Introduction

A typical example of an Asian option is a contract giving the holder the right to buy an asset for its average price over some prescribed period. Such a product is of obvious appeal to a company which must buy a commodity at a fixed time each year, yet has to sell it regularly throughout the year. In this case the underlying asset is the commodity. The same type of option is also used in foreign exchange markets by companies that have continuous sales in one currency but must purchase raw materials in a different currency and at a fixed date. Here, the underlying is the exchange rate. These options allow investors to insure against losses from adverse movements in an underlying asset without the need for continuous rehedging.

In this chapter we derive differential equations for the value of several Asian options. The common feature is that the exercise price is always some form of average of the price of the underlying over some period prior to exercise. The exercise price may depend on geometric or arithmetic averages, which may be measured either continuously or discretely. As well as deriving the equations we examine several problems in more detail, in particular the continuously sampled arithmetic average strike option with either European or American exercise features, and the European geometric average strike with either continuous or discrete sampling.