Numerical Algorithms

, Volume 75, Issue 3, pp 569–585 | Cite as

Numerical simulation of a Finite Moment Log Stable model for a European call option

Original Paper


Compared to the classical Black-Scholes model for pricing options, the Finite Moment Log Stable (FMLS) model can more accurately capture the dynamics of the stock prices including large movements or jumps over small time steps. In this paper, the FMLS model is written as a fractional partial differential equation and we will present a new numerical scheme for solving this model. We construct an implicit numerical scheme with second order accuracy for the FMLS and consider the stability and convergence of the scheme. In order to reduce the storage space and computational cost, we use a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) to solve the discrete scheme. A numerical example is presented to show the efficiency of the numerical method and to demonstrate the order of convergence of the implicit numerical scheme. Finally, as an application, we use the above numerical technique to price a European call option. Furthermore, by comparing the FMLS model with the classical B-S model, the characteristics of the FMLS model are also analyzed.


The FMLS model Riemann-Liouville fractional derivative Numerical simulation Fast Fourier transform Bi-conjugrate gradient stabilized method European option 


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  1. 1.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–659 (1973)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boyarchenko, S., Levendorskiǐ, S.: Non-Gaussian Merton-Black-Scholes Theory, vol. 9. World Scientific, Singapore (2002)Google Scholar
  3. 3.
    Carr, P., Geman, H., Madan, D.B., Yor, M.: Stochastic volatility for Lévy processes. Math. Financ. 13, 345–382 (2003)CrossRefMATHGoogle Scholar
  4. 4.
    Carr, P., Wu, L.: The finite moment log stable process and option pricing. J. Financ. 58(2), 597–626 (2003)CrossRefGoogle Scholar
  5. 5.
    Cartea, A., del-Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Physica A 374(2), 749–763 (2007)CrossRefGoogle Scholar
  6. 6.
    Chen, W.: Numerical methods for fractional Black-Scholes equations and variational inequalities governing option pricing, The University of Western Australia PHD thesis (2013)Google Scholar
  7. 7.
    Chen, W., Xu, X., Zhu, S.: Analytically pricing European-style options under the modified Black-Scholes equation with a spatial-fractional derivative. Q. Appl. Math. 72(3), 597–611 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Davis, M.H.A., Panas, V.G., Zariphopoulou, T.: European option pricing with transaction costs. SIAM J. Control. Optim. 31(2), 470–493 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gorenflo, R., Mainardi, F.: Approximation of Levy-Feller diffusion by random walk. J. Anal. Appl. 18, 231–246 (1999)MATHGoogle Scholar
  10. 10.
    Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6, 327–343 (1993)CrossRefGoogle Scholar
  11. 11.
    Huang, F., Liu, F.: The fundamental solution of the space-time fractional advection-dispersion equation. J. Appl. Math. Comput. 18, 339–350 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hull, J.C., White, A.D. : The pricing of options on assets with stochastic volatilities. J. Financ. 42, 281–300 (1987)CrossRefMATHGoogle Scholar
  13. 13.
    Hull, J.C.: Options, Futures, and Other Derivatives, the 7th ed. (With CD). Pearson Education, India (2010)Google Scholar
  14. 14.
    Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time space Caputo Riesz fractional advection diffusion equations on a finite domain. J. Math. Anal. Appl. 389, 1117–1127 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Koponen, I.: Analytic approach to the problem of convergence of truncated Levy flights towards the Gaussian stochastic process. Phys. Rev. E 52, 1197–1199 (1995)CrossRefGoogle Scholar
  16. 16.
    Li, W.: The numerical solution of fractional order equation in financial models and its application, Hangzhou University of Electronic Science and Technology, Master’s thesis (2009)Google Scholar
  17. 17.
    Liu, F., Anh, V., Turner, I., Zhuang, P.: Numerical simulation for solute transport in fractal porous media. ANZIAM J. 45(E), 461–473 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comp. 191, 12–20 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Liu, Q., Liu, F., Turner, I., Anh, V.: Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method. J. Phys. Comput. 222, 57–70 (2007)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Mandelbrot, B.: The variation of certain speculative prices. J. Bus. Univ. Chicago 36, 39–419 (1963)Google Scholar
  21. 21.
    Marom, O., Momoniat, E.: A comparison of numerical solution of fractional diffusion models in finance. Nonlinear Anal. Real World Appl. 10, 343–3442 (2009)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Merton, R.C.: Theory of Rational Option Pricing. Bell J. Econ. Manag. Sci. (The RAND Corporation) 4(1), 141–183 (1973)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3(1-2), 125–144 (1976)CrossRefMATHGoogle Scholar
  25. 25.
    Podlubny, I.: Fractional differential equations academic press (1999)Google Scholar
  26. 26.
    Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations Springer (1997)Google Scholar
  27. 27.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, 2nd ed. Springer (2007)Google Scholar
  28. 28.
    Tian, W.Y., Zhou, H., Deng, W.H. arXiv:1201.5949v3 [math.NA] (2012)
  29. 29.
    Van der Vorst, H.A.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. and Stat. Comput. 13(2), 631–644 (1992)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wang, H., Wang, K., Sircar, T.: A direct O(N log2N) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zhang, H., Liu, F., Anh, V.: Numerical approximation of Lévy-Feller diffusion equation and its probability interpretation. J. Comput. Appl. Math. 206, 1098–1115 (2007)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Zhang, H., Liu, F., Phanikumar, M.S., Meerschaert, M.M.: A novel numerical method for the time variable fractional order mobile-immobile advection-dispersion model. Comput. Math. Appl. 66, 693–701 (2013)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Zhang, H., Liu, F., Zhuang, P., Turner, I., Anh, V.: Numerical analysis of a new space-time variable fractional order advection-dispersion equation. Appl. Math. Comput. 242, 541–550 (2014)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • H. Zhang
    • 1
  • F. Liu
    • 2
  • I. Turner
    • 2
  • S. Chen
    • 3
    • 4
  • Q. Yang
    • 2
  1. 1.School of Mathematical and Computer SciencesFuzhou UniversityFuzhouChina
  2. 2.School of Mathematical SciencesQueensland University of TechnologyBrisbaneAustralia
  3. 3.School of Economic MathematicsSouthwestern University of Finance and EconomicsChengduChina
  4. 4.School of MathematicsShandong UniversityJinanChina

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