On non-Gaussianity and dependence in financial time series: a nonextensive approach

Abstract

In this article a probability density function and dependence degree analysis of financial time series, namely the Dow Jones and NYSE, is presented. The present study, which aims to give theoretical support to some stylized empirical evidence, is performed under the present non-extensive framework for which the probability distributions that optimize its fundamental information measure form, , are also the (stationary) solutions of a nonlinear Fokker–Plank equation. One determines the rescaled coefficient of the drift force and diffusion coefficient for both market indices and various aggregated times. Using a generalized form of Kullback–Leibler mutual information, I_q , one analyses the non-Gaussianity of returns using the dependence between stock market index values. The same mutual information form is used to determine the degree of dependence between returns. The analysis shows that this dependence can be considered independent from the time distance τ result that is connected with the long-range correlation in volatility.

Acknowledgments

The author wishes to express his thanks to Professor C. Tsallis for the enlightening discussions mainly on non-extensive formalism and is also grateful to E.P. Borges for comments on a first version of this manuscript. Financial support from PRONEX and CNPq (Brazilian agencies) for informatic structure is acknowledged. The author benefits from financial support provided by Fundação para a Ciĉencia e Tecnologia (Portuguese agency).

Notes

†Entropy is a physical concept closely related to information measure. For further reading consult Khinchin (1957, 1960)

†The Langevin equation , where u represents velocity, corresponds to the Brownian motion equation (Risken 1989).

‡Although market dynamics presents, in fact, time dependence, one will consider, as traditionally, that it is in a (quasi-) stationary regime.

†Even a plot of θ⁻¹ versus τ, which provides an increase of decades in the ordinate, does not allow one to characterize the type of dependence.

†As can be seen in figure 13 the autocorrelation function presents large fluctuations. The autocorrelation function can be smoothed using high-frequency data, that one could not acquire, providing reliable results.