Numerical Algorithms

, Volume 74, Issue 1, pp 59–75 | Cite as

Fourth-order compact schemes for a parabolic-ordinary system of European option pricing liquidity shocks model

  • Miglena N. KolevaEmail author
  • Walter Mudzimbabwe
  • Lubin G. Vulkov
Original Paper


This paper provides a numerical investigation for European options under parabolic-ordinary system modeling markets to liquidity shocks. Our main results concern construction and analysis of fourth order in space compact finite difference schemes (CFDS). Numerical experiments using Richardson extrapolation in time are discussed.


Option pricing High-order compact difference scheme Convergence Richardson extrapolation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing, volume 30 of Frontiers in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 297 pp (2005)Google Scholar
  2. 2.
    Atkinson, K.E.: An Introduction to Numerical Analysis, 2nd edn. Wiley, New York (1988)Google Scholar
  3. 3.
    Ciment, M., Leventhal, S., Weinberg, B.: The operator compact implicit method for parabolic equations. J. Comp. Phys. 28(2), 135–166 2 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dremkova, E., Ehrhardt, M.: A high-order compact method for nonlinear Black-Scholes option pricing equations of American Options. Int. J. Comput. Math 88(13), 2782–2797 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Düring, B., Fournié, M., Jüngel, A.: High-order compact finite dfference schemes for a nonlinear Black-Scholes equation. Intern. J. Theor. Appl. Finance 6 (7), 767–789 (2003)CrossRefzbMATHGoogle Scholar
  6. 6.
    Düring, B., Fournié, M., Jüngel, A.: Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation. Math. Mod. Num. Anal 38(2), 359–369 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Düring, B., Heuer, C.: High-order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions. SIAM J. Numer. Anal 53(5), 2113–2134 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Faragó, I., Izsák, F., Szabó, T.: An IMEX scheme combined with Richardson extrapolation methods for some reaction-diffusion equations. Q. J. Hung. Meteorol. Serv. 117(2), 201–218 (2013)Google Scholar
  9. 9.
    Gupta, M.M., Manohar, R.P., Stephenson, J.W.: A single cell high order scheme for the convection-diffusion equation with variable coeficients. Int. J. Numer. Methods Fluids 4, 641–651 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gustafson, B., Kreiss, H., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley, New York (1995)zbMATHGoogle Scholar
  11. 11.
    Karaa, S., Zhang, J.: Convergence and performance of iterative methods for solving variable coeficient convection-diffusion equation with a fourth-order compact dfference scheme. Comput. Math. Appl 44, 457–479 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kreiss, H., Oliger, J.: Methods for the approximate solutions of time-dependent problems, GARP Publication Series N10. Global Atmospheric Research Program (1973)Google Scholar
  13. 13.
    Kreiss, H.O., Thomee, V., Widlund, O.: Smoothing of initial data and rates of convergence for parabolic difference equations., Commun. Pure Appl. Math 23, 241–259 (1970)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Liao, W., Khaliq, A.Q.M.: High order compact scheme for solving nonlinear Black-Scholes equation with transaction cost. Int. J. Comput. Math 86(6), 1009–1023 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ludkovski, M, Shen, Q.: European option pricing with liquidity shocks. Int. J. of Theor. Appl. Finance 16(7), 1350043 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mudzimbabwe, W., Vulkov, L.: IMEX schemes for a parabolic-ODE system of European options with liquidity schocks. J. Comp. Appl. Math. doi: 10.1016/ (In press)
  17. 17.
    Rigal, A.: High order difference schemes for unsteady one-dimensional diffusion-convection problems. J. Comp. Phys. 114, 59–76 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Spotz, W.F., Carey, G.F.: Extension of high-order compact schemes to timedependent problems. Numer. Methods Partial Diff. Equa. 17, 657–672 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Tangman, D.Y., Gopaul, A., Bhuruth, M.: Numerical pricing of options using high-order compact finite difference schemes. J. Comp. Appl. Math 218, 270–280 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, L., Chen, W., Wang, C.: An energy-conserving second order numerical scheme for nonlinear hyperbolic equation with exponential nonlinear term. J. Appl. Math. 280, 347–366 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhao, J., Dai,W., Niu, T.: Fourth-order compact schemes of a heat conduction problem with Neumann boundary conditions. Numer. Meth. PDE 23(5), 949–959 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Miglena N. Koleva
    • 1
    Email author
  • Walter Mudzimbabwe
    • 2
  • Lubin G. Vulkov
    • 2
  1. 1.Department of MathematicsRuse UniversityRuseBulgaria
  2. 2.Department of Applied Mathematics and StatisticsRuse UniversityRuseBulgaria

Personalised recommendations