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A General Computation Scheme for a High-Order Asymptotic Expansion Method

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Author Info

  • Akihiko Takahashi

    (Faculty of Economics, The University of Tokyo)

  • Kohta Takehara

    (The University of Tsukuba)

  • Masashi Toda

    (Faculty of Economics, The University of Tokyo)

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    Abstract

    This paper presents a new computational scheme for an asymptotic expansion method of an arbitrary order. The asymptotic expansion method in finance initiated by Kunitomo and Takahashi [1992], Yoshida [1992b] and Takahashi [1995], [1999] is a widely applicable methodology for an analytic approximation of expectation of a certain functional of diffusion processes. Hence, not only academic researchers but also many practitioners have used the methodology for a variety of financial issues such as pricing or hedging complex derivatives under high-dimensional underlying stochastic environments. In practical applications of the expansion, a crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. [1995], [1999] and Takahashi and Takehara [2007] provided explicit formulas for those conditional expectations necessary for the asymptotic expansion up to the third order. This paper presents the new method for computing an arbitrary-order expansion in a general diffusion-type stochastic environment, which is powerful especially for high-order expansions: We develops a new calculation algorithm for computing coefficients of the expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations directly. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for a lambda-SABR model up to the fifth order.

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    File URL: http://www.carf.e.u-tokyo.ac.jp/pdf/workingpaper/fseries/284.pdf
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    Bibliographic Info

    Paper provided by Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo in its series CARF F-Series with number CARF-F-272.

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    Length: 22 pages
    Date of creation: Feb 2012
    Date of revision:
    Handle: RePEc:cfi:fseres:cf272

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    References

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    1. Yoshifumi Muroi, 2005. "Pricing contingent claims with credit risk: Asymptotic expansion approach," Finance and Stochastics, Springer, vol. 9(3), pages 415-427, 07.
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    Cited by:
    1. Akihiko Takahashi & Toshihiro Yamada, 2013. "On Error Estimates for Asymptotic Expansions with Malliavin Weights ï¼ Application to Stochastic Volatility Model ï¼," CIRJE F-Series CIRJE-F-897, CIRJE, Faculty of Economics, University of Tokyo.

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