Extremes of regularly varying stochastic volatility fields

Abstract

We consider a stationary stochastic volatility field YvZv with v ∈ 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 nonergodic, 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.