Optimal limit order execution in a simple model for market microstructure dynamics

Market participants that have a task to acquire a certain position in a listed security at a predetermined price on behalf of a third party with no time urgency, i.e. to ll a perpetual limit order, can optimize the protability of their trading strategy in order to accomplish this task. We study the statistical properties of the prot distribution of a particular marketmaking strategy: one which increments the inventory as the underlying price approaches the limit order price S0 and locks in prots by gradually liquidating the inventory as the market drifts away from S0. We do so by adopting a simple model of market microstructure in which an unobservable continuous stochastic process, the microprice, drives the dynamics of limit and market orders. In this model, the arrival of market orders and updates of the limit order book are determined by the microprice crossing a discrete set of n equidistant levels between the price ticks. Assuming normal dynamics for the microprice and adopting a standard meanvariance framework, we are able to derive a closed-form solution for the optimal inventory prole which is remarkably simple: the cumulative amount held when the market price is Si is inversely proportional to Si S0, the distance in price terms from the limit order price. Finally, we show that n represents a sort of micro-volatility of the market that is distinct from the diusive volatility of the microprice and is a measure of the intensity of the bid-ask bounce.