A Numerical Approach to Pricing Exchange Options under Stochastic Volatility and Jump-Diffusion Dynamics

TL;DR

This paper develops a numerical method using the method of lines to price exchange options under complex stochastic models, simplifying the problem via a change of numeraire and analyzing parameter impacts.

Contribution

It introduces a MOL-based numerical scheme for exchange options with stochastic volatility and jumps, including a transformation to reduce dimensionality and analysis of early exercise boundary behavior.

Findings

MOL scheme effectively prices exchange options under complex models.

Model parameters significantly influence option prices and early exercise boundary.

Boundary conditions impact computational efficiency and accuracy.

Abstract

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.