A Bayesian theory of market impact

The available liquidity at any time in financial markets falls largely short of the typical size of the orders that institutional investors would trade. In order to reduce the impact on prices due to the execution of large orders, traders in financial markets split large orders into a series of smaller ones, which are executed sequentially. The resulting sequence of trades is called a meta-order. Empirical studies have revealed a non-trivial set of statistical laws on how meta-orders affect prices, which include i) the square-root behaviour of the expected price variation with the total volume traded, ii) its crossover to a linear regime for small volumes, and iii) a reversion of average prices towards equilibrium, after the sequence of trades is over. Here we recover this phenomenology within a minimal theoretical framework where the market sets prices by incorporating all information on the direction and speed of trade of the meta-order in a Bayesian manner. The simplicity of this derivation lends further support to its robustness and universality, and it sheds light on its origin. In particular, it suggests that the square-root impact law originates from the over-estimation of order flows originating from meta-orders. The theory also offers a number of falsifiable predictions that may be tested on empirical data.