General Asymptotics of Wiener Functionals and Application to Mathematical Finance

Abstract

In the present paper, we give an asymptotic expansion of probability density for a component of general diffusion models. Our approach is based on infinite dimensional analysis on the Malliavin calculus and Kusuoka-Stroock's asymptotic expansion theory for general Wiener functionals. The initial term of the expansion is given by the 'energy of path' and we calculate the energy by solving Hamilton equation. We apply our approach to the problems of mathematical finance. In particular, we obtain general asymptotic expansion formulae of implied volatilities for general diffusion models, e.g. CEV model, displaced diffusion and SABR model.

Keywords: Wiener functional, stochastic volatility, Hamilton equation, Malliavin calculus, Asymptotic approximation, SABR model

JEL Classification: G12, G13