Dynamical characteristics of global stock markets based on time dependent Tsallis non-extensive statistics and generalized Hurst exponents

Abstract

We perform non-linear analysis on stock market indices using time-dependent extended Tsallis statistics. Specifically, we evaluate the q-triplet for particular time periods with the purpose of demonstrating the temporal dependence of the extended characteristics of the underlying market dynamics. We apply the analysis on daily close price timeseries of four major global markets (S&P 500, Tokyo-NIKKEI, Frankfurt-DAX, London-LSE). For comparison, we also compute time-dependentGeneralized Hurst Exponents (GHE) Hq using the GHE method, thus estimating the temporal evolution of the multiscaling characteristics of the index dynamics. We focus on periods before and after critical market events such as stock market bubbles (2000 dot.com bubble, Japanese 1990 bubble, 2008 US real estate crisis) and find that the temporal trends of q-triplet values significantly differ among these periods indicating that in the rising period before a bubble break, the underlying extended statistics of the market dynamics strongly deviates from purely stochastic behavior, whereas, after the breakdown, it gradually converges to the Gaussian-like behavior which is a characteristic of an efficient market. We also conclude that relative temporal variation patterns of the Tsallis q-triplet can be connected to different aspects of market dynamics and reveals useful information about market conditions especially those underlying the development of a stock market bubble. We found specific temporal patterns and trends in the relative variation of the indices in the q-triplet that distinguish periods just before and just after a stock-market bubble break. Differences between endogenous and exogenous stock market crises are also captured by the temporal changes in the Tsallis q-triplet. Finally, we introduce two new time-dependent empirical metrics (Q-metrics) that are functions of the Tsallis q-triplet. We apply them to the above stock market index price timeseries and discuss the significance of their temporal dependence on market dynamics and the possibility of using them, together with the relative temporal changes of the q-triplet, as signaling tools for future market events such as the development of a market bubble.