Numerical Methods for Stable Modeling in Financial Risk Management

  • Stoyan Stoyanov
  • Borjana Racheva-Jotova


The seminal work of Mandelbrot and Fama, carried out in the 1960s, suggested the class of α-stable laws as a probabilistic model of financial assets returns. Stable distributions possess several properties which make plausible their application in the field of finance — heavy tails, excess kurtosis, domains of attraction. Unfortunately working with stable laws is very much obstructed by the lack of closed-form expressions for probability density functions and cumulative distribution functions. In the current paper we review statistical and numerical techniques which make feasible the application of stable laws in practice.


Fast Fourier Transform Stable Parameter Stable Modeling Stable Distribution Financial Time Series 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R.J. Adler, R. Feldman and M. Taqqu, (Eds.) A Practical Guide to Heavy Tails: Statistical Techniques and Applications, Birkhäuser, Boston, Basel, Berlin, 1998.MATHGoogle Scholar
  2. [2]
    V. Akgiray, C. G. Lamoureux, Estimation of Stable-law Parameter: A Comparative Study, Journal of Business and Economic Statistics, 7, 85–93, (1989).Google Scholar
  3. [3]
    H. Bergström, On Some Expansions of Stable Distributions, Arkiv for Matematik II, 375–378 (1952).CrossRefGoogle Scholar
  4. [4]
    T. Doganoglu, S. Mittnik, An Approximation Procedure for Asymmetric Stable Paretian Densities, Computational Statistics, 13, 463–475 (1998).MATHGoogle Scholar
  5. [5]
    W. H. DuMouchel, Stable Distributions in Statistical Inference, Ph.D. dissertation, Department of Statistics, Yale University (1971).Google Scholar
  6. [6]
    W. H. DuMouchel, On the Asymptotic Normality of the Maximum-Likelihood Estimate when Sampling from a Stable Distribution, Annals of Statistics, 1, 948–957 (1973).MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    P. Embrechts, C. Klüppelberg, T. Mikosch, Modeling Extremal Events for Insurance and Finance, Springer, Berlin, Heidelberg, New York, 1997.CrossRefGoogle Scholar
  8. [8]
    E. Fama, The behavior of stock market prices, J. Bus. Univ. Chicago, 38, 34–105, (1965).Google Scholar
  9. [9]
    E. Fama, R. Roll, Some Properties of Symmetric Stable Distributions, Journal of the American Statistical Association, 63, 817–836, (1968).MathSciNetCrossRefGoogle Scholar
  10. [10]
    E. Fama, R. Roll, Parameter Estimates for Symmetric Stable Distributions, Journal of the American Statistical Association, 66, 331–338, (1971).MATHCrossRefGoogle Scholar
  11. [11]
    I. A. Koutrouvelis, Regression-type Estimation of the Parameters of Stable Laws, Journal of the American Statistical Association, 75, 918–928, (1980).MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    I. A. Koutrouvelis, An Iterative Procedure for the Estimation of the Parameters of Stable Laws, Communications in Statistics. Simulation and Computation, 10, 17–28, (1981).MathSciNetCrossRefGoogle Scholar
  13. [13]
    B. Mandelbrot, The variation of certain speculative prices, J. Bus. Univ. Chicago, 26, 394–419, (1963).Google Scholar
  14. [14]
    J. H. McCulloch, Simple Consistent Estimators of Stable Distribution Parameters, Communications in Statistics. Simulation and Computation, 15, 1109–1136,(1986).MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    J. P. Nolan, Numerical Computation of Stable Densities and Distribution Functions, Stochastic Models, 13, 759–774, (1997).MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    S. J. Press, Estimation in Univariate and Multivariate Stable Distribution, Journal of the American Statistical Association, 67, 842–846, (1972b).MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    S.T. Rachev and S. Mittnik Stable Paretian Models in Finance, John Wiley&Sons Ltd, 2000.Google Scholar
  18. [18]
    G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance, Chapman and Hall, New York, London, 1994.MATHGoogle Scholar
  19. 19]
    V. M. Zolotarev, On the Representations of Stable Laws by Integrals, Selected Translations in Mathematical Statistics and Probability, 6, 84–88, American Mathematical Society, Providence, Rhode Island, 1964.Google Scholar
  20. [20]
    V. M. Zolotarev, One-Dimensional Stable Distributions, Nauka (in Russian), Moscow, 1983.MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Stoyan Stoyanov
    • 1
  • Borjana Racheva-Jotova
    • 1
  1. 1.Faculty of Economics and BusinessSofia University, BulgariaBulgaria

Personalised recommendations