On Error Estimates for Asymptotic Expansions with Malliavin Weights: Application to Stochastic Volatility Model

Abstract

This paper proposes a unified method for precise estimates of the error bounds in asymptotic expansions of an option price and its Greeks (sensitivities) under a stochastic volatility model. More generally, we also derive an error estimate for an asymptotic expansion around a general partially elliptic diffusion and a more general Wiener functional, which is applicable to various important valuation and risk management tasks in the financial business such as the ones for multidimensional diffusion and nondiffusion models. In particular, we take the Malliavin calculus approach, and estimate the error bounds for the Malliavin weights of both the coefficient and the residual terms in the expansions by effectively applying the properties of Kusuoka-Stroock functions introduced by Kusuoka [Kusuoka S (2003) Malliavin calculus revisited. J. Math. Sci. Univ. Tokyo 10:261–277.] functions. Moreover, a numerical experiment under the Heston-type model confirms the effectiveness of our method.