Extreme Value Theory for GARCH Processes

Abstract

We consider the extreme value theory for a stationary GARCH process with iid innovations. One of the basic ingredients of this theory is the fact that, under general conditions, GARCH processes have power law marginal tails and, more generally, regularly varying finite-dimensional distri- butions. Distributions with power law tails combined with weak dependence conditions imply that the scaled maxima of a GARCH process converge in distribution to a Fréchet distribution. The dependence structure of aGARCH process is responsible for the clustering of exceedances of a GARCH process above high and low level exceedances. The size of these clusters can be de- scribed by the extremal index.We also consider the convergence of the point processes of exceedances of a GARCH process toward a point process whose Laplace functional can be expressed explicitly in terms of the intensity mea- sure of a Poisson process and a cluster distribution.