A τ Preconditioner for Options Pricing Problem Under Jump‐Diffusion Models

ABSTRACT Recently, the partial integro‐differential equations (PIDEs) that describe the behavior of option prices under jump‐diffusion models have been widely employed and have gained significant attention due to the ability to capture critical empirical features of financial markets. While various methods exist for solving these equations, many studies rely on the three‐time‐level method to handle the integral term. In contrast, we develop the Crank–Nicolson method for solving PIDEs under the Merton and Kou models, focusing on European and American option pricing. The stability the proposed scheme is proved. Additionally, a preconditioned generalized minimal residual (GMRES) solver using the preconditioner is employed to handle the Toeplitz system. By demonstrating that the preconditioned matrix can be expressed as the sum of the identity matrix and a small‐norm matrix, independent of the matrix size, the fast convergence analysis of the PGMRES method is established. The effectiveness is demonstrated through several numerical experiments, including an empirical case study based on the Merton model.