Jean-Philippe Bouchaud, Lecture 3: Models with Time-Varying Volatility
Ken Gosier
Investment Technology Group, Boston, MA April 19, 2001
In this lecture we study several different financial time series models which include time-varying volatility. We study the resulting effects on the autocorrelations and moments of the distribution.
1 Time-Varying Volatility with Volatility Correlation
In the first model we define the increments Sxi by (1)                                                           Sxi = elal
The cumulative increment Ax^ is given by
		N
(2)		Arc at = Yl &xi i-l
where		
(3)	<	t   rr     ^            -^   rr     ^
		O a O a    ^             <->   O A     *>            ^        |   .              . . r      J \
for v < 1. In this model,	we have for the second and fourth moments	
(4)		< Ax2N >   ~ N
(5)		< Ax% >   ~ N2 + N2-"
which implies for the kurtosis		
(6)		< Ax% > - < Ax% >2
		< Ax% >2
1
Spring 2001, Jean-Philippe Bouchaud         Real-World Lectures         Lecture #3           2
(7)                                                                 k ~ N~u Note that for Gaussian iid increments, we have
(8)                                                         <Ax%>   ~ Nql2 and more generally,
(9)                                                         <Ax%>   ~ N<®
where ((q) is a characteristic of the specific distribution.
2 Multi-Fracticality
We next study the multi-fractal model of Bacry-Delour-Muzy, which may be found online at http://xxx.lanl.gov/archive/cond-mat. In this model, we have
(10)                                                               8xi = e,em where the u>j's are Gaussian, and
(11)                                                  < WiWj >= -A2 log
N-il + i
for \i — j\ < T, where T is a parameter to be specified for the series. For \i — j\ > T, we have < WiWj >= 0. The moments of this time series have the form
(12)                                                     < Axq >   - AqN^
for 1 <C N <C T. That is, a large number of steps have been taken, but still many less steps than the correlation time T. The exponent C(q) is given by
(13)                                                     C(q) = | - X2q(q ~ 2)
for q < q*, where q* is a parameter not specified in the lecture.  For q > q*, the moments diverge.
3 Relative vs. Absolute Increments, "Retarded Volatility" Model
Time-dependent volatility models may describe the absolute or relative price increments. We may have either one of
(14)                                                                           Sxi = o%e%
ox ■
(15)                                                                                    — = (Jz€z
Jbi
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Figure 1: The example k(j — i) = (l/2)a3~l,a = 0.5. The x-axis plots the lagged time j — i, and the y-axis shows the corresponding values of k. Note that k is normalized to sum to 1.
The absolute model (14) is found to be more valid over short time scales, while the relative model (15) is better for longer time scales. A simple example of a model of the type (15) is Geometric Brownian motion
Sxi
.Li
(16)                                                                    — = ej(Jo
whose relative increments have constant volatility through time.
A generalization of the time-dependent volatility is given by the "Retarded Volatility" model,
OX<i — X^ Cj(Jj
(17)
X,
R
E «0')

1-3
J=l
where the /ťs are chosen so that Y^L\ k(í) = 1- An example k{i) is graphed in Fig. (1). The Retarded Volatility model may be used to "interpolate" between the models for time-varying volatility of absolute vs. relative price increments.