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A state-constrained differential game arising in optimal portfolio liquidation

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  • Alexander Schied
  • Tao Zhang

Abstract

We consider $n$ risk-averse agents who compete for liquidity in an Almgren--Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear-quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions for finite and infinite time horizons. We prove existence and uniqueness of Nash equilibria and give closed-form solutions in some special cases. We also analyze qualitative properties of the equilibrium strategies and provide corresponding financial interpretations.

Suggested Citation

  • Alexander Schied & Tao Zhang, 2013. "A state-constrained differential game arising in optimal portfolio liquidation," Papers 1312.7360, arXiv.org, revised Jul 2015.
    • Handle: RePEc:arx:papers:1312.7360
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    References listed on IDEAS

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    1. Charles-Albert Lehalle, 2013. "Market Microstructure Knowledge Needed for Controlling an Intra-Day Trading Process," Papers 1302.4592, arXiv.org.
    2. Markus K. Brunnermeier & Lasse Heje Pedersen, 2005. "Predatory Trading," Journal of Finance, American Finance Association, vol. 60(4), pages 1825-1863, August.
    3. Bruce Ian Carlin & Miguel Sousa Lobo & S. Viswanathan, 2007. "Episodic Liquidity Crises: Cooperative and Predatory Trading," Journal of Finance, American Finance Association, vol. 62(5), pages 2235-2274, October.
    4. Alexander Schied & Torsten Schoneborn & Michael Tehranchi, 2010. "Optimal Basket Liquidation for CARA Investors is Deterministic," Applied Mathematical Finance, Taylor & Francis Journals, vol. 17(6), pages 471-489.
    5. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
    6. Robert Almgren, 2003. "Optimal execution with nonlinear impact functions and trading-enhanced risk," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(1), pages 1-18.
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    Cited by:

    1. Saran Ahuja & George Papanicolaou & Weiluo Ren & Tzu-Wei Yang, 2016. "Limit order trading with a mean reverting reference price," Papers 1607.00454, arXiv.org, revised Nov 2016.
    2. Elias Strehle, 2016. "Optimal Execution in a Multiplayer Model of Transient Price Impact," Papers 1609.00599, arXiv.org, revised Mar 2019.
    3. Qing-Qing Yang & Jia-Wen Gu & Wai-Ki Ching & Tak-Kuen Siu, 2019. "On Optimal Pricing Model for Multiple Dealers in a Competitive Market," Computational Economics, Springer;Society for Computational Economics, vol. 53(1), pages 397-431, January.

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