Dealing with the Inventory Risk. A solution to the market making problem
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 and formalized mathematically by Avellaneda and Stoikov. The market is modeled using a reference price $S_t$ following a Brownian motion with standard deviation $\sigma$, 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 approximations based on a spectral characterization of the optimal quotes.
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Sophie Laruelle & Charles-Albert Lehalle & Gilles Pag\`es, 2009.
"Optimal split of orders across liquidity pools: a stochastic algorithm approach,"
0910.1166, arXiv.org, revised May 2010.
- Sophie Laruelle & Charles-Albert Lehalle & Gilles Pagès, 2010. "Optimal split of orders across liquidity pools: a stochastic algorithm approach," Working Papers hal-00422427, HAL.
- Ho, Thomas & Stoll, Hans R., 1981. "Optimal dealer pricing under transactions and return uncertainty," Journal of Financial Economics, Elsevier, vol. 9(1), pages 47-73, March.
- Thomas Ho & Hans Stoll, "undated". "Optimal Dealer Pricing Under Transactions and Return Uncertainty," Rodney L. White Center for Financial Research Working Papers 27-79, Wharton School Rodney L. White Center for Financial Research.
- Aurélien Alfonsi & Alexander Schied, 2010. "Optimal trade execution and absence of price manipulations in limit order book models," Post-Print hal-00397652, HAL.
- Ho, Thomas S Y & Stoll, Hans R, 1983. " The Dynamics of Dealer Markets under Competition," Journal of Finance, American Finance Association, vol. 38(4), pages 1053-1074, September.
- E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2013. "Modelling microstructure noise with mutually exciting point processes," Quantitative Finance, Taylor & Francis Journals, vol. 13(1), pages 65-77, January.
- E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2011. "Modeling microstructure noise with mutually exciting point processes," Papers 1101.3422, arXiv.org.