statistique, théorie et gestion de portefeuille - Docs at ISFA

Empirical Distributions of Log-Returns:
Exponential or Power-like?
Y. Malevergne 1,2 , V. Pisarenko 3 , and D. Sornette 1,4
1 Laboratoire de Physique de la Matière Condensée
CNRS UMR6622 and Université de Nice-Sophia Antipolis
Parc Valrose, 06108 Nice Cedex 2, France
2 Institut de Science Financière et d’Assurances - Université Lyon I
43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
3 International Institute of Earthquake Prediction Theory and Mathematical Geophysics
Russian Ac. Sci. Warshavskoye sh., 79, kor. 2, Moscow 113556, Russia
4 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science
University of California, Los Angeles, California 90095
e-mails: Yannick.Malevergne@unice.fr, Vlad@sirus.mitp.ru and sornette@unice.fr
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 close to 3. First, we show by synthetic tests performed
on time series with long-range time dependence in the volatility with both Pareto and Stretched-
Exponential distributions that standard generalized extreme value (GEV) and Generalized Pareto Distribution
(GPD) estimators are quite inefficient and cannot distinguish reliably between the two classes
of distributions, in contradiction with previous results. Then, we use a parametric representation of the
distribution of returns of 100 years of daily return of the Dow Jones Industrial Average and over 1 years
of 5-minutes returns of the Nasdaq Composite index, encompassing both a regularly varying distribution
in one limit of the parameters and rapidly varying distributions of the class of the Stretched-Exponential
(SE) distributions in other limits. Using the method of nested hypothesis testing (Wilk theorem), we
conclude that both the SE distributions and Pareto distributions provide reliable descriptions of the data
and cannot be distinguished for sufficiently high thresholds. Then, we introduced a novel encompassing
test based on the discovery that the SE contains the Pareto law in a certain limit that confirms that the SE
encompasses any power law tail distribution. However, our best estimation of the parameters of the SE
model suggests that the tails decay probably more slowly that a pure SE. Summing up all the evidence
provided by our battery of tests, it seems that the tails ultimately decay slower than any SE but probably
faster than power laws with reasonable exponents. We discuss the implications of our results on the
“moment condition failure” and for risk estimation and management.
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