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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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References

  1. Asiimwe, P., Mahera, C.W., Menoukeu-Pamen, O.: On the price of risk under a regime switching CGMY process. Asia-Pacific Finan. Markets 23, 305 (2016)

    Article  Google Scholar 

  2. Bao, J., Zhao, Y.: Option pricing in Markov-modulated exponential Lévy models with stochastic interest rates. J. Comput. Appl. Math. 357, 146–160 (2019)

    Article  MathSciNet  Google Scholar 

  3. Bertaccini, D.: Reliable preconditioned iterative linear solvers for some numerical integrators. Numer. Linear Algebra Appl. 8, 111–125 (2001)

    Article  MathSciNet  Google Scholar 

  4. Bertaccini, D., Golub, G.H., Serra-Capizzano, S.: Spectral analysis of a preconditioned iterative method for the convection-diffusion equation. SIAM J. Matrix Anal. Appl. 29, 260–278 (2007)

    Article  MathSciNet  Google Scholar 

  5. Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81(3), 637–654 (1973)

    Article  MathSciNet  Google Scholar 

  6. Boyarchenko, S., Levendorskii, S.: Non-Gaussian Merton-Black-Scholes theory. World Scientific, Singapore (2002)

    Book  Google Scholar 

  7. Cartea, A., del Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Phys. A 374(2), 749–763 (2007)

    Article  Google Scholar 

  8. Chan, R., Jin, X.: An introduction to iterative Toeplitz solvers. SIAM, Philadelphia (2007)

    Book  Google Scholar 

  9. Chen, M., Deng, W.: High order algorithms for the fractional substantial diffusion equation with truncated Lévy flights. SIAM J. Sci. Comput. 37(2), A890–A917 (2015)

    Article  Google Scholar 

  10. Chen, W., Wang, S.: A penalty method for a fractional order parabolic variational inequality governing American put option valuation. Comput. Math. Appl. 67(1), 77–90 (2014)

    Article  MathSciNet  Google Scholar 

  11. Chen, W., Xu, X., Zhu, S.: A predictor-corrector approach for pricing American options under the finite moment log-stable model. Appl. Numer. Math. 97, 15–29 (2015)

    Article  MathSciNet  Google Scholar 

  12. Chen, X., Ding, D., Lei, S., Wang, W.: An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial differential models. Numer. Algor. 87, 939–965 (2021)

    Article  MathSciNet  Google Scholar 

  13. Chen, X., Wang, W., Ding, D., Lei, S.: A fast preconditioned policy iteration method for solving the tempered fractional HJB equation governing American options valuation. Comput. Math. Appl. 73, 1932–1944 (2017)

    Article  MathSciNet  Google Scholar 

  14. Elliott, R.J., Osakwe, C.J.U.: Option pricing for pure jump processes with Markov switching compensators. Financ. Stoch. 10, 250 (2006)

    Article  MathSciNet  Google Scholar 

  15. Guo, X., Li, Y., Wang, H.: Tempered fractional diffusion equations for pricing multi-asset options under CGMYe process. Comput. Math. Appl. 76, 1500–1514 (2018)

    Article  MathSciNet  Google Scholar 

  16. Heston, S.L.: A closed-form solution for options with stochatic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993)

    Article  MathSciNet  Google Scholar 

  17. Hu, J.: The estimate of \(\Vert {M^{-1}N}\Vert _{\infty }\) and the optimally scaled matrix. J. Comput. Math. 2(2), 122–129 (1984)

    MathSciNet  Google Scholar 

  18. Khaliq, A.Q.M., Liu, R.H.: New numerical scheme for pricing American option with regime-switching. Int. J. Theor. Appl. Finance 12, 319–340 (2019)

    Article  MathSciNet  Google Scholar 

  19. Kou, S.G.: A jump-diffusion model for option pricing. Manage Sci. 48(8), 1086–1101 (2002)

    Article  Google Scholar 

  20. Lei, S., Chen, X., Zhang, X.: Multilevel circulant preconditioner for high-dimensional fractional diffusion equations. East Asian J. Appl. Math. 6, 109–130 (2016)

    Article  MathSciNet  Google Scholar 

  21. Lei, S., Wang, W., Chen, X., Ding, D.: A fast preconditioned penalty method for American options pricing under regime-switching tempered fractional diffusion models. J. Sci. Comput. 75, 1633–1655 (2018)

    Article  MathSciNet  Google Scholar 

  22. Li, C., Deng, W.: High order schemes for the tempered fractional diffusion equations. Adv. Comput. Math. 42(3), 543–572 (2016)

    Article  MathSciNet  Google Scholar 

  23. Madan, D.B., Carr, P.P., Chang, E.C.: The variance gamma process and option pricing. Eur. Financ. Rev. 2, 79–105 (1998)

    Article  Google Scholar 

  24. Meng, Q., Ding, D., Sheng, Q.: Preconditioned iterative methods for fractional diffusion models in finance. Numer. Meth. Part Differ. Equ. 31(5), 1382–1395 (2014)

    Article  MathSciNet  Google Scholar 

  25. Momeya, R.H., Morales, M.: On the price of risk of the underlying Markov chain in a regime-switching exponential Lévy model. Methodol. Comput. Appl. Probab. 18, 107 (2016)

    Article  MathSciNet  Google Scholar 

  26. Peiro, A.: Skewness in financial returns. J. Bank Financ. 23, 847–862 (1999)

    Article  Google Scholar 

  27. Rachev, S.T., Menn, C., Fabozzi, F.J.: Fat-tailed and skewed asset return distributions: implications for risk management, portfolio selection, and option pricing. John Wiley and Sons, New Jersey (2005)

    Google Scholar 

  28. Saad, Y.: Iterative methods for sparse linear systems. SIAM (2003)

  29. Serra-Capizzano, S., Bertaccini, D., Golub, G.H.: How to deduce a proper eigenvalue cluster from a proper singular value cluster in the nonnormal case. SIAM J. Matrix Anal. Appl. 27, 82–86 (2005)

    Article  MathSciNet  Google Scholar 

  30. She, Z., Lao, C., Yang, H., Lin, F.: Banded preconditioners for Riesz space fractional diffusion equations. J. Sci. Comput. 86(3), 1–22 (2021)

    Article  MathSciNet  Google Scholar 

  31. Stein, E.M., Stein, J.C.: Stock price distributions with stochastic volatility: an analytic approach. Rev. Financ. Stud. 4, 727–752 (1991)

    Article  Google Scholar 

  32. Tour, G., Thakoor, N., Khaliq, A.Q.M., Tangman, D.Y.: COS method for option pricing under a regime-switching model with time-changed Lévy processes. Quant. Financ. 18, 673–692 (2018)

    Article  Google Scholar 

  33. Wang, W., Chen, X., Ding, D., Lei, S.: Circulant preconditioning technique for barrier options pricing under fractional diffusion models. Int. J. Comput. Math. 92(12), 2596–2614 (2015)

    Article  MathSciNet  Google Scholar 

  34. Zhang, H., Liu, F., Turner, I., Chen, S.: The numerical simulation of the tempered fractional Black-choles equation for European double barrier option. Appl. Math. Model. 40, 5819–5834 (2016)

    Article  MathSciNet  Google Scholar 

  35. Zhang, H., Liu, F., Turner, I., Chen, S., Yang, Q.: Numerical simulation of a Finite Moment Log Stable model for a European call option. Numer. Algor. 75, 569–585 (2017)

    Article  MathSciNet  Google Scholar 

  36. Zhang, Q., Guo, X.: Closed-form solutions for perpetual American put options with regime switching. SIAM J. Appl. Math. 64, 2034–2049 (2004)

    Article  MathSciNet  Google Scholar 

  37. Zhou, Z., Ma, J., Sun, H.: Fast Laplace transform methods for free-boundary problems of fractional diffusion equations. J. Sci. Comput. 74, 49–69 (2018)

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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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  4. Zi-Hang 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 (2024). https://doi.org/10.1007/s11075-024-01769-0

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