A GENERAL COMPUTATION SCHEME FOR A HIGH-ORDER ASYMPTOTIC EXPANSION METHOD

Akihiko Takahashi, Kohta Takehara () and Masashi Toda
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Akihiko Takahashi: Graduate School of Economics, University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
Kohta Takehara: Graduate School of Economics, University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan
Masashi Toda: Graduate School of Economics, University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

International Journal of Theoretical and Applied Finance (IJTAF), 2012, vol. 15, issue 06, 1-25

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. Takahashi (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 λ-SABR model up to the fifth order.

Keywords: Asymptotic expansion; Malliavin calculus; approximation formula; stochastic volatility; λ-SABR model (search for similar items in EconPapers)
Date: 2012
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Citations: View citations in EconPapers (32)

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DOI: 10.1142/S0219024912500446

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