The Mathematics of Financial Modelingand Investment Management

13-Fat Tails-Scaling-Stabl Page 382 Wednesday, February 4, 2004 1:00 PM

382 The Mathematics of Financial Modeling and Investment Management

Xt = j

p

∑ L

iX

t +^

q

∑ LZt

i = 1 j = 1

The theory of ARMA processes developed in Chapters 11 and 12

can be carried over at least partially to cover the case of fat-tailed inno-

vations. In particular, an ARMA(p,q) process with IID α -stable innova-

tions admits a stationary, infinite moving average representation under

the same conditions as in the classical finite-variance case. The coeffi-

cients of the moving average satisfy the condition

∞

∑ hi

α ∞ <

i = 0

In the case of fat-tailed innovations, covariances, and autocovariances

looses their meaning. It can also be demonstrated, however, that the

empirical autocorrelation function is meaningful and is asymptotically

normal. It can be demonstrated that maximum likelihood estimates can be

extended to the infinite variance case, though through a number of ad hoc

processes.

ARCH/GARCH Processes

As we saw in Chapter 12, The simplest ARCH model can be written as

follows. Suppose that X is the random variable to be modeled, Z is a

sequence of independent standard normal variables, and σ is a hidden

variable. The ARCH(1) model is written as

Xt = σ (^) tZt
σ
2 2
t = βδ + Xt– 1
This basic model was extended by Bollerslev^17 who proposed the
GARCH(p,q) model written as
Xt = σ (^) tZt
(^17) Tim Bollerslev, “Generalized Autoregressive Conditional Heteroscedasticity,”
Journal of Econometrics 31 (1989), pp. 307–327.