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

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.