A 2nd-Order FDM for a 2D Fractional Black-Scholes Equation

  • W. Chen
  • S. WangEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10187)


We develop a finite difference method (FDM) for a 2D fractional Black-Scholes equation arising in the optimal control problem of pricing European options on two assets under two independent geometric Lévy processes. We establish the convergence of the method by showing that the FDM is consistent, stable and monotone. We also show that the truncation error of the FDM is of 2nd order. Numerical experiments demonstrate that the method produces financially meaningful results when used for solving practical problems.


Fractional Derivative Option Price Truncation Error Finite Difference Method Underlying Asset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



S. Wang’s work was partially supported by the AOARD Project #15IOA095.


  1. 1.
    Barles, G.: Convergence of numerical schemes for degenerate parabolic equations arising in finance theory. In: Rogers, L.C.G., Talay, D. (eds.) Numerical Methods in Finance. Cambridge University Press, Cambridge (1997)Google Scholar
  2. 2.
    Cartea, A., del-Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Phys. A 374, 749–763 (2007)CrossRefGoogle Scholar
  3. 3.
    Chen, W., Wang, S.: A penalty method for a fractional order parabolic variational inequality governing American put option valuation. Comput. Math. Appl. 67, 77–90 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, W., Wang, S.: A finite difference method for pricing European and American options under a geometric Lévy process. J. Ind. Manag. Optim. 11, 241–264 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes, vol. 2. Chapman & Hall, New York (2004)zbMATHGoogle Scholar
  6. 6.
    Cont, R., Voltchkova, E.: A finite difference scheme for option pricing in jump-diffusion and exponential Lévy models. Ecole Polytechnique Rapport Interne CMAP Working Paper No. 513 (2005)Google Scholar
  7. 7.
    Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Haslinger, J., Miettinen, M.: Finite Element Method for Hemivariational Inequalities. Kluwer Academic Publisher, Dordrecht-Boston-London (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Koleva, M.N., Vulkov, L.G.: Numerical solution of time-fractional BlackScholes equation. Comput. Appl. Math. (2016). doi: 10.1007/s40314-016-0330-z
  11. 11.
    Lesmana, D.C., Wang, S.: An upwind finite difference method for a nonlinear Black-Scholes equation governing European option valuation. Appl. Math. Comput. 219, 8818–8828 (2013)zbMATHGoogle Scholar
  12. 12.
    Lesmana, D.C., Wang, S.: Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs. Appl. Math. Comput. 251, 318–330 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Li, W., Wang, S.: Penalty approach to the HJB equation arising in European stock option pricing with proportional transaction costs. J. Optim. Theory Appl. 143, 279–293 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Li, W., Wang, S.: Pricing American options under proportional transaction costs using a penalty approach and a finite difference scheme. J. Ind. Manag. Optim. 9, 365–389 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lynch, V.E., Carreras, B.A., del-Castillo-Negrete, D., Ferreira-Mejias, K.M., Hicks, H.R.: Numerical methods for the solution of partial differential equations of fractional order. J. Comput. Phys. 192, 406–421 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Meerschaert, M.M., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211, 249–261 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, Cambridge (1974)zbMATHGoogle Scholar
  18. 18.
    Tadjeran, C., Meerschaert, M.M.: A second-order accurate numerical method for the two-dimensional fractional diffusion equation. J. Comput. Phys. 220, 813–823 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, S.: A novel fitted finite volume method for the Black-Scholes equation governing option pricing. IMA J. Numer. Anal. 24, 699–720 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, S., Yang, X.Q., Teo, K.L.: Power penalty method for a linear complementarity problem arising from American option valuation. J. Optim. Theory Appl. 129(2), 227–254 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wang, S., Zhang, S., Fang, Z.: A superconvergent fitted finite volume method for BlackScholes governing European and American option valuation. Numer. Methods Partial Differ. Equ. 31, 1190–1208 (2015)CrossRefzbMATHGoogle Scholar
  22. 22.
    Wang, S., Zhang, K.: An interior penalty method for a finite-dimensional linear complementarity problem in financial engineering. Optim. Lett. (2016). doi: 10.1007/s11590-016-1050-4
  23. 23.
    Wilmott, P., Dewynne, J., Howison, S.: Option Pricing: Mathematical Models and Computation. Oxford Financial Press, Oxford (1993)zbMATHGoogle Scholar
  24. 24.
    Zhang, K., Wang, S.: Pricing American bond options using a penalty method. Automatica 48, 472–479 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CSIRO Data61DocklandsAustralia
  2. 2.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia

Personalised recommendations