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Compact IMEX scheme for a moving boundary PIDE system of the regime-switching jump-diffusion Asian option pricing

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Abstract

The aim of this paper is to develop a compact implicit-explicit (IMEX) difference scheme for solving a moving boundary partial integro-differential equation (PIDE) system of the regime-switching jump-diffusion Asian option pricing. First, the IMEX scheme is proposed for temporal discretization and combined with the compact difference scheme for spatial discretization. Then the unconditional stability, unique solvability, and convergence rates of second-order in time and fourth-order in space are proved in the discrete \(L^2\) and \(L^\infty \) norms. Numerical examples are given to demonstrate the theoretical results and show the effectiveness of the proposed scheme with respect to the efficiency and accuracy.

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References

  1. Zvan, R., Forsyth, P.A., Vetzal, K.R.: Robust numerical methods for PDE models of Asian options. Journal of Computational Finance 1, 39–78 (1998)

    Article  Google Scholar 

  2. Večeř, J.: A new PDE approach for pricing arithmetic average Asian options. Journal of Computational Finance 4, 105–113 (2001)

    Article  Google Scholar 

  3. Dubois, F., Lelièvre, T.: Efficient pricing of Asian options by the PDE approach. Journal of Computational Finance 8, 55–64 (2005)

    Article  Google Scholar 

  4. Cen, Z., Le, A., Xu, A.: Finite difference scheme with a moving mesh for pricing Asian options. Appl. Math. Comput. 219, 8667–8675 (2013)

    MathSciNet  Google Scholar 

  5. Ma, J.T., Zhou, Z.: Convergence rates of moving mesh Rannacher methods for PDEs of Asian options pricing. J. Comput. Math. 34, 265–286 (2016)

    MathSciNet  Google Scholar 

  6. Roul, P.: A fourth order numerical method based on B-spline functions for pricing Asian options. Comput. Math. Appl. 80, 504–521 (2020)

    Article  MathSciNet  Google Scholar 

  7. Boyle, P., Draviam, T.: Pricing exotic options under regime switching. Insurance Math. Econom. 40, 267–282 (2007)

  8. Ma, J.T., Zhou, Z.: Moving mesh methods for pricing Asian options with regime switching. J. Comput. Appl. Math. 298, 211–221 (2016)

    Article  MathSciNet  Google Scholar 

  9. Dang, D.M., Nguyen, D., Sewell, G.: Numerical schemes for pricing Asian options under state-dependent regime-switching jump-diffusion models. Comput. Math. Appl. 71, 443–458 (2016)

    Article  MathSciNet  Google Scholar 

  10. Ma, J.T., Wang, H.: Convergence rates of moving mesh methods for moving boundary partial integro-differential equations from regime-switching jump-diffusion Asian option pricing. J. Comput. Appl. Math. 370, 1–16 (2020)

    Article  MathSciNet  Google Scholar 

  11. Chen, Y.: Second-order IMEX scheme for a system of partial integro-differential equations from Asian option pricing under regime-switching jump-diffusion models. Numer. Algorithms 89, 1823–1843 (2022)

    Article  MathSciNet  Google Scholar 

  12. Chen, Y., Hu, R.: \(L^\infty \)-norm convergence rates of an IMEX scheme for solving a partial integro-differential equation system arising from regime-switching jump-diffusion Asian option pricing. Int. J. Comput. Math. (2023). https://doi.org/10.1080/00207160.2023.2189048 (to appear)

  13. Kwon, Y., Lee, Y.: A second-order finite difference method for option pricing under jump-diffusion models. SIAM J. Numer. Anal. 49, 2598–2617 (2011)

    Article  MathSciNet  Google Scholar 

  14. Salmi, S., Toivanen, J.: IMEX schemes for pricing options under jump-diffusion models. Appl. Numer. Math. 84, 33–45 (2014)

    Article  MathSciNet  Google Scholar 

  15. Salmi, S., Toivanen, J., Von Sydow, L.: An IMEX-scheme for pricing options under stochastic volatility models with jumps. SIAM J. Sci. Comput. 36, B817–B834 (2014)

    Article  MathSciNet  Google Scholar 

  16. Sydow, L.V., Toivanen, J., Zhang, C.: Adaptive finite difference and IMEX time-stepping to price options under Bates model. Int. J. Comput. Math. 92, 2515–2529 (2015)

    Article  MathSciNet  Google Scholar 

  17. Kadalbajoo, M.K., Tripathi, L.P., Kumar, K.: Second order accurate IMEX methods for option pricing under Merton and Kou jump-diffusion model. J. Sci. Comput. 65, 979–1024 (2015)

    Article  MathSciNet  Google Scholar 

  18. Kadalbajoo, M.K., Tripathi, L.P., Kumar, K.: An error analysis of a finite element method with IMEX-time semidiscretizations for some partial integro-differential inequalities arising in the pricing of American options. SIAM J. Numer. Anal. 55, 869–891 (2017)

    Article  MathSciNet  Google Scholar 

  19. Kazmi, K.: An IMEX predictor-corrector method for pricing options under regime-switching jump-diffusion models. Int. J. Comput. Math. 96, 1137–1157 (2019)

    Article  MathSciNet  Google Scholar 

  20. Chen, Y.Z., Xiao, A.G., Wang, W.S.: An IMEX-BDF\(2\) compact scheme for pricing options under regime-switching jump-diffusion models. Math. Methods Appl. Sci. 42, 2646–2663 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  21. Wang, W., Chen, Y., Fang, H.: On the vatiable two-step IMEX BDF methods for parabolic integro-differential equations with nonsmooth initial data arising in finance. SIAM J. Numer. Anal. 57, 1289–1317 (2019)

    Article  MathSciNet  Google Scholar 

  22. Lee, S.T., Sun, H.W.: Fourth-order compact scheme with local mesh refinement for option pricing in jump-diffusion model. Numer. Methods Partial Differential Equations 28, 1079–1098 (2012)

    Article  MathSciNet  Google Scholar 

  23. Ladyzenskaja, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23. American Mathematical Society, Providence, R.I. (1968)

  24. Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)

    Article  MathSciNet  Google Scholar 

  25. Wang, Y.M.: A compact finite difference method for solving a class of time fractional convection-subdiffusion equations. BIT Numer. Math. 55, 1187–1217 (2015)

    Article  MathSciNet  Google Scholar 

  26. Samarskii, A.A., Andreev, V.B.: Difference methods for elliptic equations. Nauka, Moscow (1976)

    Google Scholar 

  27. Morton, K.W., Mayers, D.F.: Numerical solution of partial differential equations. Cambridge University Press, UK (2005)

    Book  Google Scholar 

  28. Bank, R.E., Santos, R.F.: Analysis of some moving space-time finite element methods. SIAM J. Numer. Anal. 30, 1–18 (1993)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author is sincerely grateful to the editor and anonymous referees for their valuable comments that have led to a greatly improved paper.

Funding

The work was supported by Technology and Venture Finance Research Center of Sichuan Key Research Base for Social Sciences (Grant No. KJJR2019-003).

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  1. School of Economics, Xihua University, Chengdu, 610039, P.R. China

    Yong Chen

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  1. Yong Chen
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Chen, Y. Compact IMEX scheme for a moving boundary PIDE system of the regime-switching jump-diffusion Asian option pricing. Numer Algor 95, 1055–1077 (2024). https://doi.org/10.1007/s11075-023-01599-6

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