Estimation of α-Stable Sub-Gaussian Distributions for Asset Returns

  • Sebastian Kring
  • Svetlozar T. Rachev
  • Markus Höchstötter
  • Frank J. Fabozzi
Part of the Contributions to Economics book series (CE)

Fitting multivariate α-stable distributions to data is still not feasible in higher dimensions since the (non-parametric) spectral measure of the characteristic function is extremely difficult to estimate in dimensions higher than 2. This was shown by [3] and [15]. α-stable sub-Gaussian distributions are a particular (parametric) subclass of the multivariate α-stable distributions. We present and extend a method based on [16] to estimate the dispersion matrix of an α-stable sub-Gaussian distribution and estimate the tail index α of the dis¬tribution. In particular, we develop an estimator for the off-diagonal entries of the dispersion matrix that has statistical properties superior to the normal off-diagonal estimator based on the covariation. Furthermore, this approach allows estimation of the dispersion matrix of any normal variance mixture distribution up to a scale parameter. We demonstrate the behaviour of these estimators by fitting an α-stable sub-Gaussian distribution to the DAX30 components. Finally, we conduct a stable principal component analysis and calculate the coefficient of tail dependence of the prinipal components.


Random Vector Asset Return Stable Distribution Tail Dependence Tail Index 
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.
    P., F. Delbaen, J.M. Eber and D. Heath. 1999. Coherent Measure of Risk. Mathematical Finance 9, 203–228CrossRefGoogle Scholar
  2. 2.
    Bingham, N.H., R. Kiesel and R. Schmidt. 2003. A Semi-parametric Ap¬proach to Risk Management. Quantitative Finance 3, 241–250CrossRefGoogle Scholar
  3. 3.
    Cheng, B.N., S.T. Rachev. 1995. Multivariate Stable Securities in Finan¬cial Markets. Mathematical Finance 54, 133–153CrossRefGoogle Scholar
  4. 4.
    Embrechts, P., A. McNeil and D. Straumann. 1999. Correlation: Pitfalls and Alternatives. Risk 5, 69–71Google Scholar
  5. 5.
    Fama, E. 1965. The Behavior of Stock Market Prices. Journal of Business, 38, 34–105CrossRefGoogle Scholar
  6. 6.
    Fama, E. 1965. Portfolio Analysis in a Stable Paretian market. Manage¬ment Science, 11, 404–419CrossRefGoogle Scholar
  7. 7.
    Hardin Jr., C.D. 1984. Skewed Stable Variables and Processes. Technical report 79, Center for Stochastics Processes at the University of North Carolina, Chapel HillGoogle Scholar
  8. 8.
    Höchstötter, M., F.J. Fabozzi and S.T. Rachev. 2005. Distributional Anal¬ysis of the Stocks Comprising the DAX 30. Probability and Mathematical Statistics, 25, 363–383Google Scholar
  9. 9.
    Mandelbrot, B.B. 1963. New Methods in Statistical Economics. Journal of Political Economy, 71, 421–440CrossRefGoogle Scholar
  10. 10.
    Mandelbrot, B.B. 1963. The Variation of Certain Speculative Prices. Journal of Business, 36, 394–419CrossRefGoogle Scholar
  11. 11.
    Mandelbrot, B.B. 1963. New Methods in Statistical Economics. Journal of Political Economy, 71, 421–440CrossRefGoogle Scholar
  12. 12.
    Markowitz, H.M. 1952. Portfolio Selection. Journal of Finance 7, (1), 77–91CrossRefGoogle Scholar
  13. 13.
    McCulloch, J.H. 1952. Financial Applications of Stable Distributions. Hanbbook of Statistics-Statistical Methods in Finance, 14, 393–425. Elsevier Science B.V, AmsterdamGoogle Scholar
  14. 14.
    McNeil, A.J., R. Frey and P. Embrechts. 2005. Quantitative Risk Man¬agement, Princeton University Press, PrincetonGoogle Scholar
  15. 15.
    Nolan, J.P., A.K. Panorska, J.H. McCulloch. 2000. Estimation of Stable Spectral Measure. Mathematical and Computer Modeling 34, 1113–1122CrossRefGoogle Scholar
  16. 16.
    Nolan, J.P. 2005. Multivariate Stable Densities and Distribution Func¬tions: General and Elliptical Case. Deutsche Bundesbank's 2005 Annual Fall ConferenceGoogle Scholar
  17. 17.
    Rachev, S.T., S. Mittnik. 2000. Stable Paretian Models in Finance. Wiley, New YorkGoogle Scholar
  18. 18.
    Resnick, S.I. 1987. Extreme Values, Regular Variation, and Point Pro-ce s s e s. Springer, BerlinGoogle Scholar
  19. 19.
    Samorodnitsky, G., M. Taqqu. 1994. Stable Non-Gaussian Random Pro¬cesses, Chapmann & Hall, New YorkGoogle Scholar
  20. 20.
    Schmidt, R. 2002. Tail Dependence for Elliptically Contoured Distribu¬tions. Mathematical Methods of Operations Research 55, 301–327CrossRefGoogle Scholar
  21. 21.
    Stoyanov, S.V. 2005. Optimal Portfolio Management in Highly Volatile Markets.Ph.D. thesis, University of Karlsruhe, GermanyGoogle Scholar
  22. 22.
    Stoyanov, S.V., B. Racheva-Iotova 2004. Univariate Stable Laws in the Fields of Finance-Approximations of Density and Distribution Functions. Journal of Concrete and Applicable Mathematics, 2/1, 38–57Google Scholar

Copyright information

© Physica-Verlag Heidelberg 2009

Authors and Affiliations

  1. 1.Department of Econometrics Statistics and Mathematical FinanceUniversity of KarlsruheGermany
  2. 2.Yale School of ManagementNew Haven CTUSA

Personalised recommendations