Power exchange options

In this paper we present pricing results for an option to exchange the value of one asset raised to a power (S1α1) for the value of another asset raised to a power (S2α2). We refer to such options as power exchange options since they simultaneously generalize results for both the Fischer-Margrabe-type option to exchange one asset for another and power options. We explicitly solve for the price of the European power exchange option under the assumption of risk-neutrality. We also use our results to price options paying the best or worst of powers of two assets. Finally, we establish sufficient conditions for the equivalence of the pricing problems of the American and European power exchange options.