Price fluctuations, market activity and trading volume

We investigate the relation between trading activity - measured by the number of trades [iopmath latex="$N_{Delta t}$"] Nt [/iopmath] - and the price change [iopmath latex="$G_{Delta t}$"] Gt [/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="$N_{Delta t}$"] Nt [/iopmath] in [iopmath latex="$Delta t$"] t [/iopmath] and the variance [iopmath latex="$W^2_{Delta t}$"] W2t [/iopmath] of the price changes for all transactions in [iopmath latex="$Delta t$"] t [/iopmath]. We find that [iopmath latex="$N_{Delta t}$"] Nt [/iopmath] displays power-law decaying time correlations whereas [iopmath latex="$W_{Delta t}$"] Wt [/iopmath] displays only weak time correlations, indicating that the long-range correlations previously found in [iopmath latex="$vert G_{Delta t} vert$"] &7CGt&7C [/iopmath] are largely due to those of [iopmath latex="$N_{Delta t}$"] Nt [/iopmath]. Further, we analyse the distribution [iopmath latex="$P{N_{Delta t} gt x}$"] P{Nt>x} [/iopmath] and find an asymptotic behaviour consistent with a power-law decay. We then argue that the tail-exponent of [iopmath latex="$P{N_{Delta t} gt x}$"] P{Nt>x} [/iopmath] is insufficient to account for the tail-exponent of [iopmath latex="$P{G_{Delta t} gt x}$"] P{Gt>x} [/iopmath]. Since [iopmath latex="$N_{Delta t}$"] Nt [/iopmath] and [iopmath latex="$W_{Delta t}$"] Wt [/iopmath] display only weak interdependence, we argue that the fat tails of the distribution [iopmath latex="$P{G_{Delta t} gt x}$"] P{Gt>x} [/iopmath] arise from [iopmath latex="$W_{Delta t}$"] Wt [/iopmath], which has a distribution with power-law tail exponent consistent with our estimates for [iopmath latex="$G_{Delta t}$"] Gt [/iopmath]. Further, we analyse the statistical properties of the number of shares [iopmath latex="$Q_{Delta t}$"] Qt [/iopmath] traded in [iopmath latex="$Delta t$"] t [/iopmath], and find that the distribution of [iopmath latex="$Q_{Delta t}$"] Qt [/iopmath] is consistent with a Levy-stable distribution. We also quantify the relationship between [iopmath latex="$Q_{Delta t}$"] Qt [/iopmath] and [iopmath latex="$N_{Delta t}$"] Nt [/iopmath], which provides one explanation for the previously observed volume-volatility co-movement.