Escape problem under stochastic volatility: The Heston model

Abstract

We solve the escape problem for the Heston random diffusion model from a finite interval of span L . We obtain exact expressions for the survival probability (which amounts to solving the complete escape problem) as well as for the mean exit time. We also average the volatility in order to work out the problem for the return alone regardless of volatility. We consider these results in terms of the dimensionless normal level of volatility-a ratio of the three parameters that appear in the Heston model-and analyze their form in several asymptotic limits. Thus, for instance, we show that the mean exit time grows quadratically with large spans while for small spans the growth is systematically slower, depending on the value of the normal level. We compare our results with those of the Wiener process and show that the assumption of stochastic volatility, in an apparently paradoxical way, increases survival and prolongs the escape time. We finally observe that the model is able to describe the main exit-time statistics of the Dow-Jones daily index.