A numerical approach to pricing exchange options under stochastic volatility and jump-diffusion dynamics

Abstract

We consider a method of lines (MOL) approach to determine prices of European and American exchange options when underlying asset prices are modeled with stochastic volatility and jump-diffusion dynamics. As with any other numerical scheme for partial differential equations (PDEs), the MOL becomes increasingly complex when higher dimensions are involved, so 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 numéraire. Under the equivalent martingale measure induced by this change of numéraire, we derive the exchange option pricing integro-partial differential equations (IPDEs) and investigate the early exercise boundary of the American exchange option. We then discuss a numerical solution of the IPDEs using the MOL, its implementation using computing software and possible alternative boundary conditions at the far limits of the computational domain. Our analytical and numerical investigation shows that the near-maturity behavior of the early exercise boundary of the American exchange option is significantly influenced by the dividend yields and the presence of jumps in the underlying asset prices. Furthermore, with the numerical results generated by the MOL, we are able to show that key jump and stochastic volatility parameters significantly affect the early exercise boundary and exchange option prices. Our numerical analysis also verifies that the MOL performs more efficiently, than other finite difference methods or simulation approaches for American options, since the MOL integrates the computation of option prices, greeks and the early exercise boundary, and does so with the least error.