Appl Math Optim (2013) 67:353–390
Optimal Portfolio Selection Under Concave Price
Jin Ma · Qingshuo Song · Jing Xu · Jianfeng Zhang
Published online: 15 January 2013
© Springer Science+Business Media New York 2013
Abstract In this paper we study an optimal portfolio selection problem under instan-
taneous 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.
Jin Ma is supported in part by NSF grants #DMS 0806017 and 1106853.
The research of Qingshuo Song is supported in part by the Research Grants Council of Hong Kong
Jianfeng Zhang is supported in part by NSF grant #DMS 1008873.
J. Ma · J. Zhang (
Department of Mathematics, University of Southern California, Los Angels, CA 90089, USA
Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong
School of Economics and Business Administration, Chongqing University, Chongqing 400030,