The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 383 Wednesday, February 4, 2004 1:00 PM

Fat Tails, Scaling, and Stable Laws 383

p q

σ^2 22

t = β + ∑γ iσti– + ∑δiXti–

i = 1 i = 1

The IID variables Z can be standard normal variables or other symmet-

rical, eventually fat-tailed, variables.

Let’s first observe that model parameters must be constrained in

order to guarantee the stationarity of the model. Stationarity conditions

depend on each model. No general simple expression for the stationarity

conditions is available.

Due to the multiplicative nature of noise, GARCH models are able

to generate fat-tailed distributions even if innovations have finite vari-

ance. This fact was established by Kesten^18 in 1973. The tail index can

be theoretically computed at least in the case GARCH(1,1). Suppose a

GARCH(1,1) stationary process with Gaussian innovation is given. It

can be demonstrated that

(

c

PX > x) ≈ ---x

• 2 κ
  2

where κ is the solution of an integral equation. In the generic p, q case,

the return process is still fat-tailed but no practical way to compute the

index from model parameter is known.

Subordinated Processes

Subordinated processes allow the time scale to vary. Subordinated mod-

els are, in a sense, the counterpart of stochastic volatility models insofar

as they model the change in volatility by contracting and expanding the

time scale. The first model was proposed in 1973 by Clark.^19 Subordi-

nated models have been extensively studied by Ghysels, Gourieroux,

and Josiak.^20

Subordinated models can be applied quite naturally in the context

of trading. Individual trades are randomly spaced. In modern electronic

exchanges, the time and size of trades are individually recorded thus

allowing for accurate estimates of the distributional properties of inter-

trades intervals. Consideration of random spacings between trades natu-

(^18) H. Kesten, “Random Difference Equations and Renewal Theory for Products of
Random Matrices” Acta Mathematica 131 (1973), pp. 207–248.
(^19) P.K. Clark, “A Subordinated Stochastic Process Model with Finite Variance for
Speculative Prices,” Econometrica 41 (January 1973), pp. 735–755.
(^20) E. Ghysels, C. Gourieroux, and J. Josiak, “Market Time and Asset Price Move-
ment Theory and Estimation,” Working Paper 95-32 Cyrano, Montreal, 1995.