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

34 Pages Posted: 19 Feb 2012

See all articles by Akihiko Takahashi

Akihiko Takahashi

University of Tokyo - Faculty of Economics

Toshihiro Yamada

Hitotsubashi University

Date Written: February 19, 2012

Abstract

This 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 Leandre's approach (Leandre, 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 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.

Keywords: Malliavin calculus, Bismut indentity, Integration-by-parts, Semigroup, Asymptotic expansion, Short time asymptotics, Heat kernel expansions, Derivatives pricing, Stochastic volatility, Local volatility, SABR model, lambda-SABR models, Heston model

Suggested Citation

Takahashi, Akihiko and Yamada, Toshihiro, A Remark on Approximation of the Solutions to Partial Differential Equations in Finance (February 19, 2012). Available at SSRN: https://ssrn.com/abstract=2007584 or http://dx.doi.org/10.2139/ssrn.2007584

Akihiko Takahashi

University of Tokyo - Faculty of Economics ( email )

7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033
Japan

Toshihiro Yamada (Contact Author)

Hitotsubashi University ( email )

2-1 Naka Kunitachi-shi
Tokyo 186-8601
Japan

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