Optimal Portfolio Selection under Concave Price Impact

TL;DR

This paper investigates an optimal portfolio problem considering concave price impact, revealing that optimal strategies are piecewise constant and formulating the problem as a fixed-cost-free impulse control problem.

Contribution

It models concave price impact as a key factor, reducing the problem to an impulse control framework and proving the existence and structure of optimal strategies.

Findings

Optimal strategies are piecewise constant.

The problem is formulated as an impulse control problem without fixed costs.

The value function satisfies a Quasi-Variational Inequality.

Abstract

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.