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

## Author Info

• Olivier Gu\'eant
• Charles-Albert Lehalle
• Joaquin Fernandez Tapia

## 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 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.

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File URL: http://arxiv.org/pdf/1105.3115

## Bibliographic Info

Paper provided by arXiv.org in its series Papers with number 1105.3115.

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 Length: Date of creation: May 2011 Date of revision: Aug 2012 Handle: RePEc:arx:papers:1105.3115 Contact details of provider: Web page: http://arxiv.org/

## References

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1. Sophie Laruelle & Charles-Albert Lehalle & Gilles Pagès, 2009. "Optimal split of orders across liquidity pools: a stochastic algorithm approach," Working Papers hal-00422427, HAL.
2. Aurélien Alfonsi & Alexander Schied, 2010. "Optimal trade execution and absence of price manipulations in limit order book models," Post-Print hal-00397652, HAL.
3. 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.
4. 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.
5. 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 1053-74, September.
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