Portfolio Liquidation in Dark Pools in Continuous Time

37 Pages Posted: 31 Jan 2012 Last revised: 3 Dec 2015

See all articles by Peter Kratz

Peter Kratz

Humboldt University of Berlin

Torsten Schoeneborn

AHL (Man Investments); University of Oxford - Oxford-Man Institute of Quantitative Finance

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Date Written: December 10, 2012


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

Kratz, Peter and Schoeneborn, Torsten, Portfolio Liquidation in Dark Pools in Continuous Time (December 10, 2012). Available at SSRN: https://ssrn.com/abstract=1995421 or http://dx.doi.org/10.2139/ssrn.1995421

Peter Kratz (Contact Author)

Humboldt University of Berlin ( email )

Unter den Linden 6
Berlin, Berlin 10099

Torsten Schoeneborn

AHL (Man Investments) ( email )

Sugar Quay
Lower Thames Street
London, EC3R 6DU
Great Britain

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

Eagle House
Walton Well Road
Oxford, Oxfordshire OX2 6ED
United Kingdom

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