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} approximately 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_{Deltat}(t)= summation operator_{i=1};{N}q_{i} , the total number of shares exchanged as a result of the N=N_{Deltat} trades occurring in a time interval Deltat (termed share volume). We analyze the statistical properties of trade size q identical withq_{i} and share volume Q identical withQ_{Deltat}(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) approximately 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_{Deltat} over fixed time intervals and find that for all three markets P{Q>x} approximately x;{-zeta_{Q}} with exponent zeta_{Q}<2 within the Lévy stable domain. To test the validity for Deltat=1day 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.