The escape problem under stochastic volatility: the Heston model

TL;DR

This paper derives exact solutions for the escape problem in the Heston stochastic volatility model, analyzing survival probabilities and mean exit times, and compares these with the Wiener process to reveal effects of stochastic volatility.

Contribution

It provides the first exact expressions for survival probability and mean exit time in the Heston model, including averaging over volatility and asymptotic analysis.

Findings

Mean exit time grows quadratically with large spans.

Stochastic volatility increases survival and prolongs escape time.

Results differ from classical Wiener process predictions.

Abstract

We solve the escape problem for the Heston random diffusion model. We obtain exact expressions for the survival probability (which ammounts 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 volatility. We look over 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 assymptotic 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 apparent paradoxical way, increases survival and prolongs the escape time.