IMEX–Crank–Nicolson Methods for the Merton Jump-Diffusion PIDE: Stability, Convergence, and Fast Jump Evaluation

This paper presents an efficient numerical framework for solving the Merton jump–diffusion partial integro-differential equation (PIDE) arising in European option pricing. To address the nonlocal integral term generated by asset price jumps, we employ an IMEX Crank–Nicolson time-stepping scheme that preserves the tri-diagonal structure of the resulting linear system. A nonuniform spatial grid and fast Gaussian quadrature with spline interpolation are incorporated to enhance accuracy and computational efficiency. We establish the unconditional stability and convergence of the IMEX–Crank–Nicolson scheme through a detailed theoretical analysis. Numerical experiments confirm the theoretical results and illustrate the effectiveness of the proposed method for representative jump–diffusion parameters.