Methods of non-extensive statistical physics in analysis of price returns on Polish stock market

We use methods of non-extensive statistical physics to describe quantitatively the memory effect involved in returns of companies from WIG 30 index on the Warsaw Stock Exchange. The entropic approach based on the generalization of the Boltzmann-Gibbs entropy to non-additive Tsallis q-entropy is applied to fit fat tailed distribution of returns to q-normal (Tsallis) distribution. The existence of long term memory effects in price returns generated by two-point autocorrelations are checked via calculation of the Hurst exponent within detrended fluctuation analysis approach. The results are collected for diversified frequency of data sampling. We confirm the perfect inverse cubic power law for low time-lags (≈1 min) of returns for the main WIG 30 index as well as for the most of separate stocks, however this relationship does not hold for longer time-lags. The particular emphasis is given to a study of an independent fit of probability distribution of positive and negative returns to q-normal distribution. We discuss in this context the asymmetry between tails in terms of the Tsallis parameters q^{±}. A qualitative and quantitative relationship between the frequency of data sampling, the parameters q and q^{±}, and the corresponding main Hurst exponent H is provided to analyze the effect of memory in data caused by linear and nonlinear autocorrelations. A new quantifier based on asymmetry of the Tsallis index instead of skewness of distribution is proposed which we believe is able to describe the stage of market development and its robustness to speculation.