IDEAS home Printed from
   My bibliography  Save this paper

Asymptotics for Small Nonlinear Price Impact: a PDE Approach to the Multidimensional Case


  • Erhan Bayraktar
  • Thomas Caye
  • Ibrahim Ekren


We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small non-linear price impact. In our model cross-impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around its frictionless version. The leading order correction is characterized by a nonlinear second order PDE related to an ergodic control problem and a linear parabolic PDE. We illustrate our result on a multivariate geometric Brownian motion price model.

Suggested Citation

  • Erhan Bayraktar & Thomas Caye & Ibrahim Ekren, 2018. "Asymptotics for Small Nonlinear Price Impact: a PDE Approach to the Multidimensional Case," Papers 1811.06650,, revised Jun 2020.
  • Handle: RePEc:arx:papers:1811.06650

    Download full text from publisher

    File URL:
    File Function: Latest version
    Download Restriction: no

    References listed on IDEAS

    1. Paolo Guasoni & Mikl'os R'asonyi, 2015. "Hedging, arbitrage and optimality with superlinear frictions," Papers 1506.05895,
    2. Alexander Schied & Torsten Schöneborn, 2009. "Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets," Finance and Stochastics, Springer, vol. 13(2), pages 181-204, April.
    3. Olivier Guéant & Jiang Pu, 2017. "Option Pricing And Hedging With Execution Costs And Market Impact," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 803-831, July.
    4. Ludovic Moreau & Johannes Muhle-Karbe & H. Mete Soner, 2014. "Trading with Small Price Impact," Papers 1402.5304,, revised Mar 2015.
    5. Paolo Guasoni & Marko Weber, 2017. "Dynamic Trading Volume," Mathematical Finance, Wiley Blackwell, vol. 27(2), pages 313-349, April.
    6. Bertsimas, Dimitris & Lo, Andrew W., 1998. "Optimal control of execution costs," Journal of Financial Markets, Elsevier, vol. 1(1), pages 1-50, April.
    7. Dylan Possamai & H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs: the multidimensional case," Papers 1212.6275,, revised Jan 2013.
    8. Robert Almgren, 2003. "Optimal execution with nonlinear impact functions and trading-enhanced risk," Applied Mathematical Finance, Taylor & Francis Journals, vol. 10(1), pages 1-18.
    9. Gârleanu, Nicolae & Pedersen, Lasse Heje, 2016. "Dynamic portfolio choice with frictions," Journal of Economic Theory, Elsevier, vol. 165(C), pages 487-516.
    10. Karel Janeček & Steven Shreve, 2004. "Asymptotic analysis for optimal investment and consumption with transaction costs," Finance and Stochastics, Springer, vol. 8(2), pages 181-206, May.
    11. Jan Kallsen & Johannes Muhle-Karbe, 2017. "The General Structure Of Optimal Investment And Consumption With Small Transaction Costs," Mathematical Finance, Wiley Blackwell, vol. 27(3), pages 659-703, July.
    12. Ludovic MOREAU & Johannes MUHLE-KARBE & Halil Mete SONER, 2014. "Trading with Small Price Impact," Swiss Finance Institute Research Paper Series 14-17, Swiss Finance Institute, revised Mar 2015.
    13. Peter Bank & Moritz Vo{ss}, 2018. "Optimal investment with transient price impact," Papers 1804.07392,
    14. 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.
    15. Yan Dolinsky & Halil Soner, 2013. "Duality and convergence for binomial markets with friction," Finance and Stochastics, Springer, vol. 17(3), pages 447-475, July.
    16. H. Mete Soner & Nizar Touzi, 2012. "Homogenization and asymptotics for small transaction costs," Papers 1202.6131,, revised Jun 2013.
    17. Jan Kallsen & Shen Li, 2013. "Portfolio Optimization under Small Transaction Costs: a Convex Duality Approach," Papers 1309.3479,
    18. Ibrahim Ekren & Johannes Muhle-Karbe, 2017. "Portfolio Choice with Small Temporary and Transient Price Impact," Papers 1705.00672,, revised Apr 2020.
    Full references (including those not matched with items on IDEAS)

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1811.06650. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (arXiv administrators). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.