Tsallis q - triplet and Stock Market Indices : The cases of S & P 500 and TVIX

Abstract

In this study we present results of the evaluation of q− triplet of Tsallis non-extensive statistics concerning two time stock market time series, Standart & Poor’s 500 (S & P) 500 and TVIX. The analysis of the results, support the hypothesis in which economic dynamics from physical point of view correspond to far from equilibrium spatial distributed non-linear dynamics. In particular, the analysis of the stock market time series revealed underlying complex dynamics, indicating clearly that the statistics of the dynamics in the multifractal phase space can be described by Tsallis distribution functions of power law and heavy tails forms. The non-extensive character of the underlying dynamics is related to the existence of long range interactions in space and time, as well as the interaction in many scales. In addition, the detailed analysis of S & P index unraveled the existence of non-equilibrium phase transitions depicted clearly in the variations of Tsallis q-triplet values. These phase transitions are connected with non-equilibrium stationary states of economical dynamics derived from processes of strong self organization which correspond to local maxima of Tsallis entropy, while the changes in the control parameters can induce new phase transitions and shifts to new metaequilibrium steady states of maximizing Tsallis entropy. These phase transitions lead to changes in Tsallis qtriplet values which correspond to multi-fractal changes in the formation of the phase state and an alteration in the phenomenology of the economical system. Finally, these characteristics also indicate the existence of fractional dynamics in the phase space which can be described through fractional Fokker-Planck equations and anomalous diffusion equations. The solutions of these equations are fractional spatiotemporal functions and non-Gaussian distributions functions which fall into the category of Levy distributions and Tsallis distributions.