Unravelling the trading invariance hypothesis

We confirm and substantially extend the recent empirical result of Andersen et al. \cite{Andersen2015}, 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" $I=W/N^{3/2}$ proposed by Kyle and Obizhaeva is in fact quite different for different contracts, in particular between futures and single stocks. Our analysis suggests $I/{\cal C}$ as a more natural candidate, where $\cal C$ 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 $\sigma$ and the traded volume $V$ as a function of $N$, that reveal the existence of a characteristic number of trades $N_0$ above which the expected behaviour $\sigma \sim \sqrt{N}$ and $V \sim N$ hold, but below which strong deviations appear, induced by the size of the~tick.