An Optimal Execution Problem with Market Impact

TL;DR

This paper models an optimal execution problem considering market impact, analyzing the value function's properties, and characterizing optimal strategies that vary with holdings and impact strength in continuous-time markets.

Contribution

It formulates a stochastic control framework for optimal execution with market impact and characterizes the value function as a viscosity solution, revealing how strategies depend on market impact and holdings.

Findings

Value function's right-continuity linked to market impact strength.

Semi-group property (Bellman principle) holds for the problem.

Optimal strategies shift from block to gradual liquidation depending on holdings and impact.

Abstract

We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales, otherwise the value function is continuous. Moreover, we show the semi-group property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of the optimal strategies change completely, depending on the amount of the trader's security holdings and where optimal strategies in the Black-Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral.