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Optimal investment with transient price impact


  • Peter Bank
  • Moritz Vo{ss}


We introduce a price impact model which accounts for finite market depth, tightness and resilience. Its coupled bid- and ask-price dynamics induce convex liquidity costs. We provide existence of an optimal solution to the classical problem of maximizing expected utility from terminal liquidation wealth at a finite planning horizon. In the specific case when market uncertainty is generated by an arithmetic Brownian motion with drift and the investor exhibits constant absolute risk aversion, we show that the resulting singular optimal stochastic control problem readily reduces to a deterministic optimal tracking problem of the optimal frictionless constant Merton portfolio in the presence of convex costs. Rather than studying the associated Hamilton-Jacobi-Bellmann PDE, we exploit convex analytic and calculus of variations techniques allowing us to construct the solution explicitly and to describe the free boundaries of the action- and non-action regions in the underlying state space. As expected, it is optimal to trade towards the frictionless Merton position, taking into account the initial bid-ask spread as well as the optimal liquidation of the accrued position when approaching terminal time. It turns out that this leads to a surprisingly rich phenomenology of possible trajectories for the optimal share holdings.

Suggested Citation

  • Peter Bank & Moritz Vo{ss}, 2018. "Optimal investment with transient price impact," Papers 1804.07392,
  • Handle: RePEc:arx:papers:1804.07392

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    References listed on IDEAS

    1. Aur'elien Alfonsi & Antje Fruth & Alexander Schied, 2007. "Optimal execution strategies in limit order books with general shape functions," Papers 0708.1756,, revised Feb 2010.
    2. 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.
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    7. Alexandre Roch & H. Mete Soner, 2013. "Resilient Price Impact Of Trading And The Cost Of Illiquidity," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(06), pages 1-27.
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    9. Ibrahim Ekren & Johannes Muhle-Karbe, 2017. "Portfolio Choice with Small Temporary and Transient Price Impact," Papers 1705.00672,, revised Apr 2020.
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    Cited by:

    1. Erhan Bayraktar & Thomas Cayé & Ibrahim Ekren, 2021. "Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 36-108, January.
    2. Peter Bank & Yan Dolinsky, 2018. "Continuous-time Duality for Super-replication with Transient Price Impact," Papers 1808.09807,, revised May 2019.

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