The escape problem under stochastic volatility: the Heston model

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.

29 pages, 12 figures

References in corpus (10)

• Hitting Time Distributions in Financial Markets
• Mean Escape Time in a System with Stochastic Volatility
• Role of Noise in a Market Model with Stochastic Volatility
• Comparison between the probability distribution of returns in the Heston model and empirical data for stock indexes
• Multiple time scales and the exponential Ornstein-Uhlenbeck stochastic volatility model
• Scaling and data collapse for the mean exit time of asset prices
• Extreme times in financial markets
• A path integral approach to closed-form option pricing formulas with applications to stochastic volatility and interest rate models
• Volatility: a hidden Markov process in financial time series
• Extreme times for volatility processes

Cited by in corpus (10)

• The stabilizing effect of volatility in financial markets
• First-passage and risk evaluation under stochastic volatility
• Path integral approach to Asian options in the Black-Scholes model
• Non-stationary Feller process with time-varying coefficients
• Epstein-Zin Utility Maximization on a Random Horizon
• Extreme values and the level-crossing problem. An application to the Feller process
• Quantum Neural Computation for Option Price Modelling
• Default risk modeling beyond the first-passage approximation: Position-dependent killing
• Mean first passage times of higher-dimensional velocity jump processes
• Adaptive-Wave Alternative for the Black-Scholes Option Pricing Model