Asymmetry of price returns-Analysis and perspectives from a non-extensive statistical physics point of view.

We study how the approach grounded on non-extensive statistical physics can be applied to describe and distinguish different stages of the stock and money market development. A particular attention is given to asymmetric behavior of fat tailed distributions of positive and negative returns. A new method to measure this asymmetry is proposed. It is based on the value of the non-extensive Tsallis parameter q. The new quantifier of the relative asymmetry level between tails in terms of the Tsallis parameters q± is provided 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, i.e., the Warsaw Stock Exchange (WSE) in Poland and-for comparison-data from the most mature money market (Forex). It is argued that the proposed new 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 between statistical properties of positive and negative returns for various stocks and for diversified time-lags Δt of data counting. The particular caution in this context is addressed to the difference between intraday and interday returns. Our search is extended 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 (respectively in Warsaw, Lisbon, Paris and London). The latter analysis seems to indicate quantitatively that stocks from the same economic sector traded on different markets within European Union (EU) may be a target of diversified level of speculations involved in trading independently on the true economic situation of the company. Our work thus gives indications that the statement:" where you are is more important than who you are" is true on trading markets

Results of <i>q</i>-normal distribution fit to statistics of returns for main exchange rates on Forex.

<p>Polish currency (PLN—złoty) has been added for comparison. Shown are also results of independent fit of Tsallis parameter <i>q</i>^± to the right (positive returns) and left (negative returns) tail of probability distribution for diversified time-lags.</p

Probability distribution (blue point—1 day, green square—2 days, cyan diamond—3 days, red triangle—4 days) and <i>q</i>-normal distribution fit (solid lines) to normalized and centered returns of the leading European food-industrial stores.

<p>Interday data are collected from the period July 2005—July 2015 for time-lag 1 ÷ 4 days. All plots are shifted vertically in a similar manner as in Figs <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0188541#pone.0188541.g001" target="_blank">1</a>–<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0188541#pone.0188541.g005" target="_blank">5</a> and compared with normal distribution (dashed line). Shown are results of independent fit to positive and negative returns.</p

Skewness of price returns for chosen stokcs from WIG 30 stock index.

<p>Skewness of price returns for chosen stokcs from WIG 30 stock index.</p

Dependence between skewness <i>A</i> versus the time-lag Δ<i>t</i> in log-linear scale for chosen companies from WIG 30 stock index.

<p>The results seem to be much less informative than those from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0188541#pone.0188541.g006" target="_blank">Fig 6</a>.</p

Same as in Fig 1 for the capital weighted WIG 30 stock index of Warsaw Stock Exchange (WSE).

<p>Same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0188541#pone.0188541.g001" target="_blank">Fig 1</a> for the capital weighted WIG 30 stock index of Warsaw Stock Exchange (WSE).</p

Same as in Fig 10 for USD/GBP exchange rate.

<p>Same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0188541#pone.0188541.g010" target="_blank">Fig 10</a> for USD/GBP exchange rate.</p

Same as in Fig 10 for USD/EUR exchange rate.

<p>Same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0188541#pone.0188541.g010" target="_blank">Fig 10</a> for USD/EUR exchange rate.</p

Difference of Tsallis parameters <i>δq</i> = <i>q</i>⁺ − <i>q</i>⁻, the main Tsallis parameter value <i>q</i>, the relative asymmetry ratio |<i>δq</i>|/<i>q</i> and the averaged relative asymmetry ratio 〈|<i>δq</i>|/<i>q</i>〉 for the food-industrial trading companies quoted on different European stock markets but making the same economic sector.

<p>Difference of Tsallis parameters <i>δq</i> = <i>q</i>⁺ − <i>q</i>⁻, the main Tsallis parameter value <i>q</i>, the relative asymmetry ratio |<i>δq</i>|/<i>q</i> and the averaged relative asymmetry ratio 〈|<i>δq</i>|/<i>q</i>〉 for the food-industrial trading companies quoted on different European stock markets but making the same economic sector.</p

Same as in Fig 1 for PZU stock.

<p>Same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0188541#pone.0188541.g001" target="_blank">Fig 1</a> for PZU stock.</p