Optimal investment with transient price impact

TL;DR

This paper develops a model for optimal trading considering transient price impacts, providing explicit solutions and characterizing optimal trading regions, thus advancing understanding of trading strategies under realistic market frictions.

Contribution

It introduces a novel price impact model with convex liquidity costs and explicitly solves the associated optimal control problem using convex analysis techniques.

Findings

Explicit solution for the optimal trading strategy.

Characterization of action and non-action regions.

Optimal trading towards the frictionless Merton portfolio.

Abstract

We introduce a price impact model which accounts for finite market depth, tightness and resilience. Its coupled bid- and ask-price dynamics induce convex liquidity costs. We provide existence of an optimal solution to the classical problem of maximizing expected utility from terminal liquidation wealth at a finite planning horizon. In the specific case when market uncertainty is generated by an arithmetic Brownian motion with drift and the investor exhibits constant absolute risk aversion, we show that the resulting singular optimal stochastic control problem readily reduces to a deterministic optimal tracking problem of the optimal frictionless constant Merton portfolio in the presence of convex costs. Rather than studying the associated Hamilton-Jacobi-Bellmann PDE, we exploit convex analytic and calculus of variations techniques allowing us to construct the solution explicitly and to describe the free boundaries of the action- and non-action regions in the underlying state space. As expected, it is optimal to trade towards the frictionless Merton position, taking into account the initial bid-ask spread as well as the optimal liquidation of the accrued position when approaching terminal time. It turns out that this leads to a surprisingly rich phenomenology of possible trajectories for the optimal share holdings.