First-passage and risk evaluation under stochastic volatility

Abstract

We solve the first-passage problem for the Heston random diffusion model. We obtain exact analytical expressions for the survival and the hitting probabilities to a given level of return. We study several asymptotic behaviors and obtain approximate forms of these probabilities which prove, among other interesting properties, the nonexistence of a mean-first-passage time. One significant result is the evidence of extreme deviations-which implies a high risk of default-when certain dimensionless parameter, related to the strength of the volatility fluctuations, increases. We confront the model with empirical daily data and we observe that it is able to capture a very broad domain of the hitting probability. We believe that this may provide an effective tool for risk control which can be readily applicable to real markets both for portfolio management and trading strategies.

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