BIT Numerical Mathematics

, Volume 51, Issue 1, pp 217–233 | Cite as

A multigrid preconditioner for an adaptive Black-Scholes solver

  • Alison RamageEmail author
  • Lina von Sydow


This paper is concerned with the efficient solution of the linear systems of equations that arise from an adaptive space-implicit time discretisation of the Black-Scholes equation. These nonsymmetric systems are very large and sparse, so an iterative method will usually be the method of choice. However, such a method may require a large number of iterations to converge, particularly when the timestep used is large (which is often the case towards the end of a simulation which uses adaptive timestepping). An appropriate preconditioner is therefore desirable. In this paper we show that a very simple multigrid algorithm with standard components works well as a preconditioner for these problems. We analyse the eigenvalue spectrum of the multigrid iteration matrix for uniform grid problems and illustrate the method’s efficiency in practice by considering the results of numerical experiments on both uniform grids and those which use adaptivity in space.


Finite differences Multigrid Preconditioning Option pricing Adaptive grids 

Mathematics Subject Classification (2000)

65M06 65F08 65M55 


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  1. 1.
    Christara, C., Dang, D.: Adaptive and high-order methods for valuing American options. J. Comput. Financ. (2010, to appear).
  2. 2.
    Elman, H., Ramage, A.: Fourier analysis of multigrid for a model two-dimensional convection-diffusion equation. BIT Numer. Math. 46, 283–299 (2006) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Freund, R., Nachtigal, N.: QMR: a Quasi-Minimal Residual method for non-Hermitian linear systems. Numer. Math. 60, 315–339 (1991) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Greenbaum, A.: Iterative Methods for Solving Linear Systems. SIAM, Philadelphia (1997) zbMATHGoogle Scholar
  5. 5.
    Hackbusch, W.: Multi-grid Methods and Applications. Springer, New York (1980) Google Scholar
  6. 6.
    Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer Series in Comput. Mathematics, vol. 14. Springer, New York (1996) zbMATHGoogle Scholar
  7. 7.
    Linde, G., Persson, J., von Sydow, L.: A highly accurate adaptive finite difference solver for the Black-Scholes equation. Int. J. Comput. Math. 86, 2104–2121 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Lötstedt, P., Persson, J., von Sydow, L., Tysk, J.: Space-time adaptive finite difference method for European multi-asset options. Comput. Math. Appl. 53, 1159–1180 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Lötstedt, P., Söderberg, S., Ramage, A., Hemmingsson-Frändén, L.: Implicit solution of hyperbolic equations with space-time adaptivity. BIT Numer. Math. 42, 134–158 (2002) CrossRefzbMATHGoogle Scholar
  10. 10.
    Oosterlee, C.: On multigrid for linear complementarity problems with application to American-style options. Electron. Trans. Numer. Anal. 15, 165–185 (2003) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Persson, J., von Sydow, L.: Pricing European multi-asset options using a space-time adaptive FD-method. Comput. Vis. Sci. 10, 173–183 (2007) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Persson, J., von Sydow, L.: Pricing American options using a space-time adaptive finite difference method. Math. Comput. Simul. 80, 1922–1935 (2010) CrossRefzbMATHGoogle Scholar
  13. 13.
    Ramage, A.: A multigrid preconditioner for stabilised discretisations of advection-diffusion problems. J. Comput. Appl. Math. 101, 187–203 (1999) CrossRefMathSciNetGoogle Scholar
  14. 14.
    Ramage, A., Elman, H.: Some observations on multigrid convergence for convection-diffusion equations. Comput. Vis. Sci. 10, 43–56 (2007) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Reisinger, C., Wittum, G.: Efficient hierarchical approximation of high-dimensional option pricing problems. SIAM J. Sci. Comput. 29, 440–458 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Saad, Y.: Iterative Methods for Sparse Linear Systems. PWS, Boston (1996) zbMATHGoogle Scholar
  17. 17.
    Saad, Y., Schultz, M.: GMRES: a Generalized Minimal Residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Trottenberg, U., Oosterlee, C., Schüller, A.: Multigrid. Academic Press, London (2001) zbMATHGoogle Scholar
  19. 19.
    van der Vorst, H.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992) CrossRefzbMATHGoogle Scholar
  20. 20.
    Wesseling, P.: An Introduction to Multigrid Methods. Wiley, Chichester (1991) Google Scholar
  21. 21.
    Wienands, R., Oosterlee, C., Washio, T.: Fourier analysis of GMRES(m) preconditioned by multigrid. SIAM J. Sci. Comput. 22, 582–603 (2000) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Zubair, H., Leentvaar, C., Oosterlee, C.: Efficient d-multigrid preconditioners for sparse-grid solution of high-dimensional partial differential equations. Int. J. Comput. Math. 84, 1131–1149 (2007) CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of StrathclydeGlasgowUK
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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