  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.
