A note on Almgren-Chriss optimal execution problem with geometric Brownian motion

TL;DR

This paper provides an explicit solution to the Almgren-Chriss optimal liquidation problem assuming stock prices follow a geometric Brownian motion, using functional analysis techniques, and extends to stochastic drift scenarios.

Contribution

It offers a novel explicit solution to the optimal liquidation problem under geometric Brownian motion and extends the framework to stochastic drift and portfolio liquidation.

Findings

Explicit solution for geometric Brownian motion case

Framework extension to stochastic drift

Applicable to portfolio liquidation scenarios

Abstract

We solve explicitly the Almgren-Chriss optimal liquidation problem where the stock price process follows a geometric Brownian motion. Our technique is to work in terms of cash and to use functional analysis tools. We show that this framework extends readily to the case of a stochastic drift for the price process and the liquidation of a portfolio.