This article proposes implicit–explicit (IMEX) high-order numerical methods for pricing European and American options under Merton’s jump diffusion model. We incorporate the convex combination parameter \(c\in [0,1]\) of the zeroth-order term into the IMEX third-order and fourth-order semi-implicit backward differentiation formula to improve the stability and efficiency of the methods. In order to comprehensively understand the stability of IMEX methods in time discretization, we use Fourier method for analysis and draw a stability domain graph. To further improve the accuracy of numerical solutions in spatial discretization, a fourth-order compact finite difference scheme is designed. For the linear complementarity problem of the American options, we ingeniously combined IMEX high-order methods and operator splitting methods. Several numerical experiments are provided to investigate the accuracy and convergence order of IMEX high-order methods for pricing European and American options in both time and spatial. The experimental results have verified the effectiveness and advantages of the proposed numerical schemes.
