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

Akihiko Takahashi and Toshihiro Yamada
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Akihiko Takahashi: Faculty of Economics, The University of Tokyo
Toshihiro Yamada: Mitsubishi UFJ Trust Investment Technology Institute Co.,Ltd. (MTEC)

No CARF-F-273, CARF F-Series from Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo

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 timehomogenous local volatility and local-stochastic volatility models in finance, which include Heston (Heston (1993)) and (ă-)SABR models (Hagan et.al. (2002), Labordere (2008)) as special cases. Some numerical examples are shown.

Pages: 35 pages
Date: 2012-02, Revised 2012-03
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Citations: View citations in EconPapers (7)

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