We generalize the reaction-diffusion model A + B -> 0 in order to study the impact of an excess of A (or B) at the reaction front. We provide an exact solution of the model, which shows that linear response breaks down: the average displacement of th
e reaction front grows as the square-root of the imbalance. We argue that this model provides a highly simplified but generic framework to understand the square-root impact of large orders in financial markets.
We revisit the epsilon-intelligence model of Toth et al.(2011), 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 li
quidity 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 Toth et al. 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.
Order flow in equity markets is remarkably persistent in the sense that order signs (to buy or sell) are positively autocorrelated out to time lags of tens of thousands of orders, corresponding to many days. Two possible explanations are herding, cor
responding to positive correlation in the behavior of different investors, or order splitting, corresponding to positive autocorrelation in the behavior of single investors. We investigate this using order flow data from the London Stock Exchange for which we have membership identifiers. By formulating models for herding and order splitting, as well as models for brokerage choice, we are able to overcome the distortion introduced by brokerage. On timescales of less than a few hours the persistence of order flow is overwhelmingly due to splitting rather than herding. We also study the properties of brokerage order flow and show that it is remarkably consistent both cross-sectionally and longitudinally.
We propose a dynamical theory of market liquidity that predicts that the average supply/demand profile is V-shaped and {it vanishes} around the current price. This result is generic, and only relies on mild assumptions about the order flow and on the
fact that prices are (to a first approximation) diffusive. This naturally accounts for two striking stylized facts: first, large metaorders have to be fragmented in order to be digested by the liquidity funnel, leading to long-memory in the sign of the order flow. Second, the anomalously small local liquidity induces a breakdown of linear response and a diverging impact of small orders, explaining the square-root impact law, for which we provide additional empirical support. Finally, we test our arguments quantitatively using a numerical model of order flow based on the same minimal ingredients.
We present an empirical study of the intertwined behaviour of members in a financial market. Exploiting a database where the broker that initiates an order book event can be identified, we decompose the correlation and response functions into contrib
utions coming from different market participants and study how their behaviour is interconnected. We find evidence that (1) brokers are very heterogeneous in liquidity provision -- some are consistently liquidity providers while others are consistently liquidity takers. (2) The behaviour of brokers is strongly conditioned on the actions of {it other} brokers. In contrast brokers are only weakly influenced by the impact of their own previous orders. (3) The total impact of market orders is the result of a subtle compensation between the same broker pushing the price in one direction and the liquidity provision of other brokers pushing it in the opposite direction. These results enforce the picture of market dynamics being the result of the competition between heterogeneous participants interacting to form a complicated market ecology.
We present results on simulations of a stock market with heterogeneous, cumulative information setup. We find a non-monotonic behaviour of traders returns as a function of their information level. Particularly, the average informed agents underperfor
m random traders; only the most informed agents are able to beat the market. We also study the effect of a strategy updating mechanism, when traders have the possibility of using other pieces of information than the fundamental value. These results corroborate the latter ones: it is only for the most informed player that it is rewarding to stay fundamentalist. The simulations reproduce some stylized facts of tick-by-tick stock-exchange data and globally show informational efficiency.
