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

In: Recent Advances In Financial Engineering 2011

Author

Listed:
  • Akihiko Takahashi

    (Graduate School of Economics, the University of Tokyo, Japan)

  • Toshihiro Yamada

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

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 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.

Suggested Citation

  • Akihiko Takahashi & Toshihiro Yamada, 2012. "A Remark on Approximation of the Solutions to Partial Differential Equations in Finance," World Scientific Book Chapters, in: Akihiko Takahashi & Yukio Muromachi & Hidetaka Nakaoka (ed.), Recent Advances In Financial Engineering 2011, chapter 8, pages 133-181, World Scientific Publishing Co. Pte. Ltd..
    • Handle: RePEc:wsi:wschap:9789814407335_0008
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    Cited by:

    1. Akihiko Takahashi & Toshihiro Yamada, 2023. "Solving Kolmogorov PDEs without the curse of dimensionality via deep learning and asymptotic expansion with Malliavin calculus," Partial Differential Equations and Applications, Springer, vol. 4(4), pages 1-31, August.
    2. 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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