Stochastic Volatility Processes

Abstract

In a stochastic volatility process, the positivity and mean reversion of the volatility should be enforced. The mean reversion can be achieved by the drift, equivalent to an Ornstein–Uhlenbeck process. The positivity can be enforced either by an exponential or by taming down the stochastic term by the volatility as done in the Heston process. Both classes of processes are investigated, in the simpler one time scale version, and in a multiscale generalization. Both structures lead to similar mug shots, with an exponential memory or a long-term memory. Yet, all of them are time-reversal invariant, in disagreement with the stylized facts.