The Tick-by-Tick Dynamical Consistency of Price Impact in Limit Order Books

Constant price impact functions, much used in financial literature, are shown to give rise to paradoxical outcomes as they do not allow for proper predictability removal: for instance, the exploitation of a single large trade whose size and time of execution are known in advance to some insider leaves the arbitrage opportunity unchanged, which allows arbitrage exploitation multiple times. We argue that chain arbitrage exploitation should not exist, which provides an a contrario consistency criterion. Remarkably, all the stocks investigated in the Paris Stock Exchange have dynamically consistent price impact functions. Both the bid-ask spread and the feedback of sequential same-side market orders onto both sides of the order book are essential to ensure consistency at the smallest time scale.