In finance, one usually deals not with prices but with growth rates R , defined as the difference in logarithm between two consecutive prices. Here we consider not the trading volume, but rather the volume growth rate R̃ , the difference in logarithm between two consecutive values of trading volume. To this end, we use several methods to analyze the properties of volume changes | R̃ |, and their relationship to price changes | R |. We analyze 14,981 daily recordings of the Standard and Poor's (S & P) 500 Index over the 59-year period 1950–2009, and find power-law cross-correlations between | R | and | R̃ | by using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these cross-correlations. Motivated by the relationship between | R | and | R̃ |, we estimate the tail exponent α̃ of the probability density function P (| R̃ |) ∼ | R̃ | ^−1−α̃ for both the S & P 500 Index as well as the collection of 1819 constituents of the New York Stock Exchange Composite Index on 17 July 2009. As a new method to estimate α̃, we calculate the time intervals τ _q between events where R̃ > q . We demonstrate that τ̃ _q , the average of τ _q , obeys τ̃ _q ∼ q ^α̃ . We find α̃ ≈ 3. Furthermore, by aggregating all τ _q values of 28 global financial indices, we also observe an approximate inverse cubic law.

We study long-range magnitude cross-correlations in collective modes of real-world data from finance, physiology, and genomics using time-lag random matrix theory. We find long- range magnitude cross-correlations i) in time series of price fluctuations, ii) in physiological time series, both healthy and pathological, indicating scale-invariant interactions between different physiological time series, and iii) in ChIP-seq data of the mouse genome, where we uncover a complex interplay of different DNA-binding proteins, resulting in power-law cross-correlations in xij, the probability that protein i binds to gene j, ranging up to 10 million base pairs. In finance, we find that the changes in singular vectors and singular values are largest in times of crisis. We find that the largest 500 singular values of the NYSE Composite members follow a Zipf distribution with exponent ≈ 2. In physiology, we find statistically significant differences between alcoholic and control subjects. Copyright c EPLA, 2010

We propose a modified time lag random matrix theory in order to study time-lag cross correlations in multiple time series. We apply the method to 48 world indices, one for each of 48 different countries. We find long-range power-law cross correlations in the absolute values of returns that quantify risk, and find that they decay much more slowly than cross correlations between the returns. The magnitude of the cross correlations constitutes ``bad news'' for international investment managers who may believe that risk is reduced by diversifying across countries. We find that when a market shock is transmitted around the world, the risk decays very slowly. We explain these time-lag cross correlations by introducing a global factor model (GFM) in which all index returns fluctuate in response to a single global factor. For each pair of individual time series of returns, the cross correlations between returns (or magnitudes) can be modeled with the autocorrelations of the global factor returns (or magnitudes). We estimate the global factor using principal component analysis, which minimizes the variance of the residuals after removing the global trend. Using random matrix theory, a significant fraction of the world index cross correlations can be explained by the global factor, which supports the utility of the GFM. We demonstrate applications of the GFM in forecasting risks at the world level, and in finding uncorrelated individual indices. We find ten indices that are practically uncorrelated with the global factor and with the remainder of the world indices, which is relevant information for world managers in reducing their portfolio risk. Finally, we argue that this general method can be applied to a wide range of phenomena in which time series are measured, ranging from seismology and physiology to atmospheric geophysics.