The Asymptotic Expansion Approach to the Valuation of Interest Rate Contingent Claims

We propose a new methodology for the valuation problem of financial contingent claims when the underlying asset prices follow a general class of continuous Itô processes. Our method can be applied to a wide range of valuation problems including complicated contingent claims associated with the term structure of interest rates. We illustrate our method by giving two examples: the valuation problems of swaptions and average (Asian) options for interest rates. Our method gives some explicit formulas for solutions, which are sufficiently numerically accurate for practical purposes in most cases. The continuous stochastic processes for spot interest rates and forward interest rates are not necessarily Markovian nor diffusion processes in the usual sense; nevertheless our approach can be rigorously justified by the Malliavin–Watanabe Calculus in stochastic analysis.