New Unified Computational Algorithm in a High-Order Asymptotic Expansion Scheme

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

Chapter 10 in Recent Advances in Financial Engineering 2009, 2010, pp 231-251 from World Scientific Publishing Co. Pte. Ltd.

Abstract: AbstractAn 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's 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: Financial Engineering; Mathematical Finance; Credit Risk; Real Options; Optimal Investment; Heterogeneous Beliefs (search for similar items in EconPapers)
Date: 2010
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