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 rateR, 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 |R|, 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 |R| by using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these crosscorrelations. Motivated by the relationship between |R| and |R|, we estimate the tail exponentα of the probability density function P(|R|) ∼ |R| −1−α 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 estimateα, we calculate the time intervals τ q between events whereR > q. We demonstrate thatτ q , the average of τ q , obeysτ q ∼ qα. We findα ≈ 3. Furthermore, by aggregating all τ q values of 28 global financial indices, we also observe an approximate inverse cubic law. econophysics | finance | volatility There is a saying on Wall Street that "it takes volume to move stock prices." A number of studies have analyzed the relationship between price changes and the trading volume in financial markets (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14). Some of these studies (1, 3-6) have found a positive relationship between price change and the trading volume. In order to explain this relationship, Clarke assumed that the daily price change is the sum of a random number of uncorrelated intraday price changes (3), so predicted that the variance of the daily price change is proportional to the average number of daily transactions. If the number of transactions is proportional to the trading volume, then the trading volume is proportional to the variance of the daily price change.The cumulative distribution function (cdf) of the absolute logarithmic price change |R| obeys a power lawIt is believed (15-18) that α ≈ 3 ("inverse cubic law"), outside the range α < 2 characterizing a Lévy distribution (18,19). A parallel analysis of Q, the volume traded, yields a power law (20-28)To our knowledge, the logarithmic volume change-R and its relation to the logarithmic price change R-has not been analyzed, and this analysis is our focus here. For each of the 1,819 companies and 28 indices, we calculate over the time interval of one day the logarithmic change in price S(t), Data Analyzedand also the logarithmic change in trading volume Q(t) (29),For each of the 3,694 time series, we also calculate the absolute values |R t | and |R t | and define the "price volatility" (30) and "volume volatility," respectively,andwhere σ R ≡ ( |R t | 2 − |R t | 2 ) 1/2 and σR ≡ ( |R t | 2 − |R t | 2 ) 1/2 are the respective standard deviat...

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PACS 89.75.Da -Systems obeying scaling laws PACS 89.90.+n -Other topics in areas of applied and interdisciplinary physics Abstract -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 longrange 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.

We study the temperature dependence of the pseudoscalar meson properties in a relativistic boundstate approach exhibiting the chiral behavior mandated by QCD. Concretely, we adopt the DysonSchwinger approach with a rank-2 separable model interaction. After extending the model to the strange sector and fixing its parameters at zero temperature, T = 0, we study the T -dependence of the masses and decay constants of all ground-state mesons in the pseudoscalar nonet. Of chief interest are η and η ′ . The influence of the QCD axial anomaly on them is successfully obtained through the Witten-Veneziano relation at T = 0. The same approach is then extended to T > 0, using lattice QCD results for the topological susceptibility. The most conspicuous finding is an increase of the η ′ mass around the chiral restoration temperature T Ch , which would suggest a suppression of η ′ production in relativistic heavy-ion collisions. The increase of the η ′ mass may also indicate that the extension of the Witten-Veneziano relation to finite temperatures becomes unreliable around and above T Ch . Possibilities of an improved treatment are discussed.

We discuss and propose the minimal generalization of the Witten-Veneziano relation to finite temperatures, prompted by STAR and PHENIX experimental results on the multiplicity of 0 mesons. After explaining why these results show that the zero-temperature Witten-Veneziano relation cannot be straightforwardly extended to temperatures T too close to the chiral restoration temperature T Ch and beyond, we find the quantity which should replace, at T > 0, the Yang-Mills topological susceptibility appearing in the T ¼ 0 Witten-Veneziano relation, in order to avoid the conflict with experiment at T > 0. This is illustrated through concrete T-dependences of pseudoscalar meson masses in a chirally well-behaved, Dyson-Schwinger approach, but our results and conclusions are of a more general nature and, essentially, model-independent.

Noisy signals in many real-world systems display long-range autocorrelations and long-range cross-correlations. Due to periodic trends, these correlations are difficult to quantify. We demonstrate that one can accurately quantify power-law cross-correlations between different simultaneously recorded time series in the presence of highly non-stationary sinusoidal and polynomial overlying trends by using the new technique of detrended cross-correlation analysis with varying order of the polynomial. To demonstrate the utility of this new method -which we call DCCA-(n), where n denotes the scale-we apply it to meteorological data.

The approach to the η 0-η complex employing chirally well-behaved quark-antiquark bound states and incorporating the non-Abelian axial anomaly of QCD through the generalization of the Witten-Veneziano relation is extended to finite temperatures. Employing the chiral condensate has led to a sharp chiral and U A ð1Þ symmetry restoration, but with the condensates of quarks with realistic explicit chiral symmetry breaking-which exhibit a smooth, crossover chiral symmetry restoration in qualitative agreement with lattice QCD results-we get a crossover U A ð1Þ transition with a smooth and gradual melting of anomalous mass contributions. In this way, we obtain a substantial decrease in the η 0 mass around the chiral transition temperature, but no decrease in the η mass. This is consistent with current empirical evidence.

We propose a modified time lag random matrix theory in order to study time lag crosscorrelations 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 constitute "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 auto-correlations 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 10 indices 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.