Stochastic volatility - Fast mean-reverting stochastic volatility processes in finance

Abstract

We derive a correction of the Black-Scholes price for a European option by introducing a stochastic process (namely Ornstein-Uhlenbeck process) for volatility instead of assuming that it is constant as in the Black-Scholes model. Our approach is relatively general since it does not require a specific description of the volatility process but only assumes its fast mean reversion. We also derive explicitly the volatility surface and show hedging strategies related to our model of volatility.Our work follows the models and ideas first pioneered by Fouque, Papanicolaou and Sircar, as we carry out asymptotic expansions on the mean reversion parameter. However, we go further in the order of derivation of the results as done in Conlon and Sullivan and Howison.