Dealing with the Inventory Risk. A solution to the market making problem

Olivier Guéant, Charles-Albert Lehalle and Joaquin Fernandez Tapia
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Joaquin Fernandez Tapia: LPMA - Laboratoire de Probabilités et Modèles Aléatoires - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique

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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 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, 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 PnL over a short time horizon. We show that the Hamilton-Jacobi-Bellman equations can be transformed into a system of linear ordinary differential equations and we solve the market making problem under inventory constraints. We also provide a spectral characterization of the asymptotic behavior of the optimal quotes and propose closed-form approximations.

Date: 2013
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Published in Mathematics and Financial Economics, 2013, 7 (4), pp.477-507. ⟨10.1007/s11579-012-0087-0⟩

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Working Paper: Dealing with the Inventory Risk. A solution to the market making problem (2012)
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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01393110

DOI: 10.1007/s11579-012-0087-0

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