  In this article, we provide representations of European and American exchange
option prices under stochastic volatility jump-diffusion (SVJD) dynamics
following models by Merton (1976), Heston (1993), and Bates (1996). A
Radon-Nikodym derivative process is also introduced to facilitate the shift
from the objective market measure to other equivalent probability measures,
including the equivalent martingale measure. Under the equivalent martingale
measure, we derive the integro-partial differential equation that characterizes
the exchange option prices. We also derive representations of the European
exchange option price using the change-of-numeraire technique proposed by Geman
et al. (1995) and the Fourier inversion formula derived by Caldana and Fusai
(2013), and show that these two representations are comparable. Lastly, we show
that the American exchange option price can be decomposed into the price of the
European exchange option and an early exercise premium.
