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Portfolio liquidation in dark pools in continuous time

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  • Peter Kratz
  • Torsten Schoneborn

Abstract

We consider an illiquid financial market where a risk averse investor has to liquidate a portfolio within a finite time horizon [0,T] and can trade continuously at a traditional exchange (the "primary venue") and in a dark pool. At the primary venue, trading yields a linear price impact. In the dark pool, no price impact costs arise but order execution is uncertain, modeled by a multi-dimensional Poisson process. We characterize the costs of trading by a linear-quadratic functional which incorporates both the price impact costs of trading at the primary exchange and the market risk of the position. The liquidation constraint implies a singularity of the value function of the resulting minimization problem at the terminal time T. Via the HJB equation and a quadratic ansatz, we obtain a candidate for the value function which is the limit of a sequence of solutions of initial value problems for a matrix differential equation. We show that this limit exists by using an appropriate matrix inequality and a comparison result for Riccati matrix equations. Additionally, we obtain upper and lower bounds of the solutions of the initial value problems, which allow us to prove a verification theorem. If a single asset position is to be liquidated, the investor slowly trades out of her position at the primary venue, with the remainder being placed in the dark pool at any point in time. For multi-asset liquidations this is generally not the case; it can, e.g., be optimal to oversize orders in the dark pool in order to turn a poorly balanced portfolio into a portfolio bearing less risk.

Suggested Citation

  • Peter Kratz & Torsten Schoneborn, 2012. "Portfolio liquidation in dark pools in continuous time," Papers 1201.6130, arXiv.org, revised Aug 2012.
  • Handle: RePEc:arx:papers:1201.6130
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    File URL: http://arxiv.org/pdf/1201.6130
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    References listed on IDEAS

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    1. Alexander Schied & Torsten Schöneborn, 2009. "Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets," Finance and Stochastics, Springer, vol. 13(2), pages 181-204, April.
    2. Grossman, Sanford J & Miller, Merton H, 1988. " Liquidity and Market Structure," Journal of Finance, American Finance Association, vol. 43(3), pages 617-637, July.
    3. Obizhaeva, Anna A. & Wang, Jiang, 2013. "Optimal trading strategy and supply/demand dynamics," Journal of Financial Markets, Elsevier, vol. 16(1), pages 1-32.
    4. Glosten, Lawrence R. & Milgrom, Paul R., 1985. "Bid, ask and transaction prices in a specialist market with heterogeneously informed traders," Journal of Financial Economics, Elsevier, vol. 14(1), pages 71-100, March.
    5. Kyle, Albert S, 1985. "Continuous Auctions and Insider Trading," Econometrica, Econometric Society, vol. 53(6), pages 1315-1335, November.
    6. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
    7. Aurelien Alfonsi & Antje Fruth & Alexander Schied, 2010. "Optimal execution strategies in limit order books with general shape functions," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 143-157.
    8. 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.
    9. Conrad, Jennifer & Johnson, Kevin M. & Wahal, Sunil, 2003. "Institutional trading and alternative trading systems," Journal of Financial Economics, Elsevier, vol. 70(1), pages 99-134, October.
    10. 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.
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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 Dec 2014.
    2. Olivier Gu'eant, 2012. "Execution and block trade pricing with optimal constant rate of participation," Papers 1210.7608, arXiv.org, revised Dec 2013.

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