Compact finite difference method for pricing European and American options under jump-diffusion models

TL;DR

This paper introduces a compact finite difference method for efficiently pricing European and American options under jump-diffusion models, ensuring high accuracy and stability through advanced discretization and smoothing techniques.

Contribution

It develops a novel compact finite difference scheme with proven stability and fourth-order convergence for option pricing under jump-diffusion models.

Findings

Achieves fourth-order convergence rate with smoothing operators.

Validates the method through numerical experiments under Merton jump-diffusion model.

Provides a stable and efficient approach for American and European option pricing.

Abstract

In this article, a compact finite difference method is proposed for pricing European and American options under jump-diffusion models. Partial integro-differential equation and linear complementary problem governing European and American options respectively are discretized using Crank-Nicolson Leap-Frog scheme. In proposed compact finite difference method, the second derivative is approximated by the value of unknowns and their first derivative approximations which allow us to obtain a tri-diagonal system of linear equations for the fully discrete problem. Further, consistency and stability for the fully discrete problem are also proved. Since jump-diffusion models do not have smooth initial conditions, the smoothing operators are employed to ensure fourth-order convergence rate. Numerical illustrations for pricing European and American options under Merton jump-diffusion model are presented to validate the theoretical results.