Options on the Minimum or the Maximum of Two Average Prices

This paper analyzes and values European-style options on the minimum or the maximum of two average prices. In particular, we provide a closed-form pricing formula for the option with geometric averaging starting at any time before maturity. Our numerical evidence shows that the use of the closed-form solution derived in this paper in a variance-reduction technique dramatically improves the accuracy of the simulated price of an option with arithmetic averaging. The paper also discusses some parity relationships within the family of average-rate options, and finds the upper and lower bounds for the proposed options with arithmetic averaging.Options on the minimum or the maximum of two average prices are useful in practice in many ways. They are naturally applicable when firms are concerned with a hedging problem involving two average asset prices. A typical example is hedging average foreign account payables when either of two foreign currencies is allowed for payment. Other interesting risk-management applications presented include hedging production costs when prices of two substitutable inputs are stochastic, and hedging profit markups when input and output prices are stochastic. Moreover, the use of the proposed options is not limited to risk management. We demonstrate, for instance, that an option on the minimum of two average prices appropriately enters the payoff function of incentive contracts for executive compensation, a problem in the theory of corporate finance. This study is closely related to both the literature on average-rate options, or Asian options, first analyzed and valued by Kemna and Vorst (1990), and the literature on options on the minimum or the maximum of two risky assets, or Rainbow options, pioneered by Stulz (1982). Existing published research seems to indicate that both academics and practitioners have paid greater attention to the later-developed but ever-increasingly popular average-rate options than to options on the minimum or the maximum of two risky assets. Our work in this paper, which non-trivially solves the pricing of the options with two average prices, would bridge such a gap between these two lines of literature. At the very least, the options proposed by the paper should spawn wider and more interesting applications than those offered by the existing literature