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We price European and American exchange options where the underlying assetprices are modelled using a Merton (1976) jump-diffusion with a common Heston(1993) stochastic volatility process. Pricing is performed under an equivalentmartingale measure obtained by setting the second asset yield process as thenumeraire asset, as suggested by Bjerskund and Stensland (1993). Such a choicefor the numeraire reduces the exchange option pricing problem, atwo-dimensional problem, to pricing a call option written on the ratio of theyield processes of the two assets, a one-dimensional problem. The jointtransition density function of the asset yield ratio process and theinstantaneous variance process is then determined from the correspondingKolmogorov backward equation via integral transforms. We then determineintegral representations for the European exchange option price and the earlyexercise premium and state a linked system of integral equations thatcharacterizes the American exchange option price and the associated earlyexercise boundary. Properties of the early exercise boundary near maturity arealso discussed.

A Put-Call Transformation of the Exchange Option Problem under Stochastic Volatility and Jump Diffusion Dynamics