Unravelling the trading invariance hypothesis
We confirm and substantially extend the recent empirical result of Andersen et al. 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 times scales. However, we find that the 'trading invariant' I = W/N3/2 proposed by Kyle and Obfizhaeva is in fact quite different for different contracts, in particular between futures and single stocks. Our analysis suggests I/S as a more natural candidate, where S is 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 behaviour σ ∼ √N and V ∼ N hold, but below which strong deviations appear, induced by the size of the tick.