Some Computations for Optimal Execution with Monotone Strategies

TL;DR

This paper investigates an optimal execution problem in a Black-Scholes market with linear price impact, focusing on non-negative trading strategies and providing a complete characterization of the value and optimal control through a nonlinear ODE.

Contribution

It introduces a probabilistic approach to characterize optimal strategies under non-negativity constraints via a nonlinear ODE, including explicit solutions in special cases.

Findings

Complete characterization of the value function and optimal control.

Explicit solutions to the nonlinear ODE in specific cases.

Demonstration of the probabilistic approach for constrained optimal execution.

Abstract

We study an optimal execution problem in the infinite horizon setup. Our financial market is given by the Black-Scholes model with a linear price impact. The main novelty of the current note is that we study the constrained case where the number of shares and the selling rate are non-negative processes. For this case we give a complete characterization of the value and the optimal control via a solution of a non-linear ordinary differential equation (ODE). Furthermore, we provide an example where the non-linear ODE can be solved explicitly. Our approach is purely probabilistic.