On Approximation of the Solutions to Partial Differential Equations in Finance

Abstract

This paper proposes a general approximation method for the solutions to second order parabolic partial differential equations (PDEs) by an extension of Leandre's approach and the Bismut identity in Malliavin calculus. We show two types of its applications, new approximations of derivatives prices and short-time asymptotic expansions of the heat kernel.In particular, we provide a new approximation formula for barrier option prices under a stochastic volatility model. We also derive short-time asymptotic expansions of the heat kernel under general time-homogenous local volatility and local-stochastic volatility models in finance which include Heston (Heston (1993)) and ($¥lambda$-) SABR models (Hagan et.al. (2002), Labordere (2008)) as special cases. Some numerical examples are shown.