Optimal Quantization Methods and Applications to Numerical Problems in Finance

  • Gilles Pagès
  • Huyên Pham
  • Jacques Printems


We review optimal quantization methods for numerically solving nonlinear problems in higher dimensions associated with Markov processes. Quantization of a Markov process consists in a spatial discretization on finite grids optimally fitted to the dynamics of the process. Two quantization methods are proposed: the first one, called marginal quantization, relies on an optimal approximation of the marginal distributions of the process, while the second one, called Markovian quantization, looks for an optimal approximation of transition probabilities of the Markov process at some points. Optimal grids and their associated weights can be computed by a stochastic gradient descent method based on Monte Carlo simulations. We illustrate this optimal quantization approach with four numerical applications arising in finance: European option pricing, optimal stopping problems and American option pricing, stochastic control problems and mean-variance hedging of options and filtering in stochastic volatility models.


Markov Chain Quantization Method Risky Asset Quantization Error Euler Scheme 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bally V. (2002): The Central Limit Theorem for a non-linear algorithm based on quantization, forthcoming in Proceedings of the Royal Society.Google Scholar
  2. [2]
    Bally V., Pagès G. (2000): A quantization algorithm for solving discrete time multidimensional optimal stopping problems, pre-print LPMA-628, Laboratoire de Probabilités & Modèles Aléatoires, Universités Paris 6&7 (France), to appear in Bernoulli.Google Scholar
  3. [3]
    Bally V., Pagès G. (2003): Error analysis of the quantization algorithm for obstacle problems, Stochastic Processes and their Applications, 106,n01, 1–47.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Bally V, Pagès G., Printems J. (2001): A stochastic quantization method for nonlinear problems, Monte Carlo Methods and Applications, 7,n01-2, 21–34.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Bally V, Pagès G., Printems J. (2002): A quantization method for pricing and hedging multi-dimensional American style options, pre-print LPMA-753, Laboratoire de Probabilités&Modèles Aléatoires, Université Paris 6&7 (France), to appear in Mathematical Finance.Google Scholar
  6. [6]
    Bally V., Pagès G., Printems J. (2003): First order schemes in the numerical quantization method, Mathematical Finance, 13,n01, 1–16.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Barles G., Souganidis P. (1991): Convergence of approximation schemes for fully nonlinear second-order equations, Asymptotics Analysis, 4, 271–283.MathSciNetMATHGoogle Scholar
  8. [8]
    Bucklew J., Wise G. (1982): Multidimensional Asymptotic Quantization Theory with r th Power distortion Measures, IEEE Transactions on Information Theory, Special issue on Quantization, 28,n0 2, 239–247.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Duflo, M. (1997): Random Iterative Models, Coll. Applications of Mathematics, 34, Springer-Verlag, Berlin, 1997.Google Scholar
  10. [10]
    Elliott R., Aggoun L. and J. Moore (1995): Hidden Markov Models, Estimation and Control, Springer Verlag.Google Scholar
  11. [11]
    Fort J.C., Pagès G. (2002): Asymptotics of optimal quantizers for some scalar distributions, Journal of Computational and Applied Mathematics, 146, 253–275.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Gersho A., Gray R. (eds.) (1982): IEEE Transactions on Information Theory, Special issue on Quantization, 28.Google Scholar
  13. [13]
    Graf S., Luschgy H. (2000): Foundations of Quantization for Probability Distributions, Lecture Notes in Mathematics n01730, Springer, Berlin.MATHCrossRefGoogle Scholar
  14. [14]
    Kieffer J. (1982): Exponential rate of Convergence for the Lloyd’s Method I, IEEE Transactions on Information Theory, Special issue on Quantization, 28,n02, 205–210.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Kohonen T. (1982): Analysis of simple self-organizing process, Biological Cybernetics, 44, 135–140.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Kushner H.J., Dupuis P. (2001): Numerical methods for stochastic control problems in continuous time, 2nd edition, Applications of Mathematics, 24, Stochastic Modelling and Applied Probability, Springer-Verlag, New York.MATHGoogle Scholar
  17. [17]
    Kushner H.J., Yin G.G. (1997): Stochastic Approximation Algorithms and Applications, Springer, New York.MATHGoogle Scholar
  18. [18]
    Pagès G. (1997): A space vector quantization method for numerical integration, Journal of Computational and Applied Mathematics, 89, 1–38.CrossRefGoogle Scholar
  19. [19]
    Pagès G., Pham H. (2001): A quantization algorithm for multidimensional stochastic control problems, pre-print LPMA-697, Laboratoire de Probabilités et Modèles Aléatoires, Universités Paris 6&7 (France).Google Scholar
  20. [20]
    Pagès G., Pham H. (2002): Optimal quantization methods for nonlinear filtering with discrete-time observations, pre-print LPMA-778, Laboratoire de Probabilités et modeles aléatoires, Universités Paris 6&7 (France).Google Scholar
  21. [21]
    Pagès G., Printems J. (2003): Optimal quadratic quantization for numerics: the Gaussian case, Monte Carlo Methods and Applications, 9,n 02.Google Scholar
  22. [22]
    Villeneuve S., Zanette A. (2002) Parabolic A.D.I. methods for pricing american option on two stocks, Mathematics of Operation Research, 27,n 01, 121–149.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Gilles Pagès
    • 1
  • Huyên Pham
    • 1
  • Jacques Printems
    • 2
  1. 1.Laboratoire de Probabilités et Modèles Aléatoires CNRS, UMR 7599Université Paris 6Paris Cedex 05France
  2. 2.Centre de Mathématiques Faculté de Sciences et Technologie CNRS, UMR 8050Université Paris 12Créteil CedexFrance

Personalised recommendations