  We consider a method of lines (MOL) approach to determine prices of European
and American exchange options when underlying asset prices are modelled with
stochastic volatility and jump-diffusion dynamics. As the MOL, as with any
other numerical scheme for PDEs, becomes increasingly complex when higher
dimensions are involved, we first simplify the problem by transforming the
exchange option into a call option written on the ratio of the yield processes
of the two assets. This is achieved by taking the second asset yield process as
the numeraire. We also characterize the near-maturity behavior of the early
exercise boundary of the American exchange option and analyze how model
parameters affect this behavior. Using the MOL scheme, we conduct a numerical
comparative static analysis of exchange option prices with respect to the model
parameters and investigate the impact of stochastic volatility and jumps to
option prices. We also consider the effect of boundary conditions at
far-but-finite limits of the computational domain on the overall efficiency of
the MOL scheme. Toward these objectives, a brief exposition of the MOL and how
it can be implemented on computing software are provided.
