naa a22 4500 510808468 CHVBK 20180411083429.0 cr unu---uuuuu 180411e20130901xx s 000 0 eng 10.1007/s11579-012-0087-0 doi (NATIONALLICENCE)springer-10.1007/s11579-012-0087-0 Dealing with the inventory risk: a solution to the market making problem [Elektronische Daten] [Olivier Guéant, Charles-Albert Lehalle, Joaquin Fernandez-Tapia] 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 (J Fin Econ 9(1): 47-73, 1981) and formalized mathematically by Avellaneda and Stoikov (Quant Fin 8(3):217-224, 2008). 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 approximations based on a spectral characterization of the optimal quotes. Springer-Verlag, 2012 Stochastic optimal control nationallicence High-frequency market making nationallicence Avellaneda-Stoikov problem nationallicence Guéant Olivier Laboratoire Jacques-Louis Lions, UFR de Mathématiques, Université Paris-Diderot, 175, rue du Chevaleret, 75013, Paris, France aut Lehalle Charles-Albert Head of Quantitative Research, Crédit Agricole Cheuvreux, 9, Quai du Président Paul Doumer, 92400, Courbevoie, France aut Fernandez-Tapia Joaquin LPMA, Université Pierre et Marie Curie, 4 Place Jussieu, 75005, Paris, France aut Mathematics and Financial Economics Springer Berlin Heidelberg 7/4(2013-09-01), 477-507 1862-9679 7:4<477 2013 7 11579 https://doi.org/10.1007/s11579-012-0087-0 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11579-012-0087-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Guéant Olivier Laboratoire Jacques-Louis Lions, UFR de Mathématiques, Université Paris-Diderot, 175, rue du Chevaleret, 75013, Paris, France aut NATIONALLICENCE 700 1- Lehalle Charles-Albert Head of Quantitative Research, Crédit Agricole Cheuvreux, 9, Quai du Président Paul Doumer, 92400, Courbevoie, France aut NATIONALLICENCE 700 1- Fernandez-Tapia Joaquin LPMA, Université Pierre et Marie Curie, 4 Place Jussieu, 75005, Paris, France aut NATIONALLICENCE 773 0- Mathematics and Financial Economics Springer Berlin Heidelberg 7/4(2013-09-01), 477-507 1862-9679 7:4<477 2013 7 11579 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer