Applied Mathematics & Optimization

, Volume 67, Issue 3, pp 353–390

Optimal Portfolio Selection Under Concave Price Impact


DOI: 10.1007/s00245-013-9191-7

Cite this article as:
Ma, J., Song, Q., Xu, J. et al. Appl Math Optim (2013) 67: 353. doi:10.1007/s00245-013-9191-7


In this paper we study an optimal portfolio selection problem under instantaneous price impact. Based on some empirical analysis in the literature, we model such impact as a concave function of the trading size when the trading size is small. The price impact can be thought of as either a liquidity cost or a transaction cost, but the concavity nature of the cost leads to some fundamental difference from those in the existing literature. We show that the problem can be reduced to an impulse control problem, but without fixed cost, and that the value function is a viscosity solution to a special type of Quasi-Variational Inequality (QVI). We also prove directly (without using the solution to the QVI) that the optimal strategy exists and more importantly, despite the absence of a fixed cost, it is still in a “piecewise constant” form, reflecting a more practical perspective.


Liquidity riskPrice impactTransaction costImpulse controlOptimal portfolio selectionStochastic optimization

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Jin Ma
    • 1
  • Qingshuo Song
    • 2
  • Jing Xu
    • 3
  • Jianfeng Zhang
    • 1
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelsUSA
  2. 2.Department of MathematicsCity University of Hong KongKowloon TongHong Kong
  3. 3.School of Economics and Business AdministrationChongqing UniversityChongqingChina