Extremes of regularly varying stochastic volatility fields

Abstract

We consider a stationary stochastic volatility field $Y_vZ_v$ with $v\in\mathbb{Z}^d$, where $Z$ is regularly varying and $Y$ has lighter tails and is independent of $Z$. We make - relative to existing literature - very general assumptions on the dependence structure of both fields. In particular this allows $Y$ to be non-ergodic, in contrast to the typical assumption that it is i.i.d., and $Z$ to be given by an infinite moving average. Considering the stochastic volatility field on a (rather general) sequence of increasing index sets, we show the existence and form of a $Y$-dependent extremal functional generalizing the classical extremal index. More precisely, conditioned on the field $Y$, the extremal functional shows exactly how the extremal clustering of the (conditional) stochastic volatility field is given in terms of the extremal clustering of the regularly varying field $Z$ and the realization of $Y$. Secondly, we construct two different cluster counting processes on a fixed, full-dimensional set with boundary of Lebesgue measure zero: By means of a coordinate-dependent upscaling of subsets, we systematically count the number of relevant clusters with an extreme observation. We show that both cluster processes converge to a Poisson point process with intensity given in terms of the extremal functional.