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

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

2003-05-20

arxiv.org/abs/physics/0305089v1

divided in two: Quantitative Finance 5 (4), 379-401 (2005) and Applied Financial Economics 16, 271-289 (2006)

Physics

Physics and Society

74 pages, 16 tables and 10 figures

Scientific paper

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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