Price fluctuations, market activity and trading volume

We investigate the relation between trading activity—measured by the number of trades N t—and the price change G t for a given stock over a time interval [t, t + t]. We relate the time-dependent standard deviation of price changes—volatility—to two microscopic quantities: the number of transactions N t in t and the variance W 2 t of the price changes for all transactions in t . We find that N t displays power-law decaying time correlations whereas W t displays only weak time correlations, indicating that the long-range correlations previously found in |G t | are largely due to those of N t . Further, we analyse the distribution P {N t > x} and find an asymptotic behaviour consistent with a power-law decay. We then argue that the tail-exponent of P {N t > x} is insufficient to account for the tail-exponent of P {G t > x}. Since N t and W t display only weak interdependence, we argue that the fat tails of the distribution P {G t > x} arise from W t , which has a distribution with power-law tail exponent consistent with our estimates for G t . Further, we analyse the statistical properties of the number of shares Q t traded in t , and find that the distribution of Q t is consistent with a Lévy-stable distribution. We also quantify the relationship between Q t and N t , which provides one explanation for the previously observed volume–volatility co-movement.