Fourth order compact scheme for option pricing under Merton and Kou jump-diffusion models

TL;DR

This paper introduces a third-time level compact numerical scheme for solving the integro-differential equations in option pricing models with jumps, achieving high accuracy and stability validated through numerical experiments.

Contribution

It develops a novel third-time level compact scheme with proven stability and fourth-order convergence for jump-diffusion option pricing models, incorporating smoothing for low regularity conditions.

Findings

Achieves fourth-order convergence in numerical experiments.

Validates the scheme's stability and accuracy for Merton and Kou models.

Provides efficient solution of integro-differential equations in finance.

Abstract

In this article, a three-time levels compact scheme is proposed to solve the partial integro-differential equation governing the option prices under jump-diffusion models. In the proposed compact scheme, the second derivative approximation of unknowns 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. Moreover, consistency and stability of the proposed compact scheme are proved. Due to the low regularity of typical initial conditions, the smoothing operator is employed to ensure the fourth-order convergence rate. Numerical illustrations for pricing European options under Merton and Kou jump-diffusion models are presented to validate the theoretical results.