Comparison study of global and local approaches describing critical phenomena on the Polish stock exchange market

We confront global and local methods to analyze the financial crash-like events on the Polish financial market from the critical phenomena point of view. These methods are based on the analysis of log-periodicity and the local fractal properties of financial time series in the vicinity of phase transitions (crashes). The whole history (1991-2008) of Warsaw Stock Exchange Index (WIG) describing the largest developing financial market in Europe, is analyzed in a daily time horizon. We find that crash-like events on the Polish financial market are described better by the log-divergent price model decorated with log-periodic behavior than the corresponding power-law-divergent price model. Predictions coming from log-periodicity scenario are verified for all main crashes that took place in WIG history. It is argued that crash predictions within log-periodicity model strongly depend on the amount of data taken to make a fit and therefore are likely to contain huge inaccuracies. Turning to local fractal description, we calculate the so-called local (time dependent) Hurst exponent Hloc for the WIG time series and we find the dependence between the behavior of the local fractal properties of the WIG time series and the crashes appearance on the financial market. The latter method seems to work better than the global approach - both for developing as for developed markets. The current situation on the market, particularly related to the Fed intervention in September'07 and the situation on the market immediately after this intervention is also analyzed from the fractional Brownian motion point of view.