A Remark on Approximation of the Solutions to Partial Differential Equations in Finance

Akihiko Takahashi and Toshihiro Yamada
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Akihiko Takahashi: Graduate School of Economics, the University of Tokyo, Japan
Toshihiro Yamada: Mitsubishi UFJ Trust Investment Technology Institute Co., Ltd. (MTEC), Japan

Chapter 8 in Recent Advances in Financial Engineering 2011, 2012, pp 133-181 from World Scientific Publishing Co. Pte. Ltd.

Abstract: AbstractThis paper proposes a general approximation method for the solution to a second-order parabolic partial differential equation (PDE) 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 present two types of its applications, approximations of derivatives prices and short-time asymptotic expansions of the heat kernel. In particular, we provide approximate formulas for option prices under local and stochastic volatility models. We also derive short-time asymptotic expansions of the heat kernel under general timehomogenous local volatility and local-stochastic volatility models in finance, which include Heston (Heston, 1993) and (λ-)SABR models (Hagan et al., 2002; Labordere, 2008) as special cases. Some numerical examples are shown.

Keywords: Financial Engineering; Mathematical Finance; Money & Banking; Risk Management; Real Option; Corporate Finance; Computational Finance (search for similar items in EconPapers)
Date: 2012
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