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}similar to x(R)(-zeta) 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)(t)=Sigma(N)(i=1)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 equivalent to q(i) and share volume Q equivalent to Q(Delta t)(t) by analyzing trade-by-trade data from three large databases representing three distinct markets: (i) 1000 major U.S. stocks for the 2-y period 1994-1995, (ii) 85 major U.K. stocks for the 2-y period 2001-2002, and (iii) 13 major Paris Bourse stocks for the 4.5-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)similar to x(q)(-zeta) with exponent zeta(q)< 2 within the Levy 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}similar to x(Q)(-zeta) with exponent zeta(Q)< 2 within the Levy stable domain. To test the validity for Delta t=1 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 Levy stable domain.