# Evaluation of extremal properties of GARCH(p,q) processes

## TL;DR

This paper develops new methods to analyze the extremal properties of GARCH(p,q) processes, enabling better understanding of tail behavior and clustering in financial return models, including complex cases previously uncharacterized.

## Contribution

The paper introduces reliable algorithms for evaluating tail indices, extremal index, and cluster functionals for all GARCH(p,q) processes, extending analysis to broader classes of stochastic recurrence equations.

## Key findings

- Derived the forward tail chain for all GARCH(p,q) processes.
- Provided methods to evaluate extremal index and tail behavior.
- Extended extremal property analysis to integrated and asymmetric GARCH processes.

## Abstract

Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling features commonly found in observed financial returns. The extremal properties of these processes are of considerable interest for market risk management. For the simplest GARCH(p,q) process, with max(p,q) = 1, all extremal features have been fully characterised. Although the marginal features of extreme values of the process have been theoretically characterised when max(p, q) >= 2, much remains to be found about both marginal and dependence structure during extreme excursions. Specifically, a reliable method is required for evaluating the tail index, which regulates the marginal tail behaviour and there is a need for methods and algorithms for determining clustering. In particular, for the latter, the mean number of extreme values in a short-term cluster, i.e., the reciprocal of the extremal index, has only been characterised in special cases which exclude all GARCH(p,q) processes that are used in practice. Although recent research has identified the multivariate regular variation property of stationary GARCH(p,q) processes, currently there are no reliable methods for numerically evaluating key components of these characterisations. We overcome these issues and are able to generate the forward tail chain of the process to derive the extremal index and a range of other cluster functionals for all GARCH(p, q) processes including integrated GARCH processes and processes with unbounded and asymmetric innovations. The new theory and methods we present extend to assessing the strict stationarity and extremal properties for a much broader class of stochastic recurrence equations.