CIRJE-F-815 "On Approximation of the Solutions to Partial Differential Equations in Finance"
Author Name Takahashi, Akihiko and Toshihiro Yamada
Date August 2011
Full Paper   PDF file
Remarks Revised in January 2012; subsequently published in Recent Advances in Financial Engineering 2011, pp.133-181, 2011.   

This paper proposes a general approximation method for the solutions to second-order parabolic partial differential equations (PDEs) widely used in finance through an extension of Léandre's approach(Léandre (2006,2008)) and the Bismut identiy(e.g. chapter IX-7 of Malliavin (1997)) 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 new approximation formulas for plain-vanilla and barrier option prices under stochastic volatility models. 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 (λ-)SABR models (Hagan (2002), Labordere (2008)) as special cases. Some numerical examples are shown.