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Extreme Value Theory for GARCH Processes

In: Handbook of Financial Time Series

Author

Listed:
  • Richard A. Davis

    (Columbia University, Department of Statistics)

  • Thomas Mikosch

    (University of Copenhagen, Laboratory of Actuarial Mathematics)

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 distributions. 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 a GARCH 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 described 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 measure of a Poisson process and a cluster distribution.

Suggested Citation

  • Richard A. Davis & Thomas Mikosch, 2009. "Extreme Value Theory for GARCH Processes," Springer Books, in: Thomas Mikosch & Jens-Peter Kreiß & Richard A. Davis & Torben Gustav Andersen (ed.), Handbook of Financial Time Series, chapter 8, pages 187-200, Springer.
    • Handle: RePEc:spr:sprchp:978-3-540-71297-8_8
    DOI: 10.1007/978-3-540-71297-8_8
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