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 at which 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 [17] and formalized mathematically by Avellaneda and Stoikov [3]. The market is modeled using a reference price S t following a Brownian motion with standard deviation σ, 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 P&L over a finite time horizon. We show that the Hamilton-Jacobi-Bellman equations associated to the stochastic optimal control problem can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also shed light on the asymptotic behavior of the optimal quotes and propose closed-form approxi-7 Our results would be mutatis mutandis the same if we added a penalization term −b(|q T |) for the shares remaining at time T . The rationale underlying this point is that price risk prevents the trader from having important exposure to the stock. Hence, qt should naturally mean-revert around 0.

This paper addresses portfolio liquidation using a new angle. Instead of focusing only on the scheduling aspect like Almgren and Chriss in [2], or only on the liquidity-consuming orders like Obizhaeva and Wang in [35], we link the optimal trade-schedule to the price of the limit orders that have to be sent to the limit order book to optimally liquidate a portfolio. Most practitioners address these two issues separately: they compute an optimal trading curve and they then send orders to the markets to try to follow it. The results obtained in this paper can be interpreted and used in two ways: (i) we solve simultaneously the two problems and provide a strategy to liquidate a portfolio over a few hours, (ii) we provide a tactic to follow a trading curve over slices of a few minutes. As far as the model is concerned, the interactions of limit orders with the market are modeled via a point process pegged to a diffusive "fair price". A Hamilton-Jacobi-Bellman equation is used to solve the control problem involving both non-execution risk and price risk. Backtests are carried out to exemplify the use of our results, both on long periods of time (for the entire liquidation process) and on slices of 5 minutes (to follow a given trading curve).