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Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?


  • Y. Malevergne

    (Univ. Nice and Univ. Lyon)

  • V. F. Pisarenko

    (Russian Acad. Sci.)

  • D. Sornette

    (UCLA and CNRS-Univ. Nice)


A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent close to 3. We revisit this results and use standard tests as well as develop a battery of new non-parametric and parametric tests (in particular with stretched exponential (SE) distributions) to characterize the distributions of empirical returns of financial time series, with application to the 100 years of daily return of the Dow Jones Industrial Average and over 1 years of 5-minutes returns of the Nasdaq Composite index. Based on the discovery that the SE distribution tends to the Pareto distribution in a certain limit such that the Pareto (or power law) distribution can be approximated with any desired accuracy on an arbitrary interval by a suitable adjustment of the parameters of the SE distribution, we demonstrate that Wilks' test of nested hypothesis still works for the non-exactly nested comparison between the SE and Pareto distributions. The SE distribution is found significantly better over the whole quantile range but becomes unnecessary beyond the 95% quantiles compared with the Pareto law. Similar conclusions hold for the log-Weibull model with respect to the Pareto distribution. Our main result is that the tails ultimately decay slower than any stretched exponential distribution but probably faster than power laws with reasonable exponents. Implications of our results on the ``moment condition failure'' and for risk estimation and management are presented.

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  • Y. Malevergne & V. F. Pisarenko & D. Sornette, 2003. "Empirical Distributions of Log-Returns: between the Stretched Exponential and the Power Law?," Papers physics/0305089,
  • Handle: RePEc:arx:papers:physics/0305089

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    2. Rui Vilela Mendes & M. J. Oliveira, 2006. "A data-reconstructed fractional volatility model," Papers math/0602013,, revised Jun 2007.
    3. V. F. Pisarenko & D. Sornette, 2004. "New statistic for financial return distributions: power-law or exponential?," Papers physics/0403075,
    4. Saralees Nadarajah & Samuel Kotz, 2006. "The modified Weibull distribution for asset returns," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 449-449.
    5. Y. Malevergne & V. Pisarenko & D. Sornette, 2006. "The modified weibull distribution for asset returns: reply," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 451-451.
    6. Sornette, Didier & Zhou, Wei-Xing, 2006. "Importance of positive feedbacks and overconfidence in a self-fulfilling Ising model of financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 370(2), pages 704-726.
    7. D. Sornette, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models," Papers 1404.0243,

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