We confirm and substantially extend the recent empirical result of Andersen et al. (Andersen, T. G., O. Bondarenko, A. S. Kyle and A. A. Obizhaeva, 2015, Unpublished), where it is shown that the amount of risk W exchanged in the E-mini S&P futures market (i.e., price times volume times volatility) scales like the 3/2 power of the number of trades N. We show that this 3/2-law holds very precisely across 12 futures contracts and 300 single US stocks, and across a wide range of time scales. However, we find that the “trading invariant” I=W∕N3∕2 proposed by Kyle and Obizhaeva is in fact quite different for different contracts, in particular, between futures and single stocks. Our analysis suggests I∕𝒞 as a more natural candidate, where 𝒞 is the average spread cost of a trade, defined as the average of the trade size times the bid–ask spread. We also establish two more complex scaling laws for the volatility σ and the traded volume V as a function of N, that reveal the existence of a characteristic number of trades N0 above which the expected behavior σ∼√N and V∼N hold, but below which strong deviations appear, induced by the size of the tick.
