Optimal Asset Liquidation with Multiplicative Transient Price Impact



We study a multiplicative transient price impact model for an illiquid financial market, where trading causes price impact which is multiplicative in relation to the current price, transient over time with finite rate of resilience, and non-linear in the order size. We construct explicit solutions for the optimal control and the value function of singular optimal control problems to maximize expected discounted proceeds from liquidating a given asset position. A free boundary problem, describing the optimal control, is solved for two variants of the problem where admissible controls are monotone or of bounded variation.


Singular control Finite-fuel problem Free boundary Variational inequality Illiquidity Multiplicative price impact Limit order book 

Mathematics Subject Classification

35R35 49J40 49L20 60H30 93E20 91G80 



Dirk Becherer: We thank Peter Bank for fruitful discussions on an early version of the control problem. Todor Bilarev: Support by German Science foundation DFG via Berlin Mathematical School BMS and research training group RTG1845 StoA is gratefully acknowledged.


  1. 1.
    Alfonsi, A., Fruth, A., Schied, A.: Optimal execution strategies in limit order books with general shape functions. Quant. Financ. 10(2), 143–157 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Alfonsi, A., Schied, A., Slynko, A.: Order book resilience, price manipulation, and the positive portfolio problem. SIAM J. Financ. Math. 3(1), 511–533 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Becherer, D., Bilarev, T., Frentrup, P.: Optimal liquidation under stochastic liquidity (2016). arXiv:1603.06498v3
  4. 4.
    Becherer, D., Bilarev, T., Frentrup, P.: Stability for gains from large investors’ strategies in M1/J1 topologies (2017). arXiv:1701.02167v1
  5. 5.
    Bertsimas, D., Lo, A.W.: Optimal control of execution costs. J. Financ. Mark. 1(1), 1–50 (1998)CrossRefGoogle Scholar
  6. 6.
    Bouchard, B., Loeper, G., Zou, Y.: Almost-sure hedging with permanent price impact. Financ. Stoch. 20(3), 741–771 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Cont, R., De Larrard, A.: Price dynamics in a Markovian limit order market. SIAM J. Financ. Math. 4(1), 1–25 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton (2004)MATHGoogle Scholar
  9. 9.
    Chan, L.K.C., Lakonishok, J.: The behavior of stock prices around institutional trades. J. Financ. 50(4), 1147–1174 (1995)CrossRefGoogle Scholar
  10. 10.
    Dellacherie, C., Meyer, P.-A.: Probabilities and Potential. B. North-Holland, Amsterdam (1982)MATHGoogle Scholar
  11. 11.
    Dufour, F., Miller, B.: Singular stochastic control problems. SIAM J. Control Optim. 43(2), 708–730 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312(2), 215–250 (1998)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Davis, M.H.A., Zervos, M.: A pair of explicitly solvable singular stochastic control problems. Appl. Math. Optim. 38(3), 327–352 (1998)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Forsyth, P.A., Kennedy, J.S., Tse, S.T., Windcliff, H.: Optimal trade execution: a mean quadratic variation approach. J. Econ. Dynam. Control 36(12), 1971–1991 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gatheral, J., Schied, A.: Dynamical models of market impact and algorithms for order execution. In: Handbook on Systemic Risk, pp. 579–602. Cambridge University Press, Cambridge (2013)Google Scholar
  16. 16.
    Guo, X., Zervos, M.: Optimal execution with multiplicative price impact. SIAM J. Financ. Math. 6(1), 281–306 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jack, A., Johnson, T.C., Zervos, M.: A singular control model with application to the goodwill problem. Stoch. Process. Appl. 118(11), 2098–2124 (2008)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, Berlin (2003)CrossRefMATHGoogle Scholar
  19. 19.
    Kallsen, J.: Optimal portfolios for exponential Lévy processes. Math. Methods Oper. Res. 51(3), 357–374 (2000)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kobila, T.Ø.: A class of solvable stochastic investment problems involving singular controls. Stoch. Stoch. Rep. 43(1–2), 29–63 (1993)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Kharroubi, I., Pham, H.: Optimal portfolio liquidation with execution cost and risk. SIAM J. Financ. Math. 1(1), 897–931 (2010)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Karatzas, I., Shreve, S.E.: Equivalent models for finite-fuel stochastic control. Stochastics 18(3–4), 245–276 (1986)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Klein, O., Maug, E., Schneider, C.: Trading strategies of corporate insiders. J. Financ. Mark. (2017). doi: 10.1016/j.finmar.2017.04.001
  24. 24.
    Løkka, A.: Optimal execution in a multiplicative limit order book. Preprint, London School of Economics (2012)Google Scholar
  25. 25.
    Lorenz, C., Schied, A.: Drift dependence of optimal trade execution strategies under transient price impact. Financ. Stoch. 17(4), 743–770 (2013)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Ly Vath, V., Mnif, M., Pham, H.: A model of optimal portfolio selection under liquidity risk and price impact. Financ. Stoch. 11(1), 51–90 (2007)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Obizhaeva, A., Wang, J.: Optimal trading strategy and supply/demand dynamics. J. Financ. Mark. 16, 1–32 (2013)CrossRefGoogle Scholar
  28. 28.
    Predoiu, S., Shaikhet, G., Shreve, S.: Optimal execution in a general one-sided limit-order book. SIAM J. Financ. Math. 2(1), 183–212 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Schied, A.: Robust strategies for optimal order execution in the Almgren-Chriss framework. Appl. Math. Financ. 20(3), 264–286 (2013)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Schachermayer, W., Teichmann, J.: How close are the option pricing formulas of Bachelier and Black-Merton-Scholes? Math. Financ. 18(1), 155–170 (2008)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Taksar, M.I.: Infinite-dimensional linear programming approach to singular stochastic control. SIAM J. Control Optim. 35(2), 604–625 (1997)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Institute of MathematicsHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations