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

Neda Bagheri Renani, Daniel Sevcovic

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.

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

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.
Paper Structure (26 sections, 36 equations, 1 figure, 3 tables)

This paper contains 26 sections, 36 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Stochastic Differential Equations of the Bates Model
  3. Risk–neutral dynamics.
  4. Option–pricing PIDE.
  5. Transformed Bates Model PIDE
  6. Spatial Discretization and IMEX--Crank--Nicolson
  7. Fourth-Order Compact Discretization of the Local Operator
  8. Compact line relations (1D, fourth order).
  9. Mixed derivative via sequential compact differentiation.
  10. Compact local operator (nodewise form).
  11. Nine-point compact balance (assembled stencil).
  12. Verification template (constant coefficients).
  13. Nonlocal Jump Operator and IMEX--CN Time Marching
  14. Discrete jump operator.
  15. IMEX--CN scheme.
  16. ...and 11 more sections

Figures (1)

  • 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