A General Computation Scheme for a High-Order Asymptotic Expansion Method
AbstractThis 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 , Yoshida  and Takahashi ,  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. ,  and Takahashi and Takehara  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 ƒÉ-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 InfoPaper provided by Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo in its series CARF F-Series with number CARF-F-242.
Length: 26 pages
Date of creation: Feb 2011
Date of revision: Jul 2011
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