The Valuation Of American Exchange Options Under

Margrabe provides a pricing formula for an exchange option where the distributions of both stock prices are log-normal with correlated components. Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a compound Poisson jump component, in addition to a continuous log-normally distributed component. We use Merton’s analysis to extend Margrabe’s results to the case of exchange options where both stock price processes also contain compound Poisson jump components. We show that there is a change in the distribution of the jump components in the equivalent martingale measure when jumps are present in the num´eraire process. In the case of the American version of such options, the price is shown to be the solution of a free boundary problem. We solve this problem using a modification of McKean’s incomplete Fourier transform method due to Jamshidian. The resulting integral equation for the early exercise boundary is solved numerically. We compare the numerical integration solution with a method of lines approach