  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.
