A Positivity-Preserving Splitting Method for 2D Black-Scholes Equations in Stochastic Volatility Models

  • Tatiana P. Chernogorova
  • Radoslav L. Valkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


In this paper we present a locally one-dimensional (LOD) splitting method to the two-dimensional Black-Scholes equation, arising in the Hull & White model for pricing European options with stochastic volatility. The parabolic equation degenerates on the boundary x = 0 and we apply to the one-dimensional subproblems the fitted finite-volume difference scheme, proposed in [8], in order to resolve the degeneration. Discrete maximum principle is proved and therefore our method is positivity-preserving. Numerical experiments are discussed.


Option Price Stochastic Volatility Global Error Splitting Method Stochastic Volatility Model 
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  1. 1.
    Chernogorova, T., Valkov, R.: Finite volume difference scheme for a degenerate parabolic equation in the zero-coupon bond pricing. Math. and Comp. Modeling 54, 2659–2671 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Clavero, C., Jorge, J.C., Lisbona, F.: Uniformly convergent schemes for singular perturbation problems combining alternating directions and exponential fitting techniques. In: Miller, J.J.H. (ed.) Applications of Advanced Computational Methods for Boundary and Interior Layers, pp. 33–52. Boole Press, Dublin (1993)Google Scholar
  3. 3.
    D’Yakonov, E.G.: Difference schemes with splitting operator for multidimensional non-stationary problem. Zh. Vychisl. Mat. i Mat. Fiz. 2, 549–568 (1962)Google Scholar
  4. 4.
    int’ Hout, K.J., Foulon, S.: ADI finite difference schemes for option pricing in the Heston model with correlation. Int. J. Numer. Anal. Mod. 7, 303–320 (2010)MathSciNetGoogle 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.
    Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer, Heidelberg (2003)CrossRefzbMATHGoogle Scholar
  7. 7.
    Hull, J., White, A.: The pricing of options on assets with stochastic volatilities. J. Financ. 42, 281–300 (1987)CrossRefGoogle Scholar
  8. 8.
    Wang, S.: A novel fitted finite volume method for Black-Sholes equation governing option pricing. IMA J. of Numer. Anal. 24, 699–720 (2004)CrossRefGoogle Scholar
  9. 9.
    Yanenko, N.N.: The Method of Fractional Steps. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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