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

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.
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https://hal.archives-ouvertes.fr/hal-01393110
Contributor : Olivier Guéant <>
Submitted on : Sunday, November 6, 2016 - 3:18:43 PM
Last modification on : Friday, March 27, 2020 - 3:01:58 AM

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  • HAL Id : hal-01393110, version 1

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Olivier Guéant, Charles-Albert Lehalle, Joaquin Fernandez Tapia. Dealing with the Inventory Risk. A solution to the market making problem. Mathematics and Financial Economics, Springer Verlag, 2013, 7 (4), pp.477-507. ⟨hal-01393110⟩

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