Cross-correlations between volume change and price change

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Abstract
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 R, 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 cross-correlations. Motivated by the relationship between |R| and |R|, we estimate the tail exponent alpha of the probability density function P(|R|) approximately |R|(-1-alpha) 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 alpha, we calculate the time intervals tau(q) between events where R > q. We demonstrate that tau(q), the average of tau(q), obeys tau(q) approximately q(alpha). We find alpha approximately 3. Furthermore, by aggregating all tau(q) values of 28 global financial indices, we also observe an approximate inverse cubic law.