New Unifi ed Computational Algorithm in a High-Order Asymptotic Expansion Scheme

Proceedings of 2009 KIER-TMU International Workshop on Financial Engineering, Forthcoming

17 Pages Posted: 21 Mar 2010

See all articles by Kohta Takehara

Kohta Takehara

University of Tokyo - Graduate School of Economics

Akihiko Takahashi

University of Tokyo - Faculty of Economics

Masashi Toda

University of Tokyo - Graduate School of Economics

Date Written: January 17, 2010

Abstract

An asymptotic expansion scheme in finance initiated by Kunitomo and Takahashi [6] and Yoshida [29] is a widely applicable methodology for analytic approximation of the expectation of a certain functional of diffusion processes. Mathematically, this methodology is justified by Watanabe theory([27]) in Malliavin calculus. In practical applications, it is desirable to investigate the accuracy and stability of the method especially with expansion up to high orders in situations where the underlying processes are highly volatile as seen in the recent financial markets. Although Takahashi[17], [18] and Takahashi and Takehara [20] provided explicit formulas for the expansion up to the third order, to our best knowledge a general computation scheme for an arbitrary order expansion has not been given yet. This paper proposes two general methods for computing the conditional expectations that are powerful especially for high order expansions: The first one, as an extension of the method introduced by the preceding papers, presents a unified scheme for computation of the conditional expectations. The second one develops a new calculation algorithm for computing the coefficients of the expansion through solving a system of ordinary differential equations that is equivalent to computing the conditional expectations. To demonstrate their effectiveness, the paper gives numerical examples of the approximation for λ-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.

Keywords: Asymptotic Expansion, Malliavin Calculus, Approximation Formula, Stochastic Volatility, λ-SABR Model, Libor Market Model, Currency Options

Suggested Citation

Takehara, Kohta and Takahashi, Akihiko and Toda, Masashi, New Unifi ed Computational Algorithm in a High-Order Asymptotic Expansion Scheme (January 17, 2010). Proceedings of 2009 KIER-TMU International Workshop on Financial Engineering, Forthcoming, Available at SSRN: https://ssrn.com/abstract=1574409

Kohta Takehara

University of Tokyo - Graduate School of Economics ( email )

Yayoi 1-1-1
Bunkyo-ku
Tokyo, Tokyo 113-8657
Japan

Akihiko Takahashi (Contact Author)

University of Tokyo - Faculty of Economics ( email )

7-3-1 Hongo, Bunkyo-ku
Tokyo 113-0033
Japan

Masashi Toda

University of Tokyo - Graduate School of Economics ( email )

Hongo 7-3-1
Bunkyo-ku
Tokyo, Tokyo 113-0033
Japan

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