Dealing with the Inventory Risk. A solution to the market making problem.

Summary Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization problem in which their return, based on the bid-ask spread they quote and the frequency they indeed provide liquidity, is challenged by the price risk they bear due to their inventory. In this paper, we consider a stochastic control problem similar to the one introduced by Ho and Stoll and formalized mathematically by Avellaneda and Stoikov. The market is modeled using a reference price S_t following a Brownian motion, arrival rates of buy or sell liquidity-consuming orders depend on the distance to the reference price S_t and a market maker maximizes the expected utility of its PnL over a short time horizon. We show that the Hamilton-Jacobi-Bellman equations can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also provide a spectral characterization of the asymptotic behavior of the optimal quotes and propose closed-form approximations.