Complex systems attract many researchers from various scientific fields. Financial markets are one of these widely studied complex systems. Statistical physics, which was originally developed to study large systems, provides novel ideas and powerful methods to analyze financial markets. The study of financial fluctuations characterizes market behavior, and helps to better understand the underlying market mechanism.;Our study focuses on volatility, a fundamental quantity to characterize financial fluctuations. We examine equity data of the entire U.S. stock market during 2001 and 2002. To analyze the volatility time series, we develop a new approach, called return interval analysis, which examines the time intervals between two successive volatilities exceeding a given value threshold. We find that the return interval distribution displays scaling over a wide range of thresholds. This scaling is valid for a range of time windows, from one minute up to one day. Moreover, our results are similar for commodities, interest rates, currencies, and for stocks of different countries. Further analysis shows some systematic deviations from a scaling law, which we can attribute to nonlinear correlations in the volatility time series. We also find a memory effect in return intervals for different time scales, which is related to the long-term correlations in the volatility.;To further characterize the mechanism of price movement, we simulate the volatility time series using two different models, fractionally integrated generalized autoregressive conditional heteroscedasticity (FIGARCH) and fractional Brownian motion (fBm), and test these models with the return interval analysis. We find that both models can mimic time memory but only fBm shows scaling in the return interval distribution.;In addition, we examine the volatility of daily opening to closing and of closing to opening. We find that each volatility distribution has a power law tail. Using the detrended fluctuation analysis (DFA) method, we show long-term auto-correlations in these volatility time series. We also analyze return, the actual price changes of stocks, and find that the returns over the two sessions are often anti-correlated.
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