A General Computation Scheme for a High-Order Asymptotic Expansion Method

Abstract

This paper presents a new computational scheme for an asymptotic expansion method of an arbitrary order. An asymptotic expansion method in finance initiated by Kunitomo and Takahashi[9], Yoshida[34] and Takahashi [20], [21] is a widely applicable methodology for an analytic approximation of the expectation of a certain functional of diffusion processes and 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 highdimensional underlying stochastic environments. In practical applications of the expansion, the crucial step is calculation of conditional expectations for a certain kind of Wiener functionals. [20], [21] and Takahashi and Takehara [23] provided explicit formulas of 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 a high-order expansion: This 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. To demonstrate its effectiveness, the paper gives numerical examples of the approximation for the 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.