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 $\tilde R$, 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 $|\tilde R|$, and their relationship to price changes $|R|$. We analyze $14,981$ daily recordings of the S\&P 500 index over the 59-year period 1950--2009, and find power-law {\it cross-correlations\/} between $|R|$ and $|\tilde R|$ using detrended cross-correlation analysis (DCCA). We introduce a joint stochastic process that models these cross-correlations. Motivated by the relationship between $| R|$ and $|\tilde R|$, we estimate the tail exponent ${\tildeα}$ of the probability density function $P(|\tilde R|) \sim |\tilde R|^{-1 -\tildeα}$ 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 $\tildeα$, we calculate the time intervals $τ_q$ between events where $\tilde R>q$. We demonstrate that $\barτ_q$, the average of $τ_q$, obeys $\bar τ_q \sim q^{\tildeα}$. We find $\tilde α\approx 3$. Furthermore, by aggregating all $τ_q$ values of 28 global financial indices, we also observe an approximate inverse cubic law.

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