Cross-correlations between volume change and price change

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) over tilde, 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 vertical bar(R) over tilde vertical bar, and their relationship to price changes vertical bar R vertical bar. 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 vertical bar R vertical bar and vertical bar(R) over tilde vertical bar by using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these cross-correlations. Motivated by the relationship between vertical bar R vertical bar and vertical bar(R) over tilde vertical bar, we estimate the tail exponent (alpha) over tilde of the probability density function P(vertical bar(R) over tilde vertical bar) similar to vertical bar(R) over tilde vertical bar(-1-(alpha) over tilde) 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) over tilde, we calculate the time intervals tau(q) between events where (R) over tilde > q. We demonstrate that (tau) over bar (q), the average of tau(q), obeys (tau) over bar (q) similar to q((alpha) over tilde). We find (alpha) over tilde approximate to 3. Furthermore, by aggregating all tq values of 28 global financial indices, we also observe an approximate inverse cubic law.