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Drift dependence of optimal trade execution strategies under transient price impact

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Abstract

We give a complete solution to the problem of minimizing the expected liquidity costs in the presence of a general drift when the underlying market impact model has linear transient price impact with exponential resilience. It turns out that this problem is well-posed only if the drift is absolutely continuous. Optimal strategies often do not exist, and when they do, they depend strongly on the derivative of the drift. Our approach uses elements from singular stochastic control, even though the problem is essentially non-Markovian due to the transience of price impact and the lack in Markovian structure of the underlying price process. As a corollary, we give a complete solution to the minimization of a certain cost-risk criterion in our setting.

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Notes

  1. Although the left-continuous modification of X is considered in [17], our definitions of both price process and costs coincide with the one in [17]. See Remark 2.2 for a detailed discussion of right- versus left-continuity.

  2. The requirement that X is bounded is natural from an economic point of view, because the total number of available shares is finite for every stock.

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Acknowledgements

The second author gratefully acknowledges support by Deutsche Forschungsgemeinschaft DFG.

We wish to thank Markus Hess and two anonymous referees for helpful comments on a previous version of the manuscript.

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  1. Department of Mathematics, University of Mannheim, A5, 6, 68131, Mannheim, Germany

    Christopher Lorenz & Alexander Schied

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  1. Christopher Lorenz
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  2. Alexander Schied
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Correspondence to Alexander Schied.

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Lorenz, C., Schied, A. Drift dependence of optimal trade execution strategies under transient price impact. Finance Stoch 17, 743–770 (2013). https://doi.org/10.1007/s00780-013-0211-x

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