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

Neda Bagheri Renani, Sevcovic, Daniel|arXiv (Cornell University)|Feb 22, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper develops a fourth-order compact finite-difference (HOC-FD) scheme for the transformed Bates PIDE, using an IMEX-CN time discretization and Simpson quadrature for jump integrals, and benchmarks it against second-order FD and quadratic FEM to demonstrate superior efficiency with comparable accuracy.

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.

Motivation & Objective

• Motivate accurate numerical pricing under stochastic volatility with jumps as in the Bates model.
• Develop a high-order compact finite-difference scheme for the transformed Bates PIDE.
• Assess accuracy and efficiency against standard second-order FD and quadratic FEM.
• Show that HOC-FD achieves near-fourth-order spatial accuracy with competitive computational cost.

Proposed method

• Transform Bates PIDE to a backward-time, normalized form suitable for numerics.
• Apply a fourth-order compact discretization for the local differential operator on a 3x3 stencil.
• Treat the local operator implicitly and the jump integral explicitly via an IMEX–Crank–Nicolson scheme.
• Approximate the nonlocal jump term with Simpson quadrature over a truncated interval.
• Benchmark HOC-FD against second-order FD and FEM (P2) in terms of accuracy (L2 and RMSE) and efficiency (DOF, CPU time).

Figure 1: European put price $V(S,t)$ under the Bates model for $S\in[70,130]$ . The intrinsic payoff $\max(K-S,0)$ and the no-arbitrage lower bound $\max\{Ke^{-rT}-S,0\}$ are shown as reference curves

Experimental results

Research questions

• RQ1 Can a fourth-order compact finite-difference scheme accurately solve the Bates PIDE on structured grids?
• RQ2 How does HOC-FD compare to standard second-order FD and FEM(P2) in spatial/temporal accuracy and computational cost?
• RQ3 Is the IMEX–Crank–Nicolson approach with explicit jump treatment stable and efficient for Bates-type models?
• RQ4 What are the efficiency gains of HOC-FD relative to FEM(P2) for European option pricing under stochastic volatility with jumps?

Key findings

• HOC-FD achieves near-fourth-order spatial accuracy, while FEM attains near-second-order accuracy in space.
• All time integrators yield about second-order temporal convergence.
• HOC-FD delivers substantial efficiency gains, with runtime often two orders of magnitude faster than FEM(P2) at comparable accuracy.
• Compared with second-order FD, HOC-FD and its peers show significantly better cost-accuracy balance on structured grids.