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Mathematics and Financial Economics

, Volume 7, Issue 4, pp 477–507 | Cite as

Dealing with the inventory risk: a solution to the market making problem

  • Olivier Guéant
  • Charles-Albert Lehalle
  • Joaquin Fernandez-Tapia
Article

Abstract

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.

Keywords

Stochastic optimal control High-frequency market making Avellaneda–Stoikov problem 

JEL Classification

C61 G10 G11 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Olivier Guéant
    • 1
  • Charles-Albert Lehalle
    • 2
  • Joaquin Fernandez-Tapia
    • 3
  1. 1.Laboratoire Jacques-Louis Lions, UFR de MathématiquesUniversité Paris-DiderotParisFrance
  2. 2.Head of Quantitative ResearchCrédit Agricole CheuvreuxCourbevoieFrance
  3. 3.LPMA, Université Pierre et Marie CurieParisFrance

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