We study how the approach grounded on non-extensive statistical physics can be applied for a description of the current state of the stock and money market. Particular attention is given to asymmetric behavior of fat tailed distributions of positive and negative returns. We propose a  new quantifier based on asymmetry between these tails in terms of  the Tsallis parameters q+, q-  to analyze the effect of memory in data caused by nonlinear autocorrelations. The presented analysis takes into account data of separate stocks from the main developing stock market in Europe - Warsaw Stock Exchange (WSE) in Poland and,  for comparison, data from the most mature money market (Forex). Our search is extended also to study memory effects and their dependence on the quotation frequency for similar large companies  -  owners of food-industrial retail supermarkets acting on both Polish and European markets (Eurocash, Jeronimo-Martins, Carrefour, Tesco) but traded on various European stock markets of diversified economical maturity. It is argued that  the proposed quantifier is able to describe the stage of market development and its robustness to speculation. The main strength is put on a description and interpretation of the asymmetry  of positive and negative returns in terms of Tsallis statistics for various stocks and for diversified time lags Δt of data collection.  Our search indicates that the stocks from the same economic sector acting in European Union (EU) may be a target of diversified level of speculations involved in trading independently on the true economic situation of the company.  The influence of the coming Brexit on the values of new quantifier, specifically for the British financial market, is also discussed.  

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