Global and local approaches describing critical phenomena on the developing and developed financial markets

Abstract

We define and confront global and local methods to analyze the financial crash-like events on the financial markets from the critical phenomena point of view. These methods are based respectively on the analysis of log-periodicity and on the local fractal properties of financial time series in the vicinity of phase transitions (crashes). The log-periodicity analysis is made in a daily time horizon, for the whole history (1991-2008) of Warsaw Stock Exchange Index (WIG) connected with the largest developing financial market in Europe. 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 by the power-law-divergent price model usually discussed in log-periodic scenarios for developed markets. 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. Next, this global analysis is confronted with the local fractal description. To do so, we provide calculation of the so-called local (time dependent) Hurst exponent H loc for the WIG time series and for main US stock market indices like DJIA and S&P 500. We point out dependence between the behavior of the local fractal properties of financial time series and the crashes appearance on the financial markets. We conclude that local fractal method seems to work better than the global approach - both for developing and developed markets. The very recent situation on the market, particularly related to the Fed intervention in September 2007 and the situation immediately afterwards is also analyzed within fractal approach. It is shown in this context how the financial market evolves through different phases of fractional Brownian motion. Finally, the current situation on American market is analyzed in fractal language. This is to show how far we still are from the end of recession and from the beginning of a new boom on US financial market or on other world leading stocks. © 2010 Springer-Verlag Tokyo.