Extremal behavior of stochastic volatility models

Empirical volatility changes in time and exhibits tails, which are
heavier than normal. Moreover, empirical volatility has - sometimes
quite substantial - upwards jumps and clusters on high levels. We
investigate classical and non-classical stochastic volatility
models with respect to their extreme behavior. We show that
classical stochastic volatility models driven by Brownian motion
can model heavy tails, but obviously they are not able to model
volatility jumps. Such phenomena can be modelled by Levy driven
volatility processes as, for instance, by Levy driven
Ornstein-Uhlenbeck models. They can capture heavy tails and
volatility jumps. Also volatility clusters can be found in such
models, provided the driving Levy process has regularly varying
tails. This results then in a volatility model with similarly heavy
tails. As the last class of stochastic volatility models, we
investigate a continuous time GARCH(1,1) model. Driven by an
arbitrary Levy process it exhibits regularly varying tails,
volatility upwards jumps and clusters on high levels.