Numerical analysis of American option pricing in a two-asset jump-diffusion model

TL;DR

This paper develops a new monotone numerical method for pricing American options in a complex two-asset jump-diffusion model, effectively handling cross-derivatives and jump integrals with proven convergence and validated by numerical tests.

Contribution

A novel monotone integration scheme for 2-D variational inequalities in American option pricing under jump-diffusion models, ensuring convergence to viscosity solutions.

Findings

Method effectively handles cross-derivative and jump integral terms.

Proven stability and convergence to viscosity solutions.

Numerical results demonstrate robustness and efficiency.

Abstract

This paper addresses an important gap in rigorous numerical treatments for pricing American options under correlated two-asset jump-diffusion models using the viscosity solution framework, with a particular focus on the Merton model. The pricing of these options is governed by complex two-dimensional (2-D) variational inequalities that incorporate cross-derivative terms and nonlocal integro-differential terms due to the presence of jumps. Existing numerical methods, primarily based on finite differences, often struggle with preserving monotonicity in the approximation of cross-derivatives, a key requirement for ensuring convergence to the viscosity solution. In addition, these methods face challenges in accurately discretizing 2-D jump integrals. We introduce a novel approach to effectively tackle the aforementioned variational inequalities while seamlessly handling cross-derivative terms and nonlocal integro-differential terms through an efficient and straightforward-to-implement monotone integration scheme. Within each timestep, our approach explicitly enforces the inequality constraint, resulting in a 2-D Partial Integro-Differential Equation (PIDE) to solve. Its solution is expressed as a 2-D convolution integral involving the Green's function of the PIDE. We derive an infinite series representation of this Green's function, where each term is non-negative and computable. This facilitates the numerical approximation of the PIDE solution through a monotone integration method. To enhance efficiency, we develop an implementation of this monotone scheme via FFTs, exploiting the Toeplitz matrix structure. The proposed method is proved to be both $\ell_{\infty} $-stable and consistent in the viscosity sense, ensuring its convergence to the viscosity solution of the variational inequality. Extensive numerical results validate the effectiveness and robustness of our approach.