An Efficient Sparse Grid Galerkin Approach for the Numerical Valuation of Basket Options Under Kou’s Jump-Diffusion Model

  • Michael Griebel
  • Alexander Hullmann
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 88)


We use a sparse grid approach to discretize a multi-dimensional partial integro-differential equation (PIDE) for the deterministic valuation of European put options on Kou’s jump-diffusion processes. We employ a generalized generating system to discretize the respective PIDE by the Galerkin approach and iteratively solve the resulting linear system. Here, we exploit a newly developed recurrence formula, which, together with an implementation of the unidirectional principle for non-local operators, allows us to evaluate the operator application in linear time. Furthermore, we exploit that the condition of the linear system is bounded independently of the number of unknowns. This is due to the use of the Galerkin generating system and the computation of L 2-orthogonal complements. Altogether, we thus obtain a method that is only linear in the number of unknowns of the respective generalized sparse grid discretization. We report on numerical experiments for option pricing with the Kou model in one, two and three dimensions, which demonstrate the optimal complexity of our approach.


Option Price Sparse Grid Basket Option Full Grid Jump Term 
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.


  1. 1.
    L. Andersen and J. Andreasen. Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing. Review of Derivatives Research, 4(3):231–262, 2000.CrossRefGoogle Scholar
  2. 2.
    R. Bellman. Adaptive Control Processes: A Guided Tour. Princeton University Press, 1961.Google Scholar
  3. 3.
    H. Bungartz and M. Griebel. Sparse grids. Acta Numerica, 13:1–123, 2004.MathSciNetCrossRefGoogle Scholar
  4. 4.
    H. Bungartz, A. Heinecke, D. Pflüger, and S. Schraufstetter. Option pricing with a direct adaptive sparse grid approach. Journal of Computational and Applied Mathematics, 2011.Google Scholar
  5. 5.
    G. Beylkin and M. Mohlenkamp. Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA, 99:10246–10251, 2002.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    F. Black and M. Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy, 81(3):637–654, 1973.CrossRefGoogle Scholar
  7. 7.
    H. Bungartz. Dünne Gitter und deren Anwendung bei der adaptiven Lösung der dreidimensionalen Poisson-Gleichung. Dissertation, Fakultät für Informatik, Technische Universität München, November 1992.Google Scholar
  8. 8.
    R. Balder and C. Zenger. The solution of multidimensional real Helmholtz equations on sparse grids. SIAM J. Sci. Comput., 17:631–646, May 1996.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    R. Cont and P. Tankov. Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series, 2004.Google Scholar
  10. 10.
    R. Cont and E. Voltchkova. A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal., 43:1596–1626, 2005.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    C. Feuersänger. Sparse Grid Methods for Higher Dimensional Approximation. Dissertation, Institut für Numerische Simulation, Universität Bonn, September 2010.Google Scholar
  12. 12.
    T. Gerstner and M. Griebel. Dimension–adaptive tensor–product quadrature. Computing, 71(1):65–87, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    M. Griebel and H. Harbrecht. On the construction of sparse tensor product spaces. Mathematics of Computations, 2012. to appear. Also available as INS Preprint No. 1104, University of Bonn.Google Scholar
  14. 14.
    M. Griebel and S. Knapek. Optimized general sparse grid approximation spaces for operator equations. Mathematics of Computations, 78(268):2223–2257, 2009.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    M. Griebel and P. Oswald. On additive Schwarz preconditioners for sparse grid discretizations. Numer. Math., 66:449–464, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    M. Griebel and P. Oswald. Tensor product type subspace splitting and multilevel iterative methods for anisotropic problems. Adv. Comput. Math., 4:171–206, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    M. Griebel and D. Oeltz. A sparse grid space-time discretization scheme for parabolic problems. Computing, 81(1):1–34, 2007.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    M. Griebel. Multilevelmethoden als Iterationsverfahren über Erzeugendensystemen. Teubner Skripten zur Numerik. Teubner, Stuttgart, 1994.zbMATHCrossRefGoogle Scholar
  19. 19.
    E. Kaasschieter. Preconditioned conjugate gradients for solving singular systems. Journal of Computational and Applied Mathematics, 24(1–2):265–275, 1988.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    S. Kou. A jump-diffusion model for option pricing. Management Science, 48(8):1086–1101, 2002.zbMATHCrossRefGoogle Scholar
  21. 21.
    S. Kou. Jump-diffusion models for asset pricing in financial engineering. In John R. Birge and Vadim Linetsky, editors, Financial Engineering, volume 15 of Handbooks in Operations Research and Management Science, pages 73–116. Elsevier, 2007.Google Scholar
  22. 22.
    R. Merton. Theory of rational option pricing. Bell Journal of Economics, 4(1):141–183, 1973.MathSciNetCrossRefGoogle Scholar
  23. 23.
    R. Merton. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3:125–144, 1976.zbMATHCrossRefGoogle Scholar
  24. 24.
    C. Schwab N. Hilber, S. Kehtari and C. Winter. Wavelet finite element method for option pricing in highdimensional diffusion market models. Research Report 2010-01, Seminar for Applied Mathematics, Swiss Federal Institute of Technology Zurich, 2010.Google Scholar
  25. 25.
    D. Oeltz. Ein Raum-Zeit Dünngitterverfahren zur Diskretisierung parabolischer Differentialgleichungen. Dissertation, Institut für Numerische Simulation, Universität Bonn, 2006.Google Scholar
  26. 26.
    C. Reisinger. Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben. Dissertation, Ruprecht-Karls-Universität Heidelberg, 2004.zbMATHGoogle Scholar
  27. 27.
    N. Reich, C. Schwab, and C. Winter. On Kolmogorov equations for anisotropic multivariate Lévy processes. Finance and Stochastics, 14(4):527–567, 2010.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    V. Thomée. Galerkin finite element methods for parabolic problems. Springer series in computational mathematics. Springer, 1997.zbMATHGoogle Scholar
  29. 29.
    J. Toivanen. Numerical valuation of European and American options under Kou’s jump-diffusion model. SIAM J. Sci. Comput., 30(4):1949–1970, 2008.MathSciNetCrossRefGoogle Scholar
  30. 30.
    A. Zeiser. Fast matrix-vector multiplication in the sparse-grid Galerkin method. J. Sci. Comput., 47(3):328–346, 2011.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute for Numerical SimulationUniversity of BonnBonnGermany

Personalised recommendations