Finite-Volume Difference Scheme for the Black-Scholes Equation in Stochastic Volatility Models

  • Tatiana Chernogorova
  • Radoslav Valkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6046)


We study numerically the two-dimensional Black-Scholes equation in stochastic volatility models [3]. For these models, starting from the conservative form of the equation, we construct a finite-volume difference scheme using the appropriate boundary conditions. The scheme is first order accurate in the space grid size. We also present some results from numerical experiments that confirm this.


Black-Scholes equation dynamical boundary condition finite difference finite-volume 


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  1. 1.
    Chernogorova, T., Valkov, R.: A computational scheme for a problem in the zero-coupon bond pricing. Amer. Inst. of Phys. (in press)Google Scholar
  2. 2.
    Ekstrom, E., Lotstedt, P., Tysk, J.: Boundary values and finite difference methods for the single-factor term structure equation. Appl. Math. Finance 16, 252–259 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ekstrom, E., Tysk, J.: The Black-Scholes equation in stochastic volatility models. J. Math. Anal. Appl. 368, 498–507 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Finan. Stud. 6, 327–343 (1993)CrossRefGoogle Scholar
  5. 5.
    Huang, C.-S., Hung, C.-H., Wang, S.: A fitted finite volume method for the valuation of options on assets with stochastic volatilities. Computing 77, 297–320 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lions, P.-L., Musiela, M.: Correlations and bounds for stochastic volatility models. Ann. Inst. H. Poincare Anal. Non Lineare 24, 1–16 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Oleinik, O.A., Radkevic, E.V.: Second Order Equations with Nonnegative Characteristic Form. Plenum Press, New York (1973)CrossRefGoogle Scholar
  8. 8.
    Thomas, J.W.: Numerical Partial Differential Equations. Springer, Berlin (1995)zbMATHGoogle Scholar
  9. 9.
    Wang, S.: A novel fitted finite volume method for Black-Scholes equation governing option pricing. IMA J. of Numer. Anal. 24, 699–720 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Tatiana Chernogorova
    • 1
  • Radoslav Valkov
    • 1
  1. 1.Faculty of Mathematics and InformaticsSofia UniversityBulgaria

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