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 Mertonâ€™s analysis to extend Margrabeâ€™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 numÂ´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 McKeanâ€™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
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