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.