Stochastic Volatility Models: Extremal Behavior

Stochastic volatility determines, as a rule, the extreme risk in price fluctuations. We review some of the most important stochastic volatility models concerning their extreme behaviour. This includes the tail behaviour as well as the cluster possibilities of such models. The following pattern is common for discretetime and continuous-time models. In linear models the volatility inherits the tails from the driving noise sequence, and clusters occur basically through the driving noise, whenever it is heavy-tailed. In non-linear models like GARCH and COGARCH models heavy tails and clusters in extremes occur from the feed-back structure of the underlying random recurrence equation.