  We price European and American exchange options where the underlying asset
prices are modelled using a Merton (1976) jump-diffusion with a common Heston
(1993) stochastic volatility process. Pricing is performed under an equivalent
martingale measure obtained by setting the second asset yield process as the
numeraire asset, as suggested by Bjerskund and Stensland (1993). Such a choice
for the numeraire reduces the exchange option pricing problem, a
two-dimensional problem, to pricing a call option written on the ratio of the
yield processes of the two assets, a one-dimensional problem. The joint
transition density function of the asset yield ratio process and the
instantaneous variance process is then determined from the corresponding
Kolmogorov backward equation via integral transforms. We then determine
integral representations for the European exchange option price and the early
exercise premium and state a linked system of integral equations that
characterizes the American exchange option price and the associated early
exercise boundary. Properties of the early exercise boundary near maturity are
also discussed.
