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

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  • 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
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Bibliographic Info

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

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Date of creation: May 2011
Date of revision: Aug 2012
Handle: RePEc:arx:papers:1105.3115

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Web page: http://arxiv.org/

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References

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  1. E. Bacry & S. Delattre & M. Hoffmann & J. F. Muzy, 2011. "Modeling microstructure noise with mutually exciting point processes," Papers 1101.3422, arXiv.org.
  2. Sophie Laruelle & Charles-Albert Lehalle & Gilles Pagès, 2009. "Optimal split of orders across liquidity pools: a stochastic algorithm approach," Working Papers hal-00422427, HAL.
  3. 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.
  4. Thomas Ho & Hans Stoll, . "Optimal Dealer Pricing Under Transactions and Return Uncertainty," Rodney L. White Center for Financial Research Working Papers 27-79, Wharton School Rodney L. White Center for Financial Research.
  5. Aurélien Alfonsi & Alexander Schied, 2010. "Optimal trade execution and absence of price manipulations in limit order book models," Post-Print hal-00397652, HAL.
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Cited by:
  1. Fabien Guilbaud & Huyen Pham, 2011. "Optimal High Frequency Trading with limit and market orders," Working Papers hal-00603385, HAL.
  2. Mauricio Labadie & Charles-Albert Lehalle, 2012. "Optimal starting times, stopping times and risk measures for algorithmic trading," Working Papers hal-00705056, HAL.
  3. Fabien Guilbaud & Huyen Pham, 2011. "Optimal High Frequency Trading with limit and market orders," Papers 1106.5040, arXiv.org.
  4. Christoph K\"uhn & Johannes Muhle-Karbe, 2013. "Optimal Liquidity Provision in Limit Order Markets," Papers 1309.5235, arXiv.org, revised May 2014.
  5. Mauricio Labadie & Charles-Albert Lehalle, 2012. "Optimal starting times, stopping times and risk measures for algorithmic trading: Target Close and Implementation Shortfall," Papers 1205.3482, arXiv.org, revised Dec 2013.
  6. Sophie Laruelle & Charles-Albert Lehalle & Gilles Pag\`es, 2011. "Optimal posting price of limit orders: learning by trading," Papers 1112.2397, arXiv.org, revised Sep 2012.
  7. Olivier Gu\'eant & Charles-Albert Lehalle & Joaquin Fernandez Tapia, 2011. "Optimal Portfolio Liquidation with Limit Orders," Papers 1106.3279, arXiv.org, revised Jul 2012.
  8. Robert Azencott & Arjun Beri & Yutheeka Gadhyan & Nicolas Joseph & Charles-Albert Lehalle & Matthew Rowley, 2013. "Realtime market microstructure analysis: online Transaction Cost Analysis," Papers 1302.6363, arXiv.org, revised Mar 2013.
  9. M. Alessandra Crisafi & Andrea Macrina, 2014. "Optimal Execution in Lit and Dark Pools," Papers 1405.2023, arXiv.org.

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