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Empirical distributions of stock returns: between the stretched exponential and the power law?

Author

Listed:
  • Y. Malevergne
  • V. Pisarenko
  • D. Sornette

Abstract

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 b close to 3. We develop a battery of new non-parametric and parametric tests to characterize the distributions of empirical returns of moderately large financial time series, with application to 100 years of daily returns of the Dow Jones Industrial Average, to 1 year of 5-min returns of the Nasdaq Composite index and to 17 years of 1-min returns of the Standard & Poor's 500. We propose a parametric representation of the tail of the distributions of returns encompassing both a regularly varying distribution in one limit of the parameters and rapidly varying distributions of the class of the stretched-exponential (SE) and the log-Weibull or Stretched Log-Exponential (SLE) distributions in other limits. Using the method of nested hypothesis testing (Wilks' theorem), we conclude that both the SE distributions and Pareto distributions provide reliable descriptions of the data but are hardly distinguishable for sufficiently high thresholds. Based on the discovery that the SE distribution tends to the Pareto distribution in a certain limit, 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 to be 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, with a noticeable exception concerning the very-high-frequency data. Summing up all the evidence provided by our tests, it seems that the tails ultimately decay slower than any SE but probably faster than power laws with reasonable exponents. Thus, from a practical viewpoint, the log-Weibull model, which provides a smooth interpolation between SE and PD, can be considered as an appropriate approximation of the sample distributions.

Suggested Citation

  • Y. Malevergne & V. Pisarenko & D. Sornette, 2005. "Empirical distributions of stock returns: between the stretched exponential and the power law?," Quantitative Finance, Taylor & Francis Journals, vol. 5(4), pages 379-401.
  • Handle: RePEc:taf:quantf:v:5:y:2005:i:4:p:379-401
    DOI: 10.1080/14697680500151343
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    References listed on IDEAS

    as
    1. Jondeau, Eric & Rockinger, Michael, 2003. "Testing for differences in the tails of stock-market returns," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 559-581, December.
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    Citations

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    Cited by:

    1. Broda, Simon A. & Haas, Markus & Krause, Jochen & Paolella, Marc S. & Steude, Sven C., 2013. "Stable mixture GARCH models," Journal of Econometrics, Elsevier, vol. 172(2), pages 292-306.
    2. Lucas Fievet & Zal`an Forr`o & Peter Cauwels & Didier Sornette, 2014. "Forecasting future oil production in Norway and the UK: a general improved methodology," Papers 1407.3652, arXiv.org.
    3. 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.
    4. Suárez-García, Pablo & Gómez-Ullate, David, 2014. "Multifractality and long memory of a financial index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 394(C), pages 226-234.
    5. Gu, Gao-Feng & Chen, Wei & Zhou, Wei-Xing, 2008. "Empirical distributions of Chinese stock returns at different microscopic timescales," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(2), pages 495-502.
    6. Suárez-García, Pablo & Gómez-Ullate, David, 2013. "Scaling, stability and distribution of the high-frequency returns of the Ibex35 index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(6), pages 1409-1417.
    7. Andrew Balthrop, 2016. "Power laws in oil and natural gas production," Empirical Economics, Springer, vol. 51(4), pages 1521-1539, December.
    8. Rui Vilela Mendes & M. J. Oliveira, 2006. "A data-reconstructed fractional volatility model," Papers math/0602013, arXiv.org, revised Jun 2007.
    9. repec:ris:utmsje:0199 is not listed on IDEAS
    10. Pablo Su'arez-Garc'ia & David G'omez-Ullate, 2012. "Scaling, stability and distribution of the high-frequency returns of the IBEX35 index," Papers 1208.0317, arXiv.org.
    11. Gao-Feng Gu & Wei Chen & Wei-Xing Zhou, 2007. "Empirical distributions of Chinese stock returns at different microscopic timescales," Papers 0708.3472, arXiv.org.
    12. Sandro Claudio Lera & Didier Sornette, 2017. "GDP growth rates as confined L\'evy flights," Papers 1709.05594, arXiv.org.
    13. Saralees Nadarajah & Samuel Kotz, 2006. "The modified Weibull distribution for asset returns," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 449-449.
    14. Pablo Su'arez-Garc'ia & David G'omez-Ullate, 2013. "Multifractality and long memory of a financial index," Papers 1306.0490, arXiv.org.
    15. Makoto Nirei & Theodoros Stamatiou & Vladyslav Sushko, 2012. "Stochastic Herding in Financial Markets Evidence from Institutional Investor Equity Portfolios," BIS Working Papers 371, Bank for International Settlements.
    16. Gu, Gao-Feng & Zhou, Wei-Xing, 2007. "Statistical properties of daily ensemble variables in the Chinese stock markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 383(2), pages 497-506.
    17. Jovanovic, Franck & Schinckus, Christophe, 2017. "Econophysics and Financial Economics: An Emerging Dialogue," OUP Catalogue, Oxford University Press, number 9780190205034.
    18. Vladimir Filimonov & Didier Sornette, 2014. "Power law scaling and "Dragon-Kings" in distributions of intraday financial drawdowns," Papers 1407.5037, arXiv.org, revised Apr 2015.

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