An Asymptotic Expansion with Push-Down of Malliavin Weights

Abstract

This paper derives asymptotic expansion formulas for option prices and implied volatilities as well as the density of the underlying asset price in a stochastic volatility model. In particular, the integration-by-parts formula in Malliavin calculus and the push-down of Malliavin weights are effectively applied, which provides an expansion formula for generalized Wiener functionals and the closed-form approximation formulas in stochastic volatility environment.In addition, it presents applications of the general formula to a local volatility expansion in the stochastic volatility model and expansions of option prices in the shifted log-normal and jump-diffusion models with stochastic volatilities. Finally, with an application of the Bismut identity the paper shows an expansion of the solution to the partial differential equation for pricing in a stochastic volatility model.