Mathematics Think Tank

Dealing with the Inventory Risk

J. Fernandez-Tapia, O. Guéant, C.-A. Lehalle in Mathematics and Financial Economics 2012

Market makers continuously set bid and ask quotes for the stocks they have under consideration. Hence they face a complex optimization prob- lem in which their return, based on the bid-ask spread they quote and the fre- quency 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 St 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 St 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 trans- formed 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-

This research has been conducted within the Research Initiative “Microstructure des March ́es Financiers” under the aegis of the Europlace Institute of Finance.

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