A novel banded preconditioner for coupled tempered fractional diffusion equation generated from the regime-switching CGMY model

With the growing popularity of the regime-switching Lévy processes model in option pricing, the coupled tempered fractional diffusion equation generated from this process has garnered considerable attention. However, solving this equation is challenging due to its coupling with a Markov generator matrix, which prevents the coefficient matrix derived from the fully implicit scheme from having a Toeplitz-like structure. Currently, there is no fast algorithm with guaranteed theoretical performance for this problem based on a fully implicit scheme. Therefore, this paper proposes a novel banded preconditioner specifically designed for the regime-switching Carr-Geman-Madan-Yor (CGMY) model. The effectiveness of the preconditioner is ensured by providing related theoretical analyses. It is shown that the eigenvalues of the preconditioned matrix cluster around one under specific parameter settings. Additionally, the condition number of the preconditioned matrix is bounded by a constant without any specific parameter requirements. The proposed preconditioner and theoretical analyses can be extended to the regime-switching CGMYe model as well. Finally, the accuracy of the considered models and the effectiveness of the proposed banded preconditioner are demonstrated through three numerical examples, including an empirical example.