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

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.

Keywords:

• Non-nested hypothesis testing
• Pareto distribution
• Weibull distribution

Notes

Picoli et al. (2003) have also presented fits comparing the relative merits of SE and so-called q-exponentials (which are similar to a Student distribution with power law tails) for the description of the frequency distributions of basketball baskets, cyclone victims, brand-name drugs by retail sales, and highway length.

A generalization of the SLE to the following three-parameter family also contains the SE family in some formal limit. Consider indeed 1 − F(x) = exp(−b(ln (1 + x/D)) ^c ) for x>0, which has the same tail as expression (9). Taking D → +∞ together with b = (D/d) ^c with d finite yields 1 − F(x) = exp(−(x/d)) ^c ).

See Sornette et al. (2000) and figures 3.6–3.9, pp. 81–82, of Sornette (2004), where it is shown that SE distributions are approximately stable as a family and the effect of aggregation can be seen to slowly increase the exponent c. See also Drozdz et al. (2002), which specifically studies this convergence to a Gaussian law as a function of the time scale level.

Let us stress that we are speaking of a log-normal distribution of returns, not of price! Indeed, the standard Black and Scholes model of a log-normal distribution of prices is equivalent to a Gaussian distribution of returns. Thus, a log-normal distribution of returns is much more fat tailed, and in fact bracketed by power law tails and stretched-exponential tails.