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Asymptotics for small nonlinear price impact: A PDE approach to the multidimensional case

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  • Erhan Bayraktar
  • Thomas Cayé
  • Ibrahim Ekren

Abstract

We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small nonlinear 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 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.
    • Handle: RePEc:bla:mathfi:v:31:y:2021:i:1:p:36-108
    DOI: 10.1111/mafi.12283
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    Cited by:

    1. Ibrahim Ekren & Sergey Nadtochiy, 2019. "Utility-based pricing and hedging of contingent claims in Almgren-Chriss model with temporary price impact," Papers 1910.01778, arXiv.org, revised Jun 2020.
    2. Lukas Gonon & Johannes Muhle‐Karbe & Xiaofei Shi, 2021. "Asset pricing with general transaction costs: Theory and numerics," Mathematical Finance, Wiley Blackwell, vol. 31(2), pages 595-648, April.
    3. Yan Dolinsky & Shir Moshe, 2021. "Utility Indifference Pricing with High Risk Aversion and Small Linear Price Impact," Papers 2111.00451, arXiv.org, revised Jan 2022.

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