Computation in an Asymptotic Expansion Method

CARF Working Paper Series CARF-F-149

51 Pages Posted: 8 Jun 2009 Last revised: 25 Aug 2011

See all articles by Akihiko Takahashi

Akihiko Takahashi

University of Tokyo - Faculty of Economics

Kohta Takehara

University of Tokyo - Graduate School of Economics

Masashi Toda

University of Tokyo - Graduate School of Economics

Date Written: May 31, 2009

Abstract

An asymptotic expansion scheme in finance initiated by Kunitomo and Takahashi [15] and Yoshida [68] is a widely applicable methodology for analytic approximation of the expectation of a certain functional of diffusion processes. [46], [47] and [53] provide explicit formulas of conditional expectations necessary for the asymptotic expansion up to the third order. In general, the crucial step in practical applications of the expansion is calculation of conditional expectations for a certain kind of Wiener functionals. This paper presents two methods for computing the conditional expectations that are powerful especially for high order expansions: The first one, an extension of the method introduced by the preceding papers presents a general scheme for computation of the conditional expectations and show the formulas useful for expansions up to the fourth order explicitly. 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, stochastic volatility, λ-SABR model, Libor market model, Malliavin calculus, cross currency model, foreign exchange rate option (Forex Option)

Suggested Citation

Takahashi, Akihiko and Takehara, Kohta and Toda, Masashi, Computation in an Asymptotic Expansion Method (May 31, 2009). CARF Working Paper Series CARF-F-149, Available at SSRN: https://ssrn.com/abstract=1413924 or http://dx.doi.org/10.2139/ssrn.1413924

Akihiko Takahashi (Contact Author)

University of Tokyo - Faculty of Economics ( email )

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

Kohta Takehara

University of Tokyo - Graduate School of Economics ( email )

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