. 16: Long Range Dependence and Heavy Tails 645
mean n. What Mandelbrot realized two and half decades ago was that the strange behavior
of R/S statistic on the Nile river data might be explained if the variance of the partial
sums could grow faster than linearly fast. As we already know this implies that

n=0
|n| = . (2.1)
Hence, (2.1) is often taken as the definition of the long memory; as a definition it seems to
originate with Cox (1984).
Looking over the literature on long range dependence, one realizes that the definition
(2.1) has not proved to be the most popular one. By far the most widely used definition is
the more concrete
n  cn-d
as n   for some 0 < d < 1 and c > 0. (2.2)
A stationary process satisfying (2.2) would have been called long range dependent of index
d by Cox (1984). A weaker version of (2.2) is also sometimes mentioned; it replaces the
constant c by a slowly varying function (Beran, 1994). Quite often one writes the exponent
d = 2 - 2H for some 0.5 < H < 1 [e.g., Beran (1992)], and the reasons are historical: this
is the relation between the exponent H of self-similarity of a Fractional Brownian motion
and the rate of decay of correlations of its increments. A misnomer, H = 1-d/2 is at times
referred to as the self-similarity parameter even if nothing in the model is self similar.
Less common (but still used) point of view on long range dependence is to allow d
in (2.2) to take any positive value or, indeed, a similar assumption of regular variation
of correlations (which also allows for the slowly varying case d = 0). In fact, one could
even draw the line between long and short memory by distinguishing between correlations
decaying slower than exponentially fast and those decaying at least exponentially fast.
It is difficult to justify such importance assigned to the rate of decay of correlations
(or, almost equivalently, to the rate at which the spectral density of the stationary process
grows at the origin), unless one deals with a Gaussian model (like the Fractional Gaussian
noise, the increment process of a Fractional Brownian motion), or a process that is very
close to being Gaussian. Using correlations as a measure of the length of memory becomes
untenable in the case of heavy tails. Specifically, let Xn, n = 0,1,2,..., be a stationary
stochastic process. Let F be the distribution function of X0, and F = 1 - F the (right)
distribution tail. In a tradition going back, once again, to Mandelbrot in the early 1960s
[an exhaustive list of references is in Mandelbrot (1983)] heavy tails are synonymous with
infinite variance of X0. Once again, more concrete views are prevailing in literature; it
is common to identify heavy tails with a particular tail behavior of F . Sometimes one
assumes
F(x)  cxas
x   for some 0 <  < 2 and c > 0. (2.3)
See, for example, a recent collection Park and Willinger (2000). Note the parallels between
the various definitions of heavy tails and viewing long range dependence via the rate of
646 B. Racheva-Iotova and G. Samorodnitsky
decay of correlations. Again, one allows sometimes any positive value of  in (2.3) [see,
e.g., Müller, Dacorogna and Pictet (1998), Gomez, Selman and Crato (1997)]. Here regular
variation of the tails as opposed to power-like (Pareto-type is the common expression) is
widely accepted. Finally, faster (but still slower than exponentially fast) rate of decay of
the distribution tail is sometimes also regarded as being consistent with heavy tails. Here
one does not usually go beyond the class of subexponential distributions; see Embrechts,
Klüppelberg and Mikosch (1997).
Obviously, one cannot use correlations to draw the line between short and long memory
if the variance is infinite. Several attempts have been made to use "correlation-like"
notions in that case. In the important class of stable processes notions of covariation and
codifference have been introduced and their rate of decay for various classes of stationary
stable processes computed; see, e.g., Astrauskas, Levy and Taqqu (1991). Obviously, such
"surrogate correlations" can be expected to carry even less information than the "real" correlations
do in the case when the latter are defined [although, surprisingly, codifference
turns out to characterize mixing of stationary symmetric stable processes, a fact due, essentially,
to Maruyama (1970), see also Gross (1994)].
Before finishing this section we remark that when talking about tails we are thinking
about the right tails. For as long as the left tails do not interfere with the right tails, we will
leave it that way. When right and left tails begin to interfere with one another, we will need
to say more about the left tails and how heavy they are as well.
3. Tails and rare events
Here is an alternative point of view on long range dependence in heavy tailed processes.
Most practitioners using heavy tailed models will agree that the most important feature of
such processes is precisely their tails as expressed in probabilities of various rare events.
Risk analysis, ruin probabilities, congestion and overflow analysis are just some of the key
words that name such rare events in various modern applications. To be a bit more concrete
here are several specific examples of rare events one usually deals with. Let, once again,
Xn, n = 0,1,2,..., be a stationary stochastic process.
Example 3.1. For large  > 0 the event {X0 > } is a rare event, whose probability is
clearly related to the tails. This event is so elementary that it does not tell us anything
about the memory in the process.
Example 3.2. For k 1 and large 0,1,...,k the event {X0 > 0,X1 > 1,...,Xk >
k} is a rare event whose probability can carry very important information about the dependence
in finite pieces of the process. Generally, the dependence we can measure using
such rare events is a "tail dependence". However, for specific classes of heavy tailed
processes (e.g., stable processes, linear processes, etc.) these events can provide even more
information.
Ch. 16: Long Range Dependence and Heavy Tails 647
Example 3.3. For large n 1 and a positive sequence (j )j 0 that does not converge to
zero the event {Xj > j , j = 0,1,...,n} is a rare event and its probability is a very interesting
measure of the length of memory in the process. The case j =  > 0 for all j 0
seems to be especially appealing.
A slight generalization of this example uses a triangular array (
(n)
j )n 1, 0 j n. Here
the case 
(n)
j = (n) for j n with various asymptotic rules for ((n)) is very interesting.
Example 3.4. For k 1 and large  the event {X1 ++Xk > } is a tail event. Similarly
to Example 3.2 the probability of this event can be used to clarify the "finite dimensional
dependence" in the process.
Example 3.5. Suppose that the mean  = EX0 is finite, and that the stationary process Xn,
n = 0,1,2,..., is ergodic. For large n 1 and  > 0 the event {X1 ++Xn > n(+)}
is a rare event whose probability measures the length of memory in the sense of a tendency
of being over the mean for long stretches of time. It is, obviously, related to the tails. The
effect of heavy tails is quite special, as will be discussed below.
Example 3.6. This example has a flavor similar to that of Example 3.5. Let, once again, the
process Xn, n = 0,1,2,..., be ergodic with a finite mean  = EX0. Let  > 0. For large 
the event {X1 ++Xn > n(+)+ for some n 1} is a rare event, whose probability
is sometimes referred to as ruin probability in the context of risk analysis. In the queuing
context various stationary quantities often have expressions of this kind for their probability
tails. Adopting the risk analysis term, the ruin probability can be used to measure the length
of memory; the effect of heavy tailed case is, once again, very special here.
The list of examples can be continued indefinitely, and we have omitted some very interesting
ones. Instead, let us look at some details of the interplay between the tails, memory
and rare events in the heavy tailed case, especially in the light of Examples 3.5 and 3.6.
The starting point is to adopt the lenses of large deviations: an unlikely event happens in
the most likely way. We will argue that such lenses provide a powerful way of thinking
about the length of memory in a process. It is unfortunate that this idea is not made more
explicit in many beautiful texts on large deviations (that also reserve the term "large deviation
principle" for something else); see, e.g., Deuschel and Stroock (1989) and Dembo
and Zeitouni (1993). The following statement is not a rigorous mathematical statement.
Nevertheless, it is often very useful as a guide and, in many ways, it captures the essence
of heavy tails:
the most likely way tail related rare events happen in a heavy tailed stochastic
process is because of the smallest possible number of causes.
This "smallest possible number of causes" is often equal to one.
Thus, in Example 3.4 it turns out that, if X1,...,Xk are i.i.d. and heavy tailed, then
P(X1 +  + Xk > )  kP(X1 > )
 P max(X1,...,Xk) >  as   . (3.1)
648 B. Racheva-Iotova and G. Samorodnitsky
That is, the sum X1 + + Xk is most likely to be very large due to one of the terms being
very large. In this case the possible "causes" are simply the individual terms in the sum.
The greatest generality under which (3.1) is valid is that of subexponential distributions,
introduced by Chistyakov (1964). See also Chover, Ney and Wainger (1973), and a survey
in Goldie and Klüppelberg (1998). Similarly, in Example 3.5, for every  > 0
P X1 +  + Xn > n( + )  nP(X1 > n) as n   (3.2)
for exactly the same reason as in (3.1). Indeed, one of the terms ( causes) in the sum
X1 +  + Xn has to be exceptionally large; exactly how large can be determined by
realizing that the "nonexceptional"terms in that sum add up to about n. While the domain
of heavy tails over which (3.2) is valid does not extend to all subexponential distributions,
it does extend to all distributions with regularly varying tails of index  > 1; see, e.g.,
Heyde (1968) and Nagaev (1979).
On the other hand, for distributions with "light" tails not only (3.1) and (3.2) fail, even
their spirit is false. In fact, in the case of exponentially fast decaying tails the most likely
way for the event {X1 +  + Xn > n( + )} to happen is not because of a single cause,
or a small number of causes but, rather, because most of the terms in the sum "conspire" to
be a bit bigger than they would normally be. This is, in fact, the point of the classical large
deviation principle.
When Xn, n = 0,1,2,..., is a stationary heavy tailed stochastic process with memory, it
is not, generally, the case that individual observations should be viewed as "causes" of rare
events. The nature of such causes depends on the nature of the process and it is, sometimes,
a nontrivial problem to figure out what the "right causes" are. We will see several examples
below. Moreover, and this is precisely the point why we are interested in rare events, the
causes, when found, typically have their effect distributed over time and it is in this way
that they make the rare events happen. We argue that this temporal distribution of the effect
of the "causes" on rare events is a useful way of thinking about long range dependence.
There are two important classes of heavy tailed processes for which progress has been
made in understanding the "right causes" of certain rare events and the way the effect of
these causes is distributed over time: linear processes and infinitely divisible processes. We
discuss these below. Before doing so we would like to introduce another notion related to
certain rare events with a potential of being useful, in a similar way, in studying long range
dependence.
Certain rare events should be rather viewed as sequences of events that become more
and more rare. Examples 3.3 and 3.5 are of this nature. More generally and formally, let
Aj  Rj be a Borel set, j = 1,2,..., such that
pj := P (X1,...,Xj )  Aj  0 as j  . (3.3)
For n 1 define
Rn = max j - i + 1: 1 i j n, (Xi,Xi+1 ...,Xj )  Aj-i+1 . (3.4)
Ch. 16: Long Range Dependence and Heavy Tails 649
That is, Rn is the highest dimension of an Aj observed over the first n observations
X1,...,Xn. We call Rn the functional associated with the sequence of rare events (Aj ).
It is obvious that if Xn, n = 0,1,2,..., is a mixing stationary process and pj > 0 for
an infinite sequence of j's then Rn   with probability 1 as n  . It appears to be
almost obvious that the rate at which Rn grows is related to the rate at which pj decays to
zero. Certain rigorous connections are, indeed, possible; other connections seem to require
additional information on the process. In any case, the rate of growth of Rn is, in its own
right, related to the way rare events happen and, hence, to the memory in the process.
There is a very important reason to concentrate on the probabilities of certain rare
events and on functionals associated with sequences of certain rare events, instead of
concentrating on correlations, when trying to understand the boundary between short
memory and long memory. Such rare events and functionals are often of a direct
importance on their own right, as one can see by looking at the examples above and
thinking, for instance, of applications in risk analysis and congestion control. On the
other hand, nobody is interested in correlations on their own right. We only study correlations
hoping that they are significant for whatever application we might have at
hand. Unfortunately, the information that the correlations carry is often only indirect
and very limited, as anyone familiar, for example, with ARCH and GARCH models
realizes.
4. Some classes of heavy tailed processes
4.1. Linear processes
One of the classes of heavy tailed processes we will consider is that of heavy tailed linear
processes.
Let n, n  Z, be iid random variables. A (two-sided) linear process with the noise
sequence n, n  Z, is defined by
Xn =

j=n-j
j , n = 0,1,2,..., (4.1)
where j , j  Z, is a sequence of (nonrandom) coefficients. We will assume that the noise
variables are heavy tailed, but how heavy the tails are will be left open at the moment.
It is obvious that the linear process Xn, n = 0,1,2,..., is a stationary stochastic process
as long as it is well defined, meaning that the sum defining it converges. The latter is an
assumption on the coefficients j . In particular, if E2
0 <  and E0 = 0, then a necessary
and sufficient condition for convergence of the series in (4.1) is

j=-
2
j < ; (4.2)
650 B. Racheva-Iotova and G. Samorodnitsky
a nonzero mean will require, in addition, the series 
j=- j to converge. Frequently
we will assume that the noise variables have regularly varying tails. Unless one is working
with constant sign coefficients (an assumption that we will not make in this chapter), it is
necessary to control both right and left probability tails of the noise since, say, a negative
coefficient will "translate" the left tail of the noise into the right tail of the sum in (4.1).
Therefore, a typical assumption is



P |0| >  = L()-
,
lim

P(0 > )
P(|0| > )
= p, lim

P(0 < -)
P(|0| > )
= q,
(4.3)
as   , for some  0 and 0 < p = 1 - q 1. Here L is a slowly varying (at infinity)
function. If  > 2 we are in the case of finite variance, but for  2 the precise condition
for convergence in (4.1) depends on the slowly varying function, and can be stated through
the three series theorem. In particular,

j=|j
|<
 (4.4)
for some  > 0 is a sufficient condition for convergence if 0 <  1 or if 1 <  2
and E0 = 0; a nonzero mean in the latter case will also require, as before, the series

j=- j to converge.
A rich source of information on linear processes in Brockwell and Davis (1991). This
book covers, mostly, the L2 case. For more information on the infinite variance case see,
for example, Cline (1983, 1985) and Mikosch and Samorodnitsky (2000b).
Heavy tailed linear processes are attractive to us because, in this case, the potential
"causes" of rare events appear to be evident: those are the individual noise variables n,
n  Z. This intuition has been born out in a number of situations, as will be seen below.
4.2. Infinitely divisible processes
A stochastic process Xn, n = 0,1,2,..., is infinitely divisible if for any k = 1,2,... there
is a stochastic process Y(k)
n , n = 0,1,2,..., such that the finite dimensional distributions
of Xn, n = 0,1,2,..., and of k
i=1 Y(k,i)
n , n = 0,1,2,..., coincide. Here for i = 1,...,k,
the processes Y(k,i)
n , n = 0,1,2,..., are iid copies of Y(k)
n , n = 0,1,2,.... Many important
classes of stochastic processes are, in fact, infinitely divisible. All Gaussian processes,
and all stable processes in particular, are infinitely divisible. In general, an infinitely divisible
process will have two independent components, a Gaussian one and a non-Gaussian
one. Since we are interested in heavy tails, for a vast majority of applications the Gaussian
component will have only a negligible effect on the probabilities of rare events we consider.
Therefore, we will only consider infinitely divisible processes without a Gaussian
Ch. 16: Long Range Dependence and Heavy Tails 651
component. Such processes have a characteristic function of the form
E exp i

n=0
nXn (4.5)
= exp
R
exp i

n=0
nxn - 1 - i

n=0
nxn1 |xn| 1 (dx) + i

n=0
nbn
for all n, n = 0,1,2,..., only finitely many of which are different from zero. Here 
is a -finite measure on R equipped with the product -field (the Lévy measure of the
process) and bn, n = 0,1,2,..., is a constant vector in R.
The Lévy measure of an infinitely divisible process is its most important feature. Often
an infinitely divisible process is given in the form of a stochastic integral with respect to an
infinitely divisible random measure. In that case there is a natural way to relate the Lévy
measure of the process to the basic characteristics of such an integral.
Unlike the linear processes in the previous subsection, it is less obvious what are the potential
"causes" of rare events when one deals with infinitely divisible processes as above.
There is, however, a point of view on infinitely divisible processes that turns out to be useful
here. To be able to see the essence better and not to get bogged in the technical details,
let us consider, first, a particular case, when
R
xn1 |xn| 1 (dx) <  for all n = 0,1,2 .... (4.6)
In that case one can rewrite (4.5) in the form
E exp i

n=0
nXn = exp
R
exp i

n=0
nxn - 1 (dx) + i

n=0
nbn (4.7)
with bn = bn - R xn1(|xn| 1)(dx) for n 0.
Let M be a Poisson random measure on R with mean measure . It is easy to check
that the process R xnM(dx)-bn for n 0 is well defined and has characteristic function
given by (4.7). That is, one can represent the process Xn, n = 0,1,2,..., in the sense of
equality of finite dimensional distributions in the form
Xn =
R
xnM(dx) - bn, n = 0,1,2,.... (4.8)
If (z(j) = (z
(j)
n , n 0), j = 1,2,...) is a (measurable) enumeration of the points of the
random measure M, then (4.8) means that the process Xn, n = 0,1,2,..., is the sum of
(z(j)), j = 1,2,..., (shifted by the sequence (bn)). This "discrete" structure of infinitely
652 B. Racheva-Iotova and G. Samorodnitsky
divisible processes makes the potential "causes" of certain rare events visible, and it is
precisely the Poisson points ((z(j)), j = 1,2,...) that turn out to be such "causes".
Even if the assumption (4.6) does not hold, then a representation similar to (4.8) can still
be written, but this time an appropriate centering is required to make the Poisson integral
to converge. The important point is that the discrete structure is still here, and the potential
causes of rare events are still visible.
There are various ways of summing up Poisson points to get an infinitely divisible
process. A very general description is in Rosínski (1989, 1990). Sometimes it is convenient
to order the Poisson points according to the value of a particular test functional. If the
process is originally given in the form of a stochastic integral with respect to an infinitely
divisible random measure, then one can have a more concrete structure of the Poisson
points, hence better understanding of the possible causes of rare events.
The literature on infinitely divisible processes is rich. The framework preferred by many
authors is that of infinitely divisible probability laws on Banach (or other nice) spaces.
See for example Araujo and Giné (1980) and Linde (1986). A very general treatment of
stochastic integrals with respect to infinitely divisible random measures as well as representations
of infinitely divisible processes as such stochastic integrals is in Rajput and
Rosínski (1989).
An important and reasonably well understood class of infinitely divisible processes is
that of -stable processes. The latter are characterized by the following scaling property of
their Lévy measure:
(rA) = r(A)
for all measurable A  R
and r > 0. (4.9)
Here  is a parameter with the range 0 <  < 2. See Samorodnitsky and Taqqu (1994)
for information on stable processes; the structure of stationary stable processes has been
elucidated by J. Rosinski; see, e.g., Rosínski (1998).
5. Rare events, associated functionals and long range dependence
Suppose that we are considering a parametric family of laws of a stationary stochastic
process Xn, n = 0,1,2,.... Let  be the (generally, infinite dimensional) parameter space.
We are interested in significant changes ("phase transitions") in the rate of decay of probabilities
of certain rare events and/or in the rate of growth of the functionals associated
with sequences of rare events that may occur when the parameter  crosses the boundary
between a subset 1 of  and its complement. We argue that certain phase transitions of
this kind can be viewed as transitions between short and long range dependence.
It is clear that it is not useful to view every significant change in, say, probabilities of rare
events as an indication of interesting and important things happening to the memory of the
process. Other factors may be in play as well, most significantly related to the heaviness of
the tails. If, for example, one of the components of parameter    governs how heavy the
tails of X0 are, one can very easily induce a very significant change in the probabilities of
Ch. 16: Long Range Dependence and Heavy Tails 653
certain rare events by simply changing that particular component of the parameter without
doing anything to the memory of the process. In the examples in the sequel we will be
careful to look for phase transitions that do not involve changing how heavy the tails are.
We will see several examples of such phase transitions indicating a shift from short
to long memory below. We present some known results; these are quite scarce. When
appropriate, we supplement those with conjectures. In other cases we have performed
numerical studies to try to guess whether a phase transition occurs and, if so, of what
kind.
5.1. Unusual sample mean and long strange segments for heavy tailed linear processes
Here we consider the sequence of rare events of the Example 3.5 An = {X1 +  + Xn >
n( + )} (for a fixed  > 0) and the corresponding associated functional
Rn = max j - i + 1: 1 i j n,
Xi + Xi+1 +  + Xj
j - i + 1
>  +  . (5.1)
We will keep the distribution of the noise variables n, n  Z, in the heavy tailed linear
processes of Section 4.1 fixed; it is assumed to have the regular variation property (4.3)
with  > 1. In particular, the parameter  which is responsible for the heaviness of the
tails is kept fixed. We will also assume that the E0 = 0. In this case the parameter space
is
 =  = (...,-1,0,1,2,...)  RZ
,
satisfying (4.2) if  > 2 or (4.4) if 1 <  2 . (5.2)
Let 1   be the set of all sequences   RZ satisfying

j=|j
| < . (5.3)
Note that the set 1 contains the parameter sequence j = 1(j = 0), j  Z, in which case
the linear process is an iid sequence.
It turns out that for any value of the parameters in 1 the functionals Rn defined by
(5.1) grow at the same rate, i.e., at the same rate as for an iid sequence with the same
marginal tails. This has been established in Mansfield, Rachev and Samorodnitsky (2001).
Specifically, let F be the distribution function of the noise random variable 0 and define
the usual quantile sequence
an =
1
1 - F

(n). (5.4)
654 B. Racheva-Iotova and G. Samorodnitsky
Here for a function U on [0,), U denotes its generalized inverse
U
(y) = inf s: U(s) y .
Note that, by (4.3), the sequence (an) is regularly varying at infinity with exponent 1/. See
Resnick (1987) for more information on regular varying tails and their quantile functions.
For  > 0 let Z be a Fréchet random variable with
P(Z z) = exp -z,
z > 0. (5.5)
Assume (5.3). Then the numbers



M+() = max sup
-<k<
k
j=-
j
+
, sup
-<k<

j=k
j
+
,
M-() = max sup
-<k<
k
j=-
j
,
sup
-<k<

j=k
j
-
,
(5.6)
are, obviously, finite. Then
a-1
n Rn  -1
pM+()
+ qM-() 1/
Z (weakly) as n  , (5.7)
once again as long as (5.3) holds. Here p and q are the tail weights in (4.3). See Theorem
2.1 in Mansfield, Rachev and Samorodnitsky (2001).
What happens if   c
1 (i.e., if (5.3) fails)? It is not known whether, in this case, Rn
always grows at the rate faster than an, that is whether the sequence (of the laws of)
(a-1
n Rn, n = 1,2,...) is not tight. However, the following is known. Assume that the
coefficients (j ) are themselves regularly varying and balanced. That is, there is a function
 :[0,)  [0,) such that
(t) = L2(t)t-h
(5.8)
as t   and such that
lim
j
j
(j)
= c+, lim
j
-j
(j)
= c-, (5.9)
for some c+,c- 0, at least one of which is positive. Here
1 > h > max
1

,
1
2
(5.10)
Ch. 16: Long Range Dependence and Heavy Tails 655
and L2 is a slowly varying function. Clearly any such parameter vector  is in c
1 . Define
bn =
1


(an), (5.11)
n 1 and note that sequence (bn) is regularly varying with exponent 1/(h). Then
b-1
n Rn  p1/h
(1 - h)
-1/h
c
1/h
+ + c
1/h
- Zh (weakly) as n  . (5.12)
See Theorem 2.1 in Rachev and Samorodnitsky (2001).
Since bn grows faster than an does, under the assumptions (5.9) the sequence Rn does
grow faster than an and, hence, faster than in the iid case and, more generally, faster than
it is the case for any   1.
Both results (5.7) and (5.12) are, in the final analysis, a consequence of change in the
temporal distribution of the effect of the individual "causes": exceptionally large or exceptionally
small values of the noise variables (m). In fact, the contribution of each individual
noise variable m to the sum Xi + Xi+1 +  + Xj in (5.1) is m
j-m
d=i-m d. The intuition
of heavy tailed large deviations says that it is a single m that is most likely to be
responsible for a large value of Rn. Therefore, one would expect that for large xn
P(Rn > xn)
 P for some m = ...,-1,0,1,...,
j-m
d=i-m
d m > (j - i + 1)( + )
for some 1 i j n, j - i + 1 xn . (5.13)
This turns out to be valid. Moreover, this intuition allows one, in both cases (i.e., under
(5.3) and under (5.9)) to select the right rate of growth for xn in (5.13), which is equivalent
to selecting the appropriate normalization to Rn.
It is a bit surprising that less is known about the apparently easier problem of identifying
the rate of decay of probabilities pn = P(X1 +  + Xn > ( + )n) for  > 0 as n  .
It has been checked that under the assumption

j=j|j
| <  (5.14)
which defined a proper subset of 1,
pn  n-(-1)
L(n)-
p

j=-
j

+
+ q

j=-
j

as
n  , (5.15)
656 B. Racheva-Iotova and G. Samorodnitsky
where p, q and L are defined in (4.3), and one assumes that q > 0 if 
j=- j < 0. See
Lemma A.5 in Mikosch and Samorodnitsky (2000b). It looks very plausible that (5.15)
holds for every parameter   1. The logic of large deviations indicates that, under the
assumptions (5.9), pn is regularly varying with exponent -(h - 1) at infinity, but nobody
has presented a rigorous proof so far.
5.2. Ruin probability for heavy tailed linear processes
In this subsection we consider the rare event in Example 3.6, A = {X1 +  + Xn >
n( + ) +  for some n 1}, when  > 0 is fixed and  is large. Unfortunately, the
result for the entire set 1 is not available here. However, there is a result for the subset of
1 defined by (5.14). In the latter case, the probability of the event A (commonly referred
to as the ruin probability) satisfies
P(A) 
pM(1)
+ () + qM(1)
- ()
( - 1)
-(-1)
L() as   , (5.16)
where
M
(1)
+ () = sup
-<k<
k
j=-
j
+
, M
(1)
- () = sup
-<k<
k
j=-
j
,
(5.17)
compare with (5.6). See Theorem 2.1 in Mikosch and Samorodnitsky (2000b). We conjecture
that (5.16) holds whenever   1. Once again, a good way to think of the asymptotic
behavior of the ruin probability is to think about the most likely way the "ruin" can happen.
Realizing that the ruin is, most likely, due to a single "extraordinary" value of a noise
variable m, one would expect that
P(A) 

m=-
P
n-m
d=1-m
d m > n +  for some n 1 . (5.18)
Once again, this turns out to be valid (at least, under the assumption (5.14)).
The problem of the behaviour of the ruin probability for   c
1 has not, to the best
of our knowledge, been treated. One can pursue the logic of large deviations, leading to
(5.18). This leads us to conjecture that, under the assumptions (5.9), P(A) is, as a function
of , regularly varying with exponent -(h - 1) at infinity.
Based on the above discussion (admittedly, some part of it is "hard" results, and another
part is conjectures) one can argue that a significant change occurs for heavy tailed linear
processes as parameter  crosses the boundary between 1 and its complement. Not only
the order of magnitude of the probabilities of certain rare events, and of certain functionals
associated with sequences of certain rare events, appears to change at that boundary but
another interesting phenomenon seems to happen. Various orders of magnitude do not
Ch. 16: Long Range Dependence and Heavy Tails 657
change as the parameter varies inside of 1; not only these orders of magnitude do change
at the boundary but, also, they may keep changing as the parameter varies outside of 1.
It is important to make a remark at this moment. It does appear that one should, in
fact, look at the behavior of a family of related rare events, or a family of sequences of
related rare events, if one wants to see what precisely happens at a boundary. For example,
the assumptions (5.9) do not cover the entire c
1 . We conjecture, however, that important
changes happen when one moves from 1 into c
1 and not, necessarily, into the subset of
c
1 defined by (5.9). It is likely that, in order to see these changes, one should look not only,
say, at the event An = {X1 +  + Xn > n( + )} but also at some related rare events, for
example at the event Bn = {|X1| +  + |Xn| > n(1 + )}, with 1 = E|X1|.
It is also interesting to mention that, in the case  > 2, the condition (5.3) also implies
the absolute summability of correlations (i.e., (2.1) fails).
5.3. Rare events for stationary stable processes
The situation regarding "phase transitions" for general stationary heavy tailed infinitely
divisible processes of Section 4.2 has been investigated even less than it is the case with
the heavy tailed linear processes. There are several reasons for this, including relatively
complicated structure of stationary infinitely divisible processes and its very involved parameter
space, which is a space of measures. Most of the known results are for stable
processes, whose structure is better understood. We present here the results for a subclass
of stationary stable processes, where we will be able to see a "phase transition".
Specifically, let Xn, n = 0,1,2,..., be the linear fractional symmetric -stable noise,
1 <  < 2. For a fixed  the law of the process has an important parameter H  (0,1).
That is,
Xn =
R
fn(x)M(dx), n = 0,1,2,..., (5.19)
where M is a symmetric -stable random measure on the real line with the Lebesgue
control measure, and fn(x) = f (x + n) - f (x + n + 1), n = 0,1,2,..., x  R, with
f (x) = a (-x)
H-1/
+ - (-x - 1)
H-1/
+ + b (-x)
H-1/
- - (-x - 1)
H-1/
-
(5.20)
if H  (0,1), H = 1/. Here a and b are real numbers not simultaneously equal to zero.
For H = 1/ one has two choices,
f (x) = a1 [-1,0] (x) (5.21)
and
f (x) = a ln|x| - ln|x + 1| . (5.22)
658 B. Racheva-Iotova and G. Samorodnitsky
In the latter two cases a is a real number different from zero. The resulting symmetric
-stable process in (5.19) is an ergodic stationary process. It is the increment process
of the linear fractional symmetric -stable motion if H = 1/, an iid sequence ( the
increment process of the symmetric -stable Lévy motion) under (5.21), and the increment
process of the log-fractional symmetric -stable motion under (5.22). All of these
processes are H -self-similar with stationary increments. We refer the reader to Samorodnitsky
and Taqqu (1994) for information on stable processes, their integral representations
and on self-similar processes. The parameter space  is, then, the collection of all triples
(H,a,b) with H  (0,1), H = 1/, and a,b real, a2 + b2 > 0, together with the triples
(H,a,i) with H = 1/, a real, different from zero, and i = 1,2, depending on the choice
between (5.21) and (5.22). Let 1 be the subset of  corresponding to 0 < H < 1/.
We consider, once again, the rare event in the Example 3.6, A = {X1 +  + Xn >
n( + ) +  for some n 1}, when  > 0 is fixed and  is large. Of course  = 0 here.
Then
P(A) 



K

-(-1)
if 0 < H <
1

or under (5.21),
K

-(-1)
(log)
under (5.22),
K
H
-(1-H)
if
1

< H < 1
(5.23)
as   . Here K is a finite positive constant that depends on , H , a and b, but not on .
See Proposition 4.4 in Mikosch and Samorodnitsky (2000a).
Observe that the order of magnitude of the ruin probability remains the same as H varies
in (0,1/). Furthermore, this order of magnitude is the same as under independence. On
the other hand, as H varies in the interval (1/,H), the order of magnitude of the ruin
probability is greater than that in the case of independence and, furthermore, this order
of magnitude changes with H . As we argued earlier, this gives us a reason to say that the
range H  (0,1/) corresponds to short memory, and the range H  (1/,1) corresponds
to long memory. It is interesting that, in this case, the boundary H = 1/ contains two
points, corresponding to (5.21) and to (5.22), and it makes sense to view the latter as
corresponding to long memory, while the former is the independent case.
Here is how the intuition of large deviations works here. As mentioned in Section 4.2,
the process Xn, n = 0,1,2,..., can be represented as a sum of Poisson points. In the
symmetric stable case this can be done as follows. One can write (in terms of equality of
finite dimensional distributions) the process given by (5.19) in the form
Xn = C1/


j=1
j 
-1/
j g(Vj )-1/
fn(Vj ), n = 0,1,2,..., (5.24)
where C is a finite positive constant that depends only on , g a strictly positive measurable
function such that R g(x)dx = 1, (n)n 1 is an iid sequence of Rademacher variables
Ch. 16: Long Range Dependence and Heavy Tails 659
(P(n = -1) = P(n = 1) = 1/2), (n)n 1 are the points of a unit rate Poisson process
on (0,), and (Vn)n 1 is an iid sequence of real valued random variables with common
density g. Moreover, the three sequences are mutually independent. See Samorodnitsky
and Taqqu (1994, Section 3.10).
Rewriting
P(A) = P C1/
 sup
n 1

j=1
j 
-1/
j g(Vj )-1/
n
k=1
fk(Vj ) - n > u ,
the intuition of rare events says that it is a single one of the Poisson points (in the function
space) (j 
-1/
j g(Vj )-1/ n
k=1 fk(Vj ), n = 1,2,...) that is most likely to cause the
ruin. This intuition translates into
P(A) 

j=1
P C1/
 
-1/
j g(Vj )-1/
sup
n 1
j
n
k=1
fk(Vj ) - n > u (5.25)
as   . It is the equivalence (5.25) that allows one to understand the change in the way
the effect of these Poisson points is distributed over time as the parameter H crosses the
boundary 1/.
Interestingly, the probabilities of the rare events of Example 3.5 An = {X1 +  +
Xn > n( + )} do not indicate anything interesting happening at the point H = 1/.
In fact, since the processes under considerations are the increments of H -self-similar
processes,
pn = P(X1 +  + Xn > n) = P nH
X1 > n  const  -
n-(1-H)
as n  . Hence the order of magnitude of pn changes "ordinarily" as H crosses the
boundary 1/. As mentioned at the end of Section 5.2, one should, probably, look at certain
related rare events as well. The behavior of the associated functionals in (5.1) does not seem
to have been studied so far.
5.4. High dimensional joint tails for a linear process with stable innovations
We conclude this chapter with a simulation study of a situation in which no analytical results
are yet available. Consider a heavy tailed linear process (4.1). For a fixed  > 0 we
consider the probability of the event An = {Xj > , j = 0,...,n}, when n is large. We are
within the framework of Example 3.3. The discussion above makes it possible to conjecture
that there is a phase transition at the boundary between the set 1 in (5.3) and its complement
in the set  in (5.2). To check this conjecture we ran a simulation of 107 realizations
of a linear process with symmetric -stable innovations with different . We estimated both
the probability P(An) as a function of n and the rate of growth of the associated functional
Rn = max j - i + 1: 1 i j n, min(Xi,...,Xj ) >  . (5.26)
660 B. Racheva-Iotova and G. Samorodnitsky
We simulated first an AR(1) process with j = 0 for j = 0 or 1, 0 = 1 and varying 1.
This choice of coefficients is, clearly, in 1. Then we simulated a linear process with
j = 0 for j < 0 and j = (1 + j)-0.8 for j 0 (and  > 1/0.8). This choice of parameters
is in the set c
1 .
While a simulation study of this type cannot provide a definite answer, it seems to indicate
that for the AR(1) process the probabilities P(An) decay exponentially fast with n.
We plotted in Figure 3 the ratio -(logP(An))/n over the range of n for  in the set
{0.1,0.2,0.3,0.4} for the AR(1) process with  = 1.5 and 1 = 0.5. Notice how the curves
become horizontal.
In comparison, our simulations seem to indicate that for the linear process with j =
(1 + j)-0.8, j 0, the probabilities P(An) decay hyperbolically fast with n. We plotted
Fig. 3. The ratio -(logP (An))/n for the AR(1) process with  = 1.5 and 1 = 0.5.
Fig. 4. A plot of P (An) against n for a linear process with  = 1.5 and j = (1 + j)-0.8, j 0. Log­log scale.
Ch. 16: Long Range Dependence and Heavy Tails 661
Fig. 5. A plot of (logRn)/logn for a linear process with  = 1.5 and j = (1 + j)-0.8, j 0.
in Figure 4 P(An) against n in the log scale, for the case  = 1.5. Here we use  in the set
{0.1,1,5,40}. Notice how linear the plots are. Finally, we present a plot of (logRn)/logn
for the long memory process with  = 1.5 and   {0.1,0.2,0.5,1} (Figure 5). Our intuition
tells us that in that case Rn should grow polynomially fast with n, and the simulation
appears to bear this out.
Once again, even though a simulation study is not a conclusive evidence of a phase
transition at the boundary between the set 1 and its complement, its results are consistent
with such a phase transition.
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