Asian Options with the American Early Exercise Feature

Asian options are averaging options where the terminal payoffs depend on some form of averaging prices of the underlying asset over a part or the whole life of the option. Averaging options are particularly useful for business involved in trading on thinly traded commodities. These types of options are used by traders who are interested to hedge against the average price of a commodity over a period rather than, say, the price at the end of period. A wide variety of averaging options have been proposed, and summaries of some of these options can be found in the papers by Boyle [1] and Zhang [11]. The most commonly used sampled average is the discrete arithmetic average. However, the pricing of this class of Asian options is almost analytically intractable since the sum of lognormal densities has no explicit representation. On the other hand, if the Geometric Brownian motion is assumed for the underlying asset price, the analytic derivation of the price formula for Asian options with geometric averaging is feasible since the product of lognormal prices remains to be lognormal. The analytic procedures for deriving pricing formula of geometrically averaged Asian options can be broadly classified into two types, one uses the probabilistic approach and the other uses the partial differential equation approach. In the