Price fluctuations, market activity and trading volume

We investigate the relation between trading activity - measured by the number of trades [iopmath latex="NDelta t"] N t [/iopmath] - and the price change [iopmath latex="GDelta t"] G t [/iopmath] for a given stock over a time interval [iopmath latex="$[t,~t+Delta t]$"] [t, t + t] [/iopmath]. We relate the time-dependent standard deviation of price changes - volatility - to two microscopic quantities: the number of transactions [iopmath latex="NDelta t"] N t [/iopmath] in [iopmath latex="$Delta t$"] t [/iopmath] and the variance [iopmath latex="W²Delta t"] W 2 t [/iopmath] of the price changes for all transactions in [iopmath latex="$Delta t$"] t [/iopmath]. We find that [iopmath latex="NDelta t"] N t [/iopmath] displays power-law decaying time correlations whereas [iopmath latex="WDelta t"] W t [/iopmath] displays only weak time correlations, indicating that the long-range correlations previously found in [iopmath latex="vert GDelta t vert"] |G t| [/iopmath] are largely due to those of [iopmath latex="NDelta t"] N t [/iopmath]. Further, we analyse the distribution [iopmath latex="PNDelta t gt x"] P{N t>x} [/iopmath] and find an asymptotic behaviour consistent with a power-law decay. We then argue that the tail-exponent of [iopmath latex="PNDelta t gt x"] P{N t>x} [/iopmath] is insufficient to account for the tail-exponent of [iopmath latex="PGDelta t gt x"] P{G t>x} [/iopmath]. Since [iopmath latex="NDelta t"] N t [/iopmath] and [iopmath latex="WDelta t"] W t [/iopmath] display only weak interdependence, we argue that the fat tails of the distribution [iopmath latex="PGDelta t gt x"] P{G t>x} [/iopmath] arise from [iopmath latex="WDelta t"] W t [/iopmath], which has a distribution with power-law tail exponent consistent with our estimates for [iopmath latex="GDelta t"] G t [/iopmath]. Further, we analyse the statistical properties of the number of shares [iopmath latex="QDelta t"] Q t [/iopmath] traded in [iopmath latex="$Delta t$"] t [/iopmath], and find that the distribution of [iopmath latex="QDelta t"] Q t [/iopmath] is consistent with a Lévy-stable distribution. We also quantify the relationship between [iopmath latex="QDelta t"] Q t [/iopmath] and [iopmath latex="NDelta t"] N t [/iopmath], which provides one explanation for the previously observed volume-volatility co-movement.