Finite Element Solution of the Two-Dimensional Bates Model for Option Pricing Under Stochastic Volatility and Jumps

TL;DR

This paper introduces a high-order finite difference scheme for solving a complex option pricing model that incorporates stochastic volatility and jumps, demonstrating superior accuracy and efficiency over traditional methods.

Contribution

A fourth-order compact finite-difference scheme is developed for the Bates model's PIDE, offering improved accuracy and computational efficiency compared to existing methods.

Findings

HOC-FD achieves near-fourth-order spatial accuracy.

HOC-FD is up to 100 times faster than FEM for similar accuracy.

All methods show second-order temporal convergence.

Abstract

We propose a fourth--order compact finite--difference (HOC--FD) scheme for the transformed Bates partial integro--differential equation (PIDE). The method employs an implicit--explicit (IMEX) Crank--Nicolson framework for local terms and Simpson quadrature for the jump integral. Benchmarks against second--order finite differences (FD) and quadratic finite elements (FEM, p=2) confirm near--fourth--order spatial accuracy for HOC--FD, near--second--order for FEM, and second--order temporal convergence for all time integrators. Efficiency tests show that HOC--FD achieves similar accuracy at up to two orders of magnitude lower runtime than FEM, establishing it as a practical baseline for option pricing under stochastic volatility jump--diffusion models.