Fourier transformation and the pricing of average-rate derivatives

In this article we propose a method to compute the density of the arithmetic average of a Markov process. This approach is then applied to the pricing of average rate options (Asian options). It is demonstrated that as long as a closed form formula is available for the discount bond price when the underlying process is treated as the riskless interest rate, analytical formulas for the density function of the arithmetic average and the Asian option price can be derived. This includes the affine class of term structure models. The Cox et al. (1985) square root interest rate process is used as an example. When the underlying process follows a geometric Brownian motion, a very efficient numerical method is proposed for computing the density function of the average. Extensions of the techniques to the cases of multiple state variables are also discussed.