, Volume 67, Issue 3, pp 353-390
Date: 15 Jan 2013

Optimal Portfolio Selection Under Concave Price Impact

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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.

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 No. 9041545.
Jianfeng Zhang is supported in part by NSF grant #DMS 1008873.