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

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Abstract

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.

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Acknowledgements

The author expresses gratitude for the issues and suggestions pointed out by the reviewers, which have significantly enhanced the rigor and readability of this paper.

Funding

The corresponding author is supported by the Natural Science Foundation of Guangdong Provincial Department of Education (2022KTSCX080) and National Natural Science Foundation of China (12301481). The first author is supported by Guangdong Basic and Applied Research Foundation (2020A1515110991) and National Natural Science Foundation of China (12101137).

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Authors and Affiliations

  1. School of Economics, Guangdong University of Technology, Guangzhou, People’s Republic of China

    Xu Chen & Xin-Xin Gong

  2. Key Laboratory of Digital Economy and Data Governance, Guangdong University of Technology, Guangzhou, People’s Republic of China

    Xu Chen

  3. Department of Mathematics, Hanshan Normal University, Chaozhou, People’s Republic of China

    Zi-Run She & Zi-Hang She

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  1. Xu Chen
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  2. Xin-Xin Gong
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  3. Zi-Run She
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Contributions

X. Chen wrote the main manuscript text and was responsible for debugging the program, and X. X. Gong and Z. R. She were responsible for creating the figures. Z. H. She provided the relevant theoretical analysis. All authors reviewed the manuscript.

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Correspondence to Zi-Hang She.

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Chen, X., Gong, XX., She, ZR. et al. A novel banded preconditioner for coupled tempered fractional diffusion equation generated from the regime-switching CGMY model. Numer Algor 97, 1791–1822 (2024). https://doi.org/10.1007/s11075-024-01769-0

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  • DOI: https://doi.org/10.1007/s11075-024-01769-0

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