Optimal Microstructure Trading with a Long-Term Utility Function

Abstract

We combine Almgren--Chriss optimal execution with market microstructure in a framework where passive (joining the queue in a limit order book) or aggressive (willing to cross the bid-offer spread) modes of execution are allowed. To achieve this, we represent the Almgren--Chriss strategy within the framework of Hamiltonian dynamics. We then show that if a risk-neutral agent has expected returns equal to Hamilton's generalized momenta, then such agent repeatedly solving a myopic wealth-maximization problem reproduces the Almgren and Chriss solution. Hence the vector of generalized momenta, p, represents effective microstructure alphas, and also is the gradient of the Bellman value function. We demonstrate that our formulation is computationally efficient and provide a practical algorithm, accompanied by a numerical example which illustrates what can go wrong in the naive approach.