Masashi Toda

Computation in an Asymptotic Expansion Method

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 ƒE-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.

A GENERAL COMPUTATION SCHEME FOR A HIGH-ORDER ASYMPTOTIC EXPANSION METHOD

This 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 (1992), Yoshida (1992b) and Takahashi (1995, 1999) 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. Takahashi (1995, 1999) and Takahashi and Takehara (2007) 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 λ-SABR model up to the fifth order.

NOTE ON AN EXTENSION OF AN ASYMPTOTIC EXPANSION SCHEME

This paper presents an extension of a general computational scheme for asymptotic expansions proposed in earlier works by the authors and coworkers. In the earlier works, a new method was developed for the computation of an arbitrary-order expansion with a normal benchmark distribution in a multidimensional diffusion setting. In particular, a new algorithm was proposed for calculating coefficients in an expansion by solving a system of ordinary differential equations. In the present note, by a change of variable technique, and by various ways of setting the perturbation parameters in the expansion, we provide the flexibility of setting the benchmark distribution around which the expansion is made and an automatic way for computation up to any order in the expansion. For instance, we introduce new expansions, called the lognormal expansion and the CEV expansion. We also show some concrete examples with numerical experiments, which imply that a high-order CEV expansion will produce more a precise and stable approximation for option pricing under the SABR model than other approximation methods such as the log-normal expansion and the well-known normal expansion.