We revisit the ‘e-intelligence’ model of Toth et al. (2011)[1], that was proposed as a minimal framework to understand the square-root dependence of the impact of meta-orders on volume in financial markets. The basic idea is that most of the daily liquidity is ‘latent’ and furthermore vanishes linearly around the current price, as a consequence of the diffusion of the price itself. However, the numerical implementation of Toth et al. was criticised as being unrealistic, in particular because all the ‘intelligence’ was conferred to market orders, while limit orders were passive and random. In this work, we study various alternative specifications of the model, for example allowing limit orders to react to the order flow, or changing the execution protocols. By and large, our study lends strong support to the idea that the square-root impact law is a very generic and robust property that requires very few ingredients to be valid.
We also show that the transition from super-diffusion to sub-diffusion reported in [1] is in fact a cross-over, but that the original model can be slightly altered in order to give rise to a genuine phase transition, which is of interest on its own. We finally propose a general theoretical framework to understand how a non-linear impact may appear even in the limit where the bias in the order flow is vanishingly small.