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


  • Akihiko Takahashi

    (Faculty of Economics, The University of Tokyo)

  • Toshihiro Yamada

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


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

Suggested Citation

  • Akihiko Takahashi & Toshihiro Yamada, 2012. "A Remark on Approximation of the Solutions to Partial Differential Equations in Finance," CARF F-Series CARF-F-273, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Mar 2012.
  • Handle: RePEc:cfi:fseres:cf273

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    References listed on IDEAS

    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters,in: Theory Of Valuation, chapter 8, pages 229-288 World Scientific Publishing Co. Pte. Ltd..
    2. Eric Fournié & Jean-Michel Lasry & Pierre-Louis Lions & Jérôme Lebuchoux, 2001. "Applications of Malliavin calculus to Monte-Carlo methods in finance. II," Finance and Stochastics, Springer, vol. 5(2), pages 201-236.
    3. 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.
    4. 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 637-654, May-June.
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    Cited by:

    1. Ning Cai & Chenxu Li & Chao Shi, 2014. "Closed-Form Expansions of Discretely Monitored Asian Options in Diffusion Models," Mathematics of Operations Research, INFORMS, vol. 39(3), pages 789-822, August.

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