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A novel numerical scheme for a time fractional Black–Scholes equation

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Abstract

This paper consists of two parts. On one hand, the regularity of the solution of the time-fractional Black–Scholes equation is investigated. On the other hand, to overcome the difficulty of initial layer, a modified L1 time discretization is presented based on a change of variable. And the spatial discretization is done by using the Chebyshev Galerkin method. Optimal error estimates of the fully-discrete scheme are obtained. Finally, several numerical results are given to confirm the theoretical results.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4, 141–183 (1973)

    MathSciNet  MATH  Google Scholar 

  3. Wyss, W.: The fractional Black–Scholes equation. Fract. Calc. Appl. Anal. 3, 51–61 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Liang, J., Wang, J., Zhang, W., Qiu, W., Ren, F.: Option pricing of a bi-fractional Black–Merton–Scholes model with the Hurst exponent H in \([\frac{1}{2},1]\). Appl. Math. Lett. 23, 859–863 (2010)

    MathSciNet  MATH  Google Scholar 

  5. Jena, R.M., Chakraverty, S., Baleanu, D.: A novel analytical technique for the solution of time-fractional Ivancevic option pricing model. Physica A. 550, 124380 (2020)

    MathSciNet  Google Scholar 

  6. Chen, W., Xu, X., Zhu, S.P.: Analytically pricing double barrier options based on a time-fractional Black–Scholes equation. Comput. Math. Appl. 69, 1407–1419 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Zhang, H., Liu, F., Turner, I., Yang, Q.: Numerical solution of the time fractional Black–Scholes model governing European options. Comput. Math. Appl. 71, 1772–1783 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Dubey, V.P., Kumar, R., Kumar, D.: A reliable treatment of residual power series method for time-fractional Black–Scholes European option pricing equation. Physica A. 533, 122040 (2019)

    MathSciNet  Google Scholar 

  9. Roul, P.: A high accuracy numerical method and its convergence for time-fractional Black–Scholes equation governing European options. Appl. Numer. Math. 151, 472–493 (2020)

    MathSciNet  MATH  Google Scholar 

  10. De Staelen, R.H., Hendy, A.S.: Numerically pricing double barrier options in a time-fractional Black–Scholes model. Comput. Math. Appl. 74, 1166–1175 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Kumar, S., Kumar, D., Singh, J.: Numerical computation of fractional Black–Scholes equation arising in financial market. Egyptian J. Basic Appl. Sci. 1, 177–193 (2014)

    Google Scholar 

  12. Koleva, M.N., Vulkov, L.G.: Numerical solution of time-fractional Black–Scholes equation. J. Comput. Appl. Math. 36, 1699–1715 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Yang, X., Wu, L., Sun, S., Zhang, X.: A universal difference method for time-space fractional Black–Scholes equation. Adv. Differ. Equ. 71, 1–14 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Li, D., Liao, H., Sun, W., Wang, J., Zhang, J.: Analysis of L1-Galerkin FEMs for time fractional nonlinear parabolic problems. Commun. Comput. Phys. 24, 86–103 (2018)

    MathSciNet  Google Scholar 

  15. Li, D., Wang, J., Zhang, J.: Unconditionally convergent \(L1\)-Galerkin FEMs for nonlinear time-fractional Schrödinger equations. SIAM J. Sci. Comput. 39, 3067–3088 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Zhang, Q., Ran, M., Xu, D.: Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay. Appl. Anal. 96, 1867–1884 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Sun, Z.Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    MathSciNet  MATH  Google Scholar 

  18. Roul, P.: Analytical approach for the nonlinear partial differential equations of fractional order. Commun. Theor. Phys. 60, 269–277 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Mao, Z., Karniadakis, G.E.: A spectral method (of exponential convergence) for singular solutions of the diffusion equation with general two-sided fractional derivative. SIAM J. Numer. Anal. 56, 24–49 (2018)

    MathSciNet  MATH  Google Scholar 

  20. Chen, X., Di, Y., Duan, J., Li, D.: Linearized compact ADI schems for nonlinear time-fractional Schrödinger equations. Appl. Math. Lett. 84, 160–167 (2018)

    MathSciNet  MATH  Google Scholar 

  21. Sheng, C., Shen, J.: A space-time Petrov–Galerkin spectral method for time fractional diffusion equation. Numer. Math. Theor. Meth. Appl. 11, 854–876 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Sun, H., Sun, Z., Du, R.: A linearized second-order difference scheme for the nonlinear time-fractional fourth-order reaction-diffusion equation. Numer. Math. Theor. Meth. Appl. 12, 1168–1190 (2019)

    MathSciNet  MATH  Google Scholar 

  23. Cui, M.: Finite difference schemes for the variable coefficients single and multi-term time-fractional diffusion equations with non-smooth solutions on graded and uniform meshes. Numer. Math. Theor. Meth. Appl. 12, 1004–8979 (2019)

    MathSciNet  Google Scholar 

  24. Ran, M., Zhang, C.: A high-order accuracy method for solving the fractional diffusion equations. J. Comput. Math. 38, 239–253 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Kopteva, N., Stynes, M.: An efficient collocation method for a Caputo two-point boundary value problem. BIT. 55, 1105–1123 (2015)

    MathSciNet  MATH  Google Scholar 

  26. Stynes, M., Gracia, J.L.: A finite difference method for a two- point boundary value problem with a Caputo fractional derivative. IMA J. Numer. Anal. 35, 689–721 (2015)

    MathSciNet  MATH  Google Scholar 

  27. Gracia, J.L., Stynes, M.: Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems. J. Comput. Appl. Math. 273, 103–115 (2015)

    MathSciNet  MATH  Google Scholar 

  28. Li, L., Li, D.: Exact solutions and numerical study of time fractional Burgers’ equations. Appl. Math. Lett. 100, 106011 (2020)

    MathSciNet  MATH  Google Scholar 

  29. Jin, B., Li, B., Zhou, Z.: Numerical analysis of nonlinear subdiffusion equations. SIAM J. Numer. Anal. 56, 1–23 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Li, D., Sun, W., Wu, C.: A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions. Numer. Math. Theor. Meth. Appl. (2021). https://doi.org/10.4208/nmtma.OA-2020-0129

    Article  MathSciNet  Google Scholar 

  31. Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time fractional diffusion equation. SIAM J. Numer. Anal. 55, 1057–1079 (2017)

    MathSciNet  MATH  Google Scholar 

  32. Cen, Z., Huang, J., Xu, A., Le, A.: Numerical approximation of a time-fractional Black–Scholes equation. Comput. Math. Appl. 75, 2874–2887 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Li, D., Wu, C., Zhang, J.: Linearized Galerkin FEMs for nonlinear time fractional parabolic problems with non-smooth solutions in time direction. J. Sci. Comput. 80, 403–419 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Kopteva, N.: Error analysis of the L1 method on graded and uniform meshes for a fractional-derivative problem in two and three dimensions. Math. Comput. 88, 2135–2155 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Cao, W., Zeng, F., Zhang, Z., Karniadakis, G.E.: Implicit-Explicit difference schemes for nonlinear fractional differential equations with nonsmooth solutions. SIAM J. Sci. Comput. 38, 3070–3093 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Jin, B., Li, B., Zhou, Z.: Correction of high-order BDF convolution quadrature for fractional evolution equations. SIAM J. Sci. Comput. 39, 3129–3152 (2017)

    MathSciNet  Google Scholar 

  37. Courant, R., Hilbert, D., Bergmann, P.: Methods of Mathematical Physics. Interscience Publishers Inc, New York (1953)

    MATH  Google Scholar 

  38. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  39. Shen, J., Tang, T., Wang, L.L.: Spectral Methods. Springer, Heidelberg (2011)

    MATH  Google Scholar 

  40. Canuto, C., Quarteroni, A.: Approximation results for orthognal polynomials in Sobolev Spaces. Math. Comput. 38, 67–86 (1982)

    MATH  Google Scholar 

  41. Bressan, N., Quarteroni, A.: Analysis of Chebyshev collocation methods for parabolic equations. SIAM J. Numer. Anal. 23, 1138–1154 (1986)

    MathSciNet  MATH  Google Scholar 

  42. Thomée, V.: Galerkin finite element methods for parabolic problems. Springer, Berlin (2006)

    MATH  Google Scholar 

  43. Zhou, B., Chen, X., Li, D.: Nonuniform Alikhanov linearized Galerkin finite element methods for nonlinear time-fractional parabolic equations. J. Sci. Comput. 85, 39 (2020)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Mianfu She, Lili Li, Renxuan Tang & Dongfang Li

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Mianfu She & Dongfang Li

Authors
  1. Mianfu She
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  2. Lili Li
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  3. Renxuan Tang
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  4. Dongfang Li
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Correspondence to Renxuan Tang.

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This work is supported by NSFC (Grant No.11771162)

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She, M., Li, L., Tang, R. et al. A novel numerical scheme for a time fractional Black–Scholes equation. J. Appl. Math. Comput. 66, 853–870 (2021). https://doi.org/10.1007/s12190-020-01467-9

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  • DOI: https://doi.org/10.1007/s12190-020-01467-9

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