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On Approximation of the Solutions to Partial Differential Equations in Finance

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  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Toshihiro Yamada

    (Mitsubishi UFJ Trust Investment Technology Institute Co.,Ltd.)

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    Abstract

    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 et.al. (2002), Labordere (2008)) as special cases. Some numerical examples are shown.

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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2011/2011cf815.pdf
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    Bibliographic Info

    Paper provided by CIRJE, Faculty of Economics, University of Tokyo in its series CIRJE F-Series with number CIRJE-F-815.

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    Length: 44 pages
    Date of creation: Aug 2011
    Date of revision:
    Handle: RePEc:tky:fseres:2011cf815

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    1. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
    2. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux & Nizar Touzi, 1999. "Applications of Malliavin calculus to Monte Carlo methods in finance," Finance and Stochastics, Springer, vol. 3(4), pages 391-412.
    3. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
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