First-passage and risk evaluation under stochastic volatility

TL;DR

This paper derives exact solutions for the first-passage problem in the Heston stochastic volatility model, revealing high-risk scenarios and the absence of a mean first-passage time, with implications for risk management in financial markets.

Contribution

It provides the first analytical expressions for first-passage probabilities in the Heston model and explores their asymptotic behaviors, highlighting extreme risk deviations.

Findings

Exact formulas for survival and hitting probabilities

High risk of default linked to volatility fluctuations

No finite mean first-passage time in certain regimes

Abstract

We solve the first-passage problem for the Heston random diffusion model. We obtain exact analytical expressions for the survival and 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 non-existence 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 believe that this may provide an effective tool for risk control which can be readily applicable to real markets.