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