Użycie formalizmu anomalnej dyfuzji oraz funkcji typu Kopuła do badania auto-korelacji i korelacji występujących w słabo przewidywalnych danych fizycznych i finansowych

Abstract

In this work the results of time series examination are presented. Geophysial data (maximum storm tides and the concentration of isotopes and ions in arctic cores) and financial data (shares prices traded on the Warsaw Stock Exchange) were examined. The analysis of the analogy between those two sorts of data is also presented. The goal of this work is to evaluate the general formalism that can be used to analyze correlations and auto-correlations of proceeded data. The Hurst exponent calculated by the Detrended Fluctuation Analysis (DFA), was used to investigate auto-correlations of time series. Two dimensional copula functions were used to investigate correlations between time series. It is worth mentioning here, that the procedure of the construction of the multidimensional frequency distribution was used by the means of the copula function applied after the fitting one dimensional frequency distributions to data. It was shown, that partially deterministic and partially stochastic model can be used to analyze discussed time series. This model concerns the influence of both global and local factors. Global factors cause correlations of all data, while local factors cause correlations of similar data (lying nearby) and randomness of data lying far away. As an example financial data can be discussed. Global factors are the macroeconomic condition and the general market condition connected with the collective investors behavior. Local factors concern particular companies and sectors where they belong. Local factors concern also individual investors strategies. During the investigation the similar behavior of the Hurst exponent values was shown for some of geophysical data (the historical concentration of sea originated ions – Na+, Mg2+ in arctic cores) and financial data (shares prices traded on the Warsaw Stock Exchange). While analyzing the concentration of discussed ions the anti-correlation signal was detected in the period of last 0.8 – 1.5 thousand years. The signal was detected by the means of the drop of the Hurst exponent value below the threshold value 𝐻𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑=0.47<0.5. Higher ions concentrations were detected after the anti-correlation signal than in the previous period. Analogically the drop of the Hurst exponent, calculated for shares prices, below some threshold value 𝐻𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑=0.45, is the anti-correlation signal after which the drop of shares prices values may occur. The difference in the threshold value, for geophysical and financial data can result from the fact that for financial data the Hurst exponent is more variable and less predictable than for physical data. The meaningful result of this work concerns the use of copula functions, supported by the use of the Hurst exponent, to evaluate the general model that can be applied to estimate the risk of the maximum storm tides or the maximum drops in shares prices. Here the Hurst exponent is an indicator in the copula selection procedure. In the case of financial data the risk of the simultaneous drop in values of two shares is concerned. In the case of geophysical data the risk of extreme storm tides in two localizations is concerned. Financial data were well modeled by the Gumbel copula and the reversed Clayton copula. Maximum storm tides were well modeled by the reversed Gumbel copula and the Clayton copula. In both cases investigation of auto-correlations was useful in the copula selection procedure. Discussed here analysis shown many analogies between financial and physical phenomena. However, the difference of these two phenomena are worth mentioning here. This is why, the further investigation is necessary, to answer the question to what extent the analogy between physics and economy can be used to model economical phenomena.