Space‐Time Spectral Element Method for PIDEs Arising in Finance

ABSTRACT This paper proposes an efficient space‐time spectral element method for option pricing under the Merton jump‐diffusion model. The model gives rise to a partial integro‐differential equation (PIDE) with a nonlocal integral term and temporal singularities due to nonsmooth initial data. To resolve these singularities and ensure high‐order accuracy, we employ an ‐version of continuous Galerkin scheme with graded time steps. For the spatial discretization, we employ a Laguerre–Legendre composite spectral element method on unbounded domains, which attains spectral accuracy with relatively few degrees of freedom for solutions with slope discontinuities. The resulting space‐time discretization gives rise to highly ill‐conditioned and nonsymmetric linear systems. To address this difficulty, we design a stable and efficient solver based on QZ decomposition and demonstrate its superiority over tensor‐product direct solvers and eigenvalue decomposition methods. Numerical experiments confirm the predicted spectral convergence and highlight the accuracy and efficiency of the proposed method in comparison with existing approaches.