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Mean--Variance Optimal Adaptive Execution


  • Julian Lorenz
  • Robert Almgren


Electronic trading of equities and other securities makes heavy use of ‘arrival price’ algorithms that balance the market impact cost of rapid execution against the volatility risk of slow execution. In the standard formulation, mean--variance optimal trading strategies are static: they do not modify the execution speed in response to price motions observed during trading. We show that substantial improvement is possible by using dynamic trading strategies and that the improvement is larger for large initial positions. We develop a technique for computing optimal dynamic strategies to any desired degree of precision. The asset price process is observed on a discrete tree with an arbitrary number of levels. We introduce a novel dynamic programming technique in which the control variables are not only the shares traded at each time step but also the maximum expected cost for the remainder of the program; the value function is the variance of the remaining program. The resulting adaptive strategies are ‘aggressive-in-the-money’: they accelerate the execution when the price moves in the trader's favor, spending parts of the trading gains to reduce risk.

Suggested Citation

  • Julian Lorenz & Robert Almgren, 2011. "Mean--Variance Optimal Adaptive Execution," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(5), pages 395-422, January.
  • Handle: RePEc:taf:apmtfi:v:18:y:2011:i:5:p:395-422 DOI: 10.1080/1350486X.2011.560707

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    References listed on IDEAS

    1. J. Michael Harrison & Stanley R. Pliska, 1981. "Martingales and Stochastic Integrals in the Theory of Continous Trading," Discussion Papers 454, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    2. David B. Colwell & Robert J. Elliott, 1993. "Discontinuous Asset Prices And Non-Attainable Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 3(3), pages 295-308.
    3. Mark Broadie & Jérôme Detemple, 1997. "The Valuation of American Options on Multiple Assets," Mathematical Finance, Wiley Blackwell, vol. 7(3), pages 241-286.
    4. Chandrasekhar Reddy Gukhal, 2001. "Analytical Valuation of American Options on Jump-Diffusion Processes," Mathematical Finance, Wiley Blackwell, vol. 11(1), pages 97-115.
    5. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
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    Cited by:

    1. repec:eee:ejores:v:264:y:2018:i:3:p:1159-1171 is not listed on IDEAS
    2. Elias Strehle, 2016. "Are Order Anticipation Strategies Harmful? A Theoretical Approach," Papers 1609.00599,, revised Sep 2017.
    3. repec:exl:2manag:v:17:y:2016:i:2:p:241-260 is not listed on IDEAS
    4. Damiano Brigo & Clement Piat, 2016. "Static vs adapted optimal execution strategies in two benchmark trading models," Papers 1609.05523,
    5. Olivier Gu'eant & Jean-Michel Lasry & Jiang Pu, 2014. "A convex duality method for optimal liquidation with participation constraints," Papers 1407.4614,, revised Dec 2014.
    6. Ryuichi Yamamoto, 2015. "Dynamic predictor selection and order splitting in a limit order market," Working Papers 1514, Waseda University, Faculty of Political Science and Economics.
    7. Du, Bian & Zhu, Hongliang & Zhao, Jingdong, 2016. "Optimal execution in high-frequency trading with Bayesian learning," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 767-777.
    8. Henryk Gurgul & Robert Syrek & Christoph Mitterer, 2016. "Price duration versus trading volume in high-frequency data for selected DAX companies," Managerial Economics, AGH University of Science and Technology, vol. 17(2), pages 241-260, December.
    9. Tim Leung & Yoshihiro Shirai, 2015. "Optimal Derivative Liquidation Timing Under Path-Dependent Risk Penalties," Papers 1502.00358,

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