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Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets

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  • Schied, Alexander
  • Schoeneborn, Torsten

Abstract

We consider the infinite-horizon optimal portfolio liquidation problem for a von Neumann-Morgenstern investor in the liquidity model of Almgren (2003). Using a stochastic control approach, we characterize the value function and the optimal strategy as classical solutions of nonlinear parabolic partial differential equations. We furthermore analyze the sensitivities of the value function and the optimal strategy with respect to the various model parameters. In particular, we find that the optimal strategy is aggressive or passive in-the-money, respectively, if and only if the utility function displays increasing or decreasing risk aversion. Surprisingly, only few further monotonicity relations exist with respect to the other parameters. We point out in particular that the speed by which the remaining asset position is sold can be decreasing in the size of the position but increasing in the liquidity price impact.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 7105.

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Date of creation: 08 Feb 2008
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Handle: RePEc:pra:mprapa:7105

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Keywords: Liquidity; illiquid markets; optimal liquidation strategies; dynamic trading strategies; algorithmic trading; utility maximization;

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References

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  1. Umut Çetin & L. C. G. Rogers, 2007. "Modeling Liquidity Effects In Discrete Time," Mathematical Finance, Wiley Blackwell, vol. 17(1), pages 15-29.
  2. Aur\'elien Alfonsi & Antje Fruth & Alexander Schied, 2007. "Optimal execution strategies in limit order books with general shape functions," Papers 0708.1756, arXiv.org, revised Feb 2010.
  3. Jean-Philippe Bouchaud & Yuval Gefen & Marc Potters & Matthieu Wyart, 2003. "Fluctuations and response in financial markets: the subtle nature of `random' price changes," Science & Finance (CFM) working paper archive 0307332, Science & Finance, Capital Fund Management.
  4. Schied, Alexander & Schöneborn, Torsten, 2007. "Optimal Portfolio Liquidation for CARA Investors," MPRA Paper 5075, University Library of Munich, Germany.
  5. Jean-Philippe Bouchaud & Yuval Gefen & Marc Potters & Matthieu Wyart, 2004. "Fluctuations and response in financial markets: the subtle nature of 'random' price changes," Quantitative Finance, Taylor and Francis Journals, vol. 4(2), pages 176-190.
  6. Kyle, Albert S, 1985. "Continuous Auctions and Insider Trading," Econometrica, Econometric Society, vol. 53(6), pages 1315-35, November.
  7. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
  8. P. Weber & B. Rosenow, 2005. "Order book approach to price impact," Quantitative Finance, Taylor and Francis Journals, vol. 5(4), pages 357-364.
  9. Amos Tversky & Daniel Kahneman, 1979. "Prospect Theory: An Analysis of Decision under Risk," Levine's Working Paper Archive 7656, David K. Levine.
  10. Robert Almgren, 2003. "Optimal execution with nonlinear impact functions and trading-enhanced risk," Applied Mathematical Finance, Taylor and Francis Journals, vol. 10(1), pages 1-18.
  11. Potters, Marc & Bouchaud, Jean-Philippe, 2003. "More statistical properties of order books and price impact," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 133-140.
  12. 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.
  13. Ajay Subramanian & Robert A. Jarrow, 2001. "The Liquidity Discount," Mathematical Finance, Wiley Blackwell, vol. 11(4), pages 447-474.
  14. Marc Potters & Jean-Philippe Bouchaud, 2002. "More statistical properties of order books and price impact," Science & Finance (CFM) working paper archive 0210710, Science & Finance, Capital Fund Management.
  15. Schoeneborn, Torsten & Schied, Alexander, 2007. "Liquidation in the Face of Adversity: Stealth Vs. Sunshine Trading, Predatory Trading Vs. Liquidity Provision," MPRA Paper 5548, University Library of Munich, Germany.
  16. He, Hua & Mamaysky, Harry, 2005. "Dynamic trading policies with price impact," Journal of Economic Dynamics and Control, Elsevier, vol. 29(5), pages 891-930, May.
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Citations

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Cited by:
  1. Olivier Gu\'eant, 2012. "Optimal execution and block trade pricing: a general framework," Papers 1210.6372, arXiv.org, revised Oct 2012.
  2. Fabien Guilbaud & Mohamed Mnif & Huy\^en Pham, 2010. "Numerical methods for an optimal order execution problem," Papers 1006.0768, arXiv.org.
  3. Tim Leung & Peng Liu, 2011. "Risk Premia and Optimal Liquidation of Credit Derivatives," Papers 1110.0220, arXiv.org, revised Oct 2012.
  4. Florian Kl\"ock & Alexander Schied & Yuemeng Sun, 2012. "Price manipulation in a market impact model with dark pool," Papers 1205.4008, arXiv.org.
  5. Aurélien Alfonsi & Alexander Schied, 2010. "Optimal trade execution and absence of price manipulations in limit order book models," Post-Print hal-00397652, HAL.
  6. Olivier Gu\'eant, 2012. "Execution and block trade pricing with optimal constant rate of participation," Papers 1210.7608, arXiv.org, revised Apr 2013.
  7. Olivier Gu\'eant & Charles-Albert Lehalle, 2012. "General Intensity Shapes in Optimal Liquidation," Papers 1204.0148, arXiv.org, revised Apr 2013.
  8. Peter Bank & Dmitry Kramkov, 2011. "A model for a large investor trading at market indifference prices. I: single-period case," Papers 1110.3224, arXiv.org, revised Oct 2011.
  9. Felix Naujokat & Ulrich Horst, 2011. "When to Cross the Spread: Curve Following with Singular Control," SFB 649 Discussion Papers SFB649DP2011-053, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  10. Alexander Weiss, 2009. "Executing large orders in a microscopic market model," Papers 0904.4131, arXiv.org, revised Jan 2010.
  11. Olivier Gu\'eant & Charles-Albert Lehalle & Joaquin Fernandez Tapia, 2011. "Optimal Portfolio Liquidation with Limit Orders," Papers 1106.3279, arXiv.org, revised Jul 2012.
  12. Peter Kratz & Torsten Sch\"oneborn, 2012. "Portfolio liquidation in dark pools in continuous time," Papers 1201.6130, arXiv.org, revised Aug 2012.
  13. Mauricio Junca, 2011. "Stochastic impulse control on optimal execution with price impact and transaction cost," Papers 1103.3482, arXiv.org, revised Jan 2013.
  14. Matteo Del Vigna, 2011. "Market equilibrium with heterogeneous behavioural and classical investors' preferences," DiMaD Working Papers 2011-09, Dipartimento di Matematica per le Decisioni, Universita' degli Studi di Firenze.
  15. Ulrich Horst & Felix Naujokat, 2010. "Illiquidity and Derivative Valuation," SFB 649 Discussion Papers SFB649DP2010-011, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  16. Ricardo Josa-Fombellida & Juan Pablo Rincón-Zapatero, 2008. "On one-dimensional stochastic control problems: applications to investment models," Economics Working Papers we086630, Universidad Carlos III, Departamento de Economía.
  17. Takashi Kato, 2011. "An Optimal Execution Problem in Geometric Ornstein-Uhlenbeck Price Process," Papers 1107.1787, arXiv.org, revised May 2012.

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