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A compact quadratic spline collocation method for the time-fractional Black–Scholes model

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Abstract

A compact quadratic spline collocation (QSC) method for the time-fractional Black–Scholes model governing European option pricing is presented. Firstly, after eliminating the convection term by an exponential transformation, the time-fractional Black–Scholes equation is transformed to a time-fractional sub-diffusion equation. Then applying \(L1 - 2\) formula for the Caputo time-fractional derivative and using a collocation method based on quadratic B-spline basic functions for the space discretization, we establish a higher accuracy numerical scheme which yields \(3-\alpha \) order convergence in time and fourth-order convergence in space. Furthermore, the uniqueness of the numerical solution and the convergence of the algorithm are investigated. Finally, numerical experiments are carried out to verify the theoretical order of accuracy and demonstrate the effectiveness of the new technique. Moreover, we also study the effect of different parameters on option price in time-fractional Black–Scholes model.

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References

  1. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier, Boston (2006)

    MATH  Google Scholar 

  2. Metzler, R., Klafter, J.: The restaurant at the end of random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A. 37, 161–208 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Sun, H.G., Zhang, Y., Chen, W., Reeves, D.M.: Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J. Contam. Hydrol. 157, 47–58 (2014)

    Google Scholar 

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

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Zhao, H., Tian, H.J.: Finite difference methods of the spatial fractional Black–Schloes equation for a European call option. IMA J. Appl. Math. 82, 836–848 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  9. Jumarie, G.: Derivation and solutions of some fractional Black–Scholes equations in coarse-grained space and time: Application to Mertons optimal portfolio. Comput. Math. Appl. 59, 1142–1164 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Liang, J.R., Wang, J., Zhang, W.J., Qiu, W.Y., Ren, F.Y.: The solutions to a bi-fractional Black–Scholes–Merton differential equation. Int. J. Pure Appl. Math. 58, 99–112 (2010)

    MathSciNet  Google Scholar 

  11. Chen, W.T., 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 

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

    MathSciNet  MATH  Google Scholar 

  13. 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 

  14. Zhou, Z.Q., Gao, X.M.: Numerical methods for pricing American options with time-fractional PDE models, Math. Prob. Eng., Article ID 5614950, pp. 1–8 (2016)

  15. Cen, Z.D., Huang, J., Xu, A.M., Le, A.B.: Numerical approximation of a time-fractional Black–Scholes equation. Comput. Math. Appl. 8, 2874–2887 (2018)

    MathSciNet  MATH  Google Scholar 

  16. 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 

  17. Tian, Z.W., Zhai, S.Y., Weng, Z.F.: Compact finite difference schemes of the time fractional Black–Scholes model. J. Appl. Anal. Comput. 10, 904–919 (2020)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Luo, W.H., Huang, T.Z., Wu, G.C., Gu, X.M.: Quadratic spline collocation method for the time fractional subdiffusion equation. Appl. Math. Comput. 276, 252–265 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Liu, J., Fu, H.F., Chai, X.C., Sun, Y.N., Guo, H.: Stability and convergence analysis of the quadratic spline collocation method for time-dependent fractional diffusion equations. Appl. Math. Comput. 346, 633–648 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Liu, J., Fu, H.F., Wang, H., Chai, X.C.: A preconditioned fast quadratic spline collocation method for two-sided space-fractional partial differential equations. J. Comput. Appl. Math. 360, 138–156 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Liu, J., Fu, H.F., Zhang, J.S.: A QSC method for fractional subdiffusion equations with fractional bounding conditions and its application in parameters identification. Math. Comput. Simulat. 174, 153–174 (2020)

    MATH  Google Scholar 

  23. Liao, W.Y.: A compact high-order finite difference method for unsteady convection-diffusion equation. Int. J. Comput. Meth. Eng. Sci. Mech. 13, 135–145 (2012)

    MathSciNet  Google Scholar 

  24. Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259, 33–50 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Christara, C.C.: Quadratic spline collocation methods for elliptic partial differential equations. BIT. Numer. Math. 34, 33–61 (1994)

    MathSciNet  MATH  Google Scholar 

  26. Houstis, E.N., Christara, C.C., Rice, J.R.: Quadratic-spline collocation methods for two point boundary value problems. Int. J. Numer. Methods Eng. 26, 935–952 (1988)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11701197 and 11701196), the Fundamental Research Funds for the Central Universities (No. ZQN-702), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(No. ZQN-YX502). Thanks to the editor and reviewers for their valuable comments and suggestions which helped us to improve the results of this paper.

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  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, Fujian, China

    Zhaowei Tian, Shuying Zhai & Zhifeng Weng

  2. School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, Jiangsu, China

    Haifeng Ji

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  1. Zhaowei Tian
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  2. Shuying Zhai
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Correspondence to Zhifeng Weng.

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Tian, Z., Zhai, S., Ji, H. et al. A compact quadratic spline collocation method for the time-fractional Black–Scholes model. J. Appl. Math. Comput. 66, 327–350 (2021). https://doi.org/10.1007/s12190-020-01439-z

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