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

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  • Akihiko Takahashi

    (Faculty of Economics, University of Tokyo)

  • Kohta Takehara

    (Graduate School of Economics, University of Tokyo)

  • Masashi Toda

    (Graduate School of Economics, University of Tokyo)

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    Abstract

    This paper presents a new computational scheme for an asymptotic expansion method of an arbitrary order. An asymptotic expansion method in finance initiated by Kunitomo and Takahashi[9], Yoshida[34] and Takahashi [20], [21] is a widely applicable methodology for an analytic approximation of the expectation of a certain functional of diffusion processes and 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 highdimensional underlying stochastic environments. In practical applications of the expansion, the crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. [20], [21] and Takahashi and Takehara [23] provided explicit formulas of 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 a high-order expansion: This 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. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for the -SABR model up to the fifth order and a cross-currency Libor market model with a general stochastic volatility model of the spot foreign exchange rate up to the fourth order.

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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-787.

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    Length: 22 pages
    Date of creation: Feb 2011
    Date of revision:
    Handle: RePEc:tky:fseres:2011cf787

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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.
    2. Akihiko Takahashi & Nakahiro Yoshida, 2005. "Monte Carlo Simulation with Asymptotic Method," CIRJE F-Series CIRJE-F-335, CIRJE, Faculty of Economics, University of Tokyo.
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