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 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 numéraire. 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.

Valuing Exchange Options under an Ornstein-Uhlenbeck Covariance Model

“…Researches differ in the modeling of the correlation, volatilities and jumps, as well as in their numerical approaches. In Garces and Cheang (2021) for example the method of lines is successfully implemented. See also Alos and Rheinlander (2017); Kim and Park (2017); Pasricha and Goel (2022) for other techniques.…”

Spread Option Pricing in Regime-Switching Jump Diffusion Models

“…In [11], a jump diffusion process is also assumed in which the systematic shocks can impact each underlying asset with random and correlated time delays. The pricing problem for the exchange option under stochastic volatility and jump diffusion dynamics was considered in [12] using a method-of-lines approach for solving the integro-partial differential equation.…”

Approximate pricing of American exchange options with jumps

“…Cheang and Chiarella (2011) presented a probabilistic representation for the American style exchange option under jump-diffusion dynamics, but their representation is yet to be solved to provide numerical values which would assist trading or investment decisions. Garces and Cheang (2021) recently presents a numerical method (method of lines) to determine prices of European and American exchange options when underlying asset prices are modeled with stochastic volatility and jump-diffusion dynamics. This article proposes a new technique to derive an efficient, analytical pricing formula for finite-lived American exchange options, when the underlying asset prices follow jump-diffusion processes.…”