Tests of scaling and universality of the distributions of trade size and share volume: Evidence from three distinct markets

Empirical evidence for scale-invariant distributions in financial data has attracted the research interest of physicists. While the power-law tails of the distribution of stock returns $P\{R>x\}\sim x^{-{\zeta }_R}$ are becoming increasingly well documented, less understood are the statistics of other closely related microstructural variables such as $q_i$, the number of shares exchanged in trade $i$ (termed the trade size) and $Q_{\Delta t}\left(t\right)={\sum }_{i=1}^N{q}_i$, the total number of shares exchanged as a result of the $N=N_{\Delta t}$ trades occurring in a time interval $\Delta t$ (termed share volume). We analyze the statistical properties of trade size $q\equiv q_i$ and share volume $Q\equiv Q_{\Delta t}\left(t\right)$ by analyzing trade-by-trade data from three large databases representing three distinct markets: (i) 1000 major U.S. stocks for the $2\text{−}\mathrm{y}$ period 1994–1995, (ii) 85 major U.K. stocks for the $2\text{−}\mathrm{y}$ period 2001–2002, and (iii) 13 major Paris Bourse stocks for the $4.5\text{−}\mathrm{y}$ period 1994–1999. We find that, for all three markets analyzed, the cumulative distribution of trade size displays a power-law tail $P(q>x)\sim x^{-{\zeta }_q}$ with exponent ${\zeta }_q<2$ within the Lévy stable domain. Our analysis of the exponent estimates of ${\zeta }_q$ suggests that the exponent value is universal in the following respects: (a) ${\zeta }_q$ is consistent across stocks within each of the three markets analyzed, and also across different markets, and (b) ${\zeta }_q$ does not display any systematic dependence on market capitalization or industry sector. We next analyze the distributions of share volume $Q_{\Delta t}$ over fixed time intervals and find that for all three markets $P\{Q>x\}\sim x^{-{\zeta }_Q}$ with exponent ${\zeta }_Q<2$ within the Lévy stable domain. To test the validity for $\Delta t=1\text{day}$ of the power-law distributions found from tick-by-tick data, we analyze a fourth large database containing daily U.S. data, and confirm a value for the exponent ${\zeta }_Q$ within the Lévy stable domain.