RE S E A R C H PA P E R QUANTITATIVE FINANCE VO L U M E 1 (2001) 124­130
quant.iop.org INSTITUTE O F PHYSICS PUBLISHING
Scaling in financial prices:
II. Multifractals and the star equation
Benoit B Mandelbrot
Sterling Professor of Mathematical Sciences, Yale University, New Haven,
CT 06520-8283, USA
Received 17 November 2000
Abstract
This is a direct continuation of the preceding paper, with which it shares the
front material and the numbering of the sections. A little repetition makes it
possible to read this paper, part II, by itself. It describes the progression of
the formalism from the financial model the author introduced in 1963, with
independence and  < 2, to the financial model the author developed 1997,
with multifractal dependence and 1 <  < , and on to current
developments.
The long informal discussion in section 3 of the preceding paper is
rephrased in formal fashion and extended. The presentation describes the
original and the multifractal forms of a `star equation' and then moves
beyond it.
4. Formal discussion: I. The functional
`star equation' in Cauchy (1853) only
allows values of  smaller than 2
This section will show how the functional star equation arose
implicitly in the distant past, in conjunction with the Gaussian.
Itwasexplicitlystated, generalized, andsolved(butnotnamed)
by Cauchy. Broader solutions were provided by Paul L´evy
who showed that, asymptotically, they follow a power-law with
0 <  < 2.
4.1. Non-random weights and the property of
scaling under addition that has long been known to
be satisfied by the Gaussian
Denote by G the reduced Gaussian random variable, meaning
that EG = 0 and EG2
= 1. Select b 2 independent values
Gn of G. This b will be called `base'. It has been known for
centuries that `the weighted sum b
n=1 mnGn is itself Gaussian
for all values of b and all non-random weights mn'.
4.2. Exact renormalizability and the `base-free' `star
equation' as it enters in the Gaussian case
In preparation for generalizations to come, it is useful to
introduce the normalized weights Wn = mn/ m2
n, which
satisfy W2
n = 1. The statement in section 4.1 now takes the
following form:
`the non-randomly weighted b
n=1 WnGn is itself a
reduced Gaussian'.
`Star equation'. This term will denote the following
identity, where  denotes identity in distribution and
~W(W1, W2, . . . , Wb(bu)) is a function of the weights:
b
n=1
WnXn  ~W(W1, W2, . . . , Wb)X. (1)
The inputs are the base, b and the non-random weights Wn.
The unknown is the random variable X, in other words, the
distribution function Pr{X < x}. Therefore, equation (1) is
a functional equation. Its solutions include the Gaussian aG.
Conversely, it can be shown that under wide conditions there
are no other solutions.
`Base-bound' versus `base-free'. When the star equation
only holds for one value of the base b, it will be called `basebound'.
When it holds irrespective of the base, it will be called
`base-free'. Given a choice, the latter alternative is preferable,
but we shall see that the former is easier to generalize.
124 1469-7688/01/010124+07$30.00  2001 IOP Publishing Ltd PII: S1469-7688(01)19127-X
QUANTITATIVE FINANCE Scaling in financial prices: II. Multifractals and the star equation
4.3. A star equation put forward by Cauchy in 1853;
symmetric solutions due to Cauchy and skew
solutions due to L´evy
In 1853, Cauchy, aged 64, introduced the base-free star
equation for independent Xm and without auxiliary conditions.
This work also had the distinction of being the first to study
random variables through their Fourier transforms, now called
characteristic functions (s). With all weights equal to 1, the
right-hand side of the star equation becomes ~W(n) and the 
must satisfy the `dual star equation'
n
(s) = [ ~W(n)s].
Cauchy advanced the reduced solution
R(s) = exp(-|s|
), where  is positive real.
Adding a scale factor  and a position factor  yields
(s) = exp(-is -  |s|
). Real characteristic functions
correspond to symmetric densities. Much later, Polya showed
that for  > 2 the Fourier transform of exp(-|s|
) fails to
be 0, hence cannot be a probability density. Therefore, the
fundamental restriction  < 2, which is at the core of this
paper, first appeared for a purely mathematical reason.
The problem was completely solved in L´evy (1925) which
describes complex-valued solutions (s) that correspond to
asymmetric distributions. In addition to ,  and  between
0 and 2, they require a fourth parameter  that satisfies
-1  1. It is  = 0 in Cauchy's original symmetric
case,  = 1 being the most asymmetric cases.
My main reason to credit the stable distributions to L´evy
rather than to Cauchy is that the cases required in applications
are most often asymmetric. Furthermore, the term `Cauchy
distribution' has already been assigned (by L´evy himself!) to
a distribution that Cauchy had credited to Poisson. Also, I
knew L´evy and consider that, while he did not father the idea,
unquestionably he deserves credit for having mothered it in the
mathematical community.
4.4. Cauchy's star equation `generates', `explains'
or `accounts for' the power-law distributions with
 < 2; it is not an example of `power law in, power
law out, with nothing concerning the bell' (see
section 2.5); it is an example of `scaling in and a bell
and a power law simultaneously coming out'
As is well known by now, the stable solutions of Cauchy's
star equation have power-law tails whose critical exponent 
can range from 0 to 2. Therefore, the desire to `explain' or,
more modestly, to `account for them', has baffled science since
Pareto. In the Cauchy equation, a power law is absent from
the input and (except for  = 2) present in the output.
5. Formal discussion: II. Random
weights W and the multifractal star
equation put forward in Mandelbrot
(1974); its solutions allow 1 <  < 
The material in this section is in part published but is little
known and section 6 reports on the application to finance of
very recent material.
The inequality  < 2 is imposed by Cauchy's star
equation relative to scaling combined with independence.
Many writers concluded that, whenever data yield  > 2,
scaling is inadequate and should be abandoned. My alternative
proposal is to foresake independence and generalize scaling
into multiscaling.
This proposal is best carried out in the context of the
M1972/1997 model sketched in section 1.7. It takes the form
of the compounded function P(t) = BH [(t)]. This is an
oscillating multifractal function, where BH () is a fractional
Brownian motion and (t) is the integral of a multifractal
measure, that is, an increasing (non-oscillating) multifractal
function. If (t) had been a function with independent
increments, compounding would have reduced to a special case
called subordination. In finance, subordination was pioneered
in 1967 by Mandelbrot and Taylor (reproduced in chapter
E21 of Mandelbrot (1997)). More recently, many authors
changed (t) but preserved independent increments. What is
needed is not a better subordination but a well-chosen general
compounding; my offering is multifractals.
The multifractal star equation's most striking property is
that it can generate a power-law distribution with an exponent
between 1 and . This happens under delicate conditions
that were dismissed as anomalous. It shall be argued that this
perception is especially clearly unwarranted in the context of
price variation. Let us elaborate.
5.1. Fractional Brownian motion of a multifractal
time; the exponents H,  and qcrit and the relation
 = qcrit/H, which allows 1 <  < 
Given 0 < H < 1, the fractional Brownian motion BH ()
is the oscillating random process defined as follows. Its
increments are Gaussian and satisfy E[BH () - BH (0)] = 0
and E[BH () - BH (0)]2
= 2H
. It follows that for q > -1,
E[|BH ( - BH (0))|q
] =  (q)qH
.
The prefactor  (q) is positive and finite.
A multifractal time transform (t) is a far more subtle
notion. The random multifractal measure in a time interval dt
will be introduced in section 5.4 and denoted as (dt). The
function (t) will be taken to be a non-decreasing function
defined as the total measure ([0, t]) in the interval [0, t].
In somewhat rough terms to be discussed in section 5.4 and
developed in section 6, (t) is characterized by the fact that its
moments take the form
E[(t)]q
= E q
t(q)+1
.
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B B Mandelbrot QUANTITATIVE FINANCE
Here the exponent (q) is a basic function in the
multifractal formalism. When different multifractal measures
share the same (q), they may differ but the differences lie
beyond the reach of the theory of multifractals.
The prefactor is ordinarily disregarded but was an
essential focus of interest in Mandelbrot (1974). It will be
motivated in section 5.3.
Up to this point, the argument assumed that all the
moments are finite otherwise meticulous scientists make this
assumption all the time, without even a thought, but in this
context it happens to lead to a paradox.
Indeed, the study of the specific cascade described in
section 5.4 revealed a possibility that a fully general approach
may have missed. It is possible to have (q) < 0 for q > 1.
If so, as t   the exponential factor t(q)+1
increases more
slowly than t; this is a conclusion I viewed as impossible.
The only way out that came to mind is that (q) <
0 must imply E q
= . This hunch was confirmed, first
heuristically and later rigorously by Kahane and Peyri`ere (see
chapter N17 of Mandelbrot (1999)).
In the present context, infinite moments are not as strange
as it seemed in 1974. In fact, their time has come because
we are concerned with price changes that follow a power law
distribution of exponent  for which the q-moments are infinite
for q > . With multifractals, the plainest way to achieve the
result is if the q-moments of are infinite for q > qcrit = H.
From 1 < qcrit <  and  < H < 1, it follows that
1 <  < . As announced, the range of possible values of
 has been extended beyond 2, to 1 <  < . The range
0 <  < 1 is not needed for the present purposes. It is not
attainable in this fashion but requires a further generalization.
Assuming q < qcrit, combine the two preceding equations
and write ~(q) = (qH). This yields
E[|BH [(t)] - BH (0)|q
] =  (q)E qH
(qH)+1
=  (q)E qH
t ~(q)+1
.
The experimental evidence provided by figure 5. Sections
5 and 6 are largely theoretical and the data that motivate them
are described in Calvet and Fisher (2001) and the third part
of Mandelbrot, Calvet and Fisher (1997). Figure 5 shows the
quality of fit that the last formula provided for one of the most
important among financial price series.
The contrast between uni- and multifractal bevaviour.
The exponents in the last three displayed formulae rule the
rates of growth of three basic expectations.
For BH (), the qth scale factor is  H
, independently of
q. For this reason, BH () is called uniscaling or unifractal.
For (t) this scale factor is  t[(q)+1]/q
, which depends
on q. For that reason, (t) is called multifractal. The same is
true of BH [(t)].
The expression [(q) + 1]/q is fundamentally more
directly relevant than the alternative expression (q)/(q - 1)
which is often found in the literature of multifractals.
Goals. The remainder of this section and section 6 concern
specific multifractal mechanisms for which the preceding
heuristic is exactly correct, and (t) has a power-law
distribution whose exponent is a critical qcrit.
0
1
2
3
4
4 4.5 5 5.5 6 6.5 7
log10 (d1) (seconds)
Figure 5. Doubly logarithmic plot of the partition function of the
data as a function of dt in the case of the US Dollar/Deutschmark
exchange rate; the data were provided by Richard Olsen in Zurich.
The main observations are (i) the fact that the plots are straight from
dt of the order of one hour to dt of more than a hundred days; the
slopes of the plots define the function ~(q); (ii) the fact that the
value of q = 1/H for which ~(q) = 0 is close to 2. Observation (i)
is a symptom of multifractality. Observation (ii) is a symptom that
the process is close to being a Wiener­Brownian motion that is
followed in multifractal time. The true value of H is a little above
1/2. Alternative statistical tests suggest the same inequality. If
further confirmed, this would be a token of persistent fractional
Brownian motion in multifractal time.
5.2. Three kinds of multiplicative cascades in b-adic
grids (Mandelbrot 1974); and the multifractal
measures they generate
A base b is prescribed, and a b-adic grid is constructed by
dividing the unit interval [0, 1] into b subintervals of equal
lengths 1/b, then each of those subintervals into b equals one,
and so on.
The notion of multiplicative cascade comes in several
forms to be distinguished momentarily. Every one begins with
a measure 0(dt) whose density is uniform on [0, 1] and equal
to one. They are generated by the weights W and the masses
M = W/b, to be introduced presently. Both will be random
and identically distributed random variables, their respective
expectations being EW = 1 and EM = 1/b. This section
could be simplified by proceeding in terms of M, but section 6
must be written in terms of W, so it is best to carry W and M
together in this section
The first cascade stage ends with a measure (t) whose
density is uniform in each of the b subintervals of [0, 1]
and equal to W1, . . . , Wb. The compounding masses are
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QUANTITATIVE FINANCE Scaling in financial prices: II. Multifractals and the star equation
M1, . . . , Mb. The second (respectively, kth) stage ends on
measuring denoted by 2(dt) (respectively, k(dt)). They
treat each cell of length b-1
(respectively, b-k+1
), just as the
first stage treated [0, 1]. After the kth stage, a cell of length
dt = b-k
constrains the mass
k(dt) = M(1)M(1, 2) . . . M(1, . . . , k) = M
of density
W(1)W(1, 2) . . . W(1 . . . k) = W.
In these representations, a term Wh
(1, . . . , h) only depends
on the first h digits in the development in the base b of the
left end point of our interval. We are interested in the suitably
defined limit (dt).
The next task is to specify the rules of statistical
dependence between weights. Three cases have been
considered in greatest detail.
In the most important and least constrained case, called
canonical, the W are independent random variables and one
postulates EW = 1/b, hence EM = 1. This identity
expresses conservation on the average. Section 5.5 will argue
that this is the most appropriate assumption.
The next more constrained case, called microcanonical,
strictly obeys the conservation relation M = 1. Therefore,
it is also called conservative. In an interesting subcase,  can
take one of b mutually distinct values. Then the only possible
randomness consists in shuffling.
In the fully constrained case, called multinomial, the
sequence M1, . . . , Mb is prescribed. When b = 2, one has the
binomial measure, which is the (increasingly) distant prototype
of all multifractals.
Observe that the last two cases impose the natural bound
M < 1; in the canonical case, in contrast, M has no natural
bound; this fact proves rich in consequences, as will be seen
presently.
5.3. The quantity  = measure [0, 1] and the
multifractal star equation put forward in
Mandelbrot (1974)
As defined in section 5.1, the multinomial and conservative
cascades redistribute mass but leave its sum constant.
Therefore, the measure k that the cascade creates after k
stages in the interval [0, 1] satisfies k = 1. Its limit for
k   also satisfies  1.
In contrast, the canonical cascade--or other properly
random cascades--preserves mass only on average. Hence
the multifractal measure they generate is itself random. A
priori, all that is known is that, because of EW, satisfies the
condition of conservation on average, namely, E = 1. Let
us show that it also satisfies a generalized star equation.
Indeed, the first cascade stage creates measures equal to
M = W/b. The remaining cascade stages multiply those
quantities by independent factors . Therefore, the total
is intrinsically divided by the cascade into the sum of b parts
of the form M  = W /b. Therefore, is the solution
of the following multifractal form of the star equation
M   or
1
b
W   .
Auxiliary conditions on the W (for example, exact
conservation or conservation on average) are those specified
by the generating cascade.
5.4. Derivation of the function (q); the relation
(qcrit) = 0 between the moments of W and the
critical exponent qcrit of .
The multifractal star equation of section 5.3 has attracted a
substantial literature, for example Durrett and Liggett (1983)
which is discussed in Mandelbrot (1999, p 370). Its solutions
surely include the multiplicative multifractal measures of
section 5.2; those solutions are stable. Other solutions are
unstable and need not be considered here.
Under the influence of historical myths and the needs of
smooth exposition, an equation is always written down first
and solved next. In gritty history, in contrast, an equation is
often devised after the fact to be satisfied by an already known
solution. Equations obtained in this way may be helpful; some
deserve to be called `explanatory', but not all.
5.4.1. Derivation of (q). Let us resume the theory in
section 5.2.
After k cascade stages, the interval [0, 1] has been
subdividedintobk
intervalsoflengthdt = b-k
andthemeasure
(dt) in such a b-adic interval is the product of two terms, one
a product of weights acting on intervals of length not less than
dt and the other a product of weights acting on smaller lengths.
The first term is the above-written k(dt); it is a `lowfrequency'
term corresponding to the first k cascade stages.
Therefore it is the product of k identically distributed and
independent random variables, Writing
(q) = - logb EWq
+ q - 1 = - logb EMq
- 1,
one has
E[
q
k ] = [EWq
]k
bqk
= (dt)(q)+1
.
This power law has already been invoked in section 5.1
for . But here it applies only to b-adic intervals. It will be
extended to all intervals in section 6.
Low wavelengths are only half of the story. One
must also multiply each k(dt) by a `high-frequency' term
corresponding to all the stages beyond the kth. Thanks to the
cascade structure, the high frequency terms are independent
and identical in distribution to . Therefore,
E[q
(dt)] = E q
[EWq
]k
bqk
= E q
[dt](q)+1
.
It follows that the so-called `partition function', defined
as (q, dt) = 
q
, has the expectation
E(q, dt) = (1/dt)E[
q
k (dt)] = E q
(dt)(q)
.
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B B Mandelbrot QUANTITATIVE FINANCE
This formula introduced (q) in Mandelbrot (1974). The
approach differs in two important ways from the way used
in the heuristic restatement of multifractals that is widely
known to physicists. That restatement treats the prefactor E q
without being concerned about its being finite. It also takes it
for granted that (q) > 0 for all q > 1. But this last property
is neither obvious nor universal and the cases where it is not
true are the crucial ones for the present discussion. Therefore,
we have reached a very critical point where, not only does
my original approach differ from the heuristics, but also goes
further in an essential direction.
5.4.2. Occurrences and consequences of qcrit < . The
first example concerned the limit log-normal multifractals
studied in Mandelbrot (1972) (Mandelbrot (1999), chapter
N14). They yield (q)  -(q - 1)(q - qcrit), hence (q) <
0 for q > qcrit.
Soon afterwards, Mandelbrot (1974) observed that the
log-normal case is not a rare anomaly. The reason why
(q) > 0 is not necessarily the case in cascades is perfectly
straightforward. Writing in terms of M = W/b and assuming
M to be bounded by max M, the function (q) is well known to
behave for large q as -q logb[max M]. Hence, the anticipated
rule `(q) > 0 for all q' cannot be taken for granted and
holds if and only if max M < 1. This inequality is indeed
necessarily satisfied in the classical elementary examples
and the more general conservative cascade model. But the
canonical cascade model allows max M > 1, hence need not
satisfy (q) > 0.
I recognized that there is only one way to avoid the
paradoxes due to (q) < 0: the combination of q > 1 with
(q) < 0 occurs only when E q
= . This hunch was
buttressed by physical heuristics and soon confirmed in full
rigor by Kahane and Peyri`ere, as reported in chapter N17 of
Mandelbrot (1999).
The restricted case. To sum up, the measures supported by
[0, 1] satisfy (1) = 0 and split into two categories, according
to whether or not (q) = 0 also has a root greater than 1.
In conservative cascades, mass is strictly preserved and
(q) > 0 for all q > 1, therefore the star equation has
no solution other than  1. More generally, under the
necessary and sufficient condition M < 1 or W < b, one
writes qcrit =  by definition. This condition implies that
E q <  for all q.
The power-law distribution case. If (q) = 0 has a second
solution > 1, that solution defines a finite qcrit. If so, has a
critical exponent qcrit, whose value is deduced from (q). That
is, like the Cauchy star equation, the multifractal star equation
yields a power-law probability output without a power-law
probability input being involved. This is the most distinctive
feature of the solution of the multifractal star equation
Conclusion. The generation of the power-law
distribution as a solution of the multifractal star
equation echoes the title of section 4.4. It is not an
incomplete and logically circular example of `power
lawinandpowerlawout, withnothingconcerningthe
bell' (see section 2.5). In contrast, it is an example of
`scaling in and a bell and a power law simultaneously
coming out'.
5.5. When the observed time series is one coordinate
of a highly multidimensional series, the `generic'
situation is qcrit < 
In the original context of turbulence, the strict conservation
rule EM = 1 that defines the microcanonical multifractals
had a physical meaning for the full three-dimensional process.
However, wind-tunnel or atmosphere observations were both
necessarily limited to linear cross-sections. Along those cross
sections, Mandelbrot (1974) argued that conservation could at
best hold on the average.
Similarly, and even more strongly, the whole financial
or economic system adds up to a highly multidimensional
process. The canonical cascade can be rationalized by
assuming that investigating a financial time series by itself
amounts to extracting a linear cross section from that full
system. As section 6 will elaborate, multiplication by a weight
is meant to model the effects of a cause. Contrary to the case
of turbulence, there is no a priori reason to assume a cause's
effects on the whole economy to be conservative. Along the
cross sections, even less is known. The canonical cascade with
independent weights is a good bet because it is described in the
literature and does make the desired point. This was the reason
to begin the study as I did. A slight turn to greater realism will
be taken in section 6 and make the same point more strongly.
6. Formal discussion: III. Limitations of
the star equation; newly-developed
base-free multifractals
This section goes beyond cascade-generated multiplicative
multifractals, therefore beyond the star equation. The novelty
is that now the inequality 1 <  <  becomes the rule rather
than a special case whose relevance must be defended.
Once again, many people whose knowledge of
multifractals is limited to the basic heuristics have never heard
of the existence and role of qcrit. Those who know that a
cascade can satisfy qcrit <  tend to dismiss this possibility
as an exaggerated response to long tails of W that may be
meaningless, or, at best, unreliable. My constant attempts to
draw attention to qcrit did bear fruit in the original context of
the dissipation of turbulence and of geophysical quantities.
But elsewhere the best-known examples of multifractals
constructed by multiplicative processes continued to suggest
that qcrit <  is an anomaly.
I have always believed the opposite: that the anomaly
resides in qcrit =  and is brought in by the strong constraints
inherent in all b-adic cascades. It is a pleasure to report that
this belief is buttressed by a strong new argument that has
materialized very recently, to which we proceed.
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QUANTITATIVE FINANCE Scaling in financial prices: II. Multifractals and the star equation
6.1. From cascades to independent cylindrical pulses
After k stages, a cascade generates a measure k(t) whose
density, as written in section 5.2, is
k[dt]bk
= W(1)W(1, 2) . . . W(1 . . . 2 . . . k).
Let us reinterpret this density in terms of a random function
W(t) defined for t > 0 as follows. It is constant in every
open interval between integers and its values in different interinteger
intervals are statistically independent. Define W(p)
as being statistically independent realizations of W(t). This
notation yields the alternative representation
k[dt]bk
= W(1)
(bt)W(2)
(b2
t) . . . W(k)
(bk
t).
In this representation, the term W(1)
(bt) only depends on
the first digit  in the development of t in the base b; the term
W(k)
(bk
t)onlydependsonthefirstk digitsinthatdevelopment.
An `integer-bound cylindrical pulse' will now be defined
as equal to 1 except in one inter-integer interval, where it is
equal to W. Each W(k)
(bk
t), and therefore also the density
of k in a b-adic interval, can now be restated as a product of
pulses bound to intervals between successive multiples of b-k
.
For every t, the number of pulses = 1 is exactly k.
While the restriction of the pulses to a b-adic grid is a
basic feature of the cascades of section 5, let us say once again
that it is completely artificial. It was introduced to follow the
old example of the binomial measure and preserved solely for
the sake of convenience. Indeed, both description and rigorous
mathematical proof are simplified by the fact that every b-adic
interval dt of length b-k
is affected by the same number k of
`low-frequency' pulses of length b-h
with h < k.
By now, these advantages have been exhausted and the
time has come to `unleash' the pulses from the constraints of
the b-adic grid.
6.2. Multifractal products of cylindrical pulses,
MPCP (Barral and Mandelbrot 2000)
To prepare for a weakening of the convenient but arbitrary
constraints due to the b-adic grid, let us rethink the nature of
the pulses implied in the restatement of a cascade in section
6.1. In a first step, a pulse of starting point t and length
2 = b-k
is represented in the plane by the `address point'
of coordinates t + b-k
/2 = t +  and b-k
/2 = . This is a
point in the `address plane' that is defined as the half square
{0 < x < 1; 0 <  < 1/2}.
In a cascade, the pattern of address points is extremely
strict and regular. To minimize artificiality, it should be
`softened' and randomized. I proposed one way that leads
to a `multifractal product of cylindrical pulses'. It consists
in replacing the cascade's very strict pattern with a Poisson
random pattern whose density is more or less the same but
is smooth and also extends to the unboundedly wide strip
0 <  < 1. Experience acquired in other problems made
me select the density

22
dtdx,
where the new parameter  is a counterpart of 1/ loge b.
An immediate consequence concerns the number of pulses
of length > dt that affect an interval of length dt. In section 5.2
it was the same k for all b-adic intervals of length b-k
. Now,
far more realistically, it is made into a Poisson random variable.
The next issue is the choice of rules to govern the weights
of the eddies. The only possibility is to follow the canonical
cascades and make the weights into independent random
variables. The resulting `MPCP process' was worked out,
first heuristically, then rigorously as reported in Barral and
Mandelbrot (2000).
The first principal novelty is that the condition EW = 1,
which was needed for cascades, is no longer necessary.
The second principal novelty resides in the form of the
function (q) to which MPCP leads. It takes the thoroughly
new form
(q) = -1 + q[1 - (EW - 1)] - [EWq
- 1].
-EWq
 -q max W as q  . Therefore, the
existence of qcrit <  is no longer an odd possibility due
to max W being very large and possibly questionable. For
(q) to become negative for large q, a broad sufficient and
very natural condition is max W > 1. A narrower sufficient
condition is EW = 1 and W  1.
For example, consider the weight W that can only take
the non-zero values 1 + W and 1 - W. In a cascade, this
weight leads to a binomial measure that has been randomized.
Not only does it preserve the property that qcrit = , but it
is a prototype of that standard property. In contrast, in the
corresponding MPCP process, even this W leads to qcrit < .
An especially novel possibility opened by pulses is
that W itself can be non-random! In a cascade, nonrandomness
implies W  1, hence the measure remains
uniform throughout the cascade. For MPCP, uniformity
follows only under much stronger conditions. The change over
from the canonical cascade to MPCP also changes the source
of randomness: the fact that the number of pulses that affect
an interval follows the Poisson distribution suffices to generate
an interesting multifractal characterized by qcrit < .
6.3. Beyond the MPCP
Pulses with a constant value are called `cylindrical' in sections
6.1 and 6.2 in view of extensions to higher dimensions. The
MPCP have already been generalized by considering pulses of
more varied form. A description of those elaborations would
be out of place here.
6.4. Remarks
The reader familiar with details of statistical thermodynamics
will observe that the MPCP model introduced the Poisson
distribution to create a resemblance with `grand canonical
ensembles', thereby closing an earlier progression from
microcanonical to canonical.
The sequence from `microcanonical' to `canonical' and
on to MPCP, teaches several lessons. As the processes'
randomness becomes increasingly unconstrained, qcrit < 
becomes an increasingly general rule with increasingly special
exceptions.
129
B B Mandelbrot QUANTITATIVE FINANCE
The second lesson concerns the value of generality. It
might deserve to be cherished by mathematicians but there are
many concrete cases where it proves counter-productive. The
general theory of multifractals proved significantly less rich in
structure than special cases that were designed for the needs
of turbulence and later extended, most recently to finance.
7. Concluding comments
Brownian motion and the resulting ideal market hypothesis has
a beautiful mathematical theory, but predicts unrealistically
low risks. My multifractal model is not the last word but looks
promising.
Implications of multifractality. The good fit of data to
the fractional Brownian motion in multifractal time raises an
endless string of hard conceptual issues. Enough mathematical
properties of the multifractals are already known and the
statistical procedures are sufficiently developed to allow the
investigation of price records to proceed well beyond pictures
like figures 2 and 3 (in part I) and to show that many price
series are indeed multifractal. Without waiting for more
mathematics, one can use Monte Carlo calculations to help
assess portfolio risks.
Louis Bachelier. This man's story is told in Mandelbrot
(1982)pages392and408. Hisnameismysterioustophysicists
and his work did not influence physics. But his PhD thesis in
mathematics was both the first work in quantitative finance and
the first treatment of Brownian motion. This last term came up
only after this process was rediscovered in statistical physics,
by Einstein in 1905, and explored mathematically by Wiener
in the 1920s. This is why the term `Wiener Brownian motion'
provides a good contrast to the fractional Brownian motion
that I introduced much later.
Question: `Granted that only a few physicists bothered
about economics and finance until recently, has statistical
physics contributed to economics and finance?' The
answer is yes, but what this question takes for granted is not
true: physics (statistical or not) has influenced economics on
innumerable occasions in the past and very deeply. Bachelier
rightly prided himself in having extended the notion of
diffusion to probability, but diffusion is not part of `statistical
physics'. My work around 1960 was informed of the statistical
physics and stood before the flowering of the study of critical
phenomena. In its own way, it used the powerful tools provided
by renormalization, fixed points and scaling.
Converse question: `Has the study of finance based on
scaling etc contributed notions of its own that might
in the future be transferred to core physics?' The
answer is yes, on several accounts. The distinction between
mild, slow and wild variability and randomness arose in
economics/finance, but there is increasing evidence that
it may help in core physics. It is still little known
but deserves close attention on the part of scientists. I
think that by identifying which phenomena are `wildly
variable', science will be better able to feel the boundary
of what seems a frontier of sharply increased difficulty.
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