Optimal Execution with Multiplicative Price Impact and Incomplete Information on the Return

TL;DR

This paper develops a mathematical model for optimal asset liquidation considering multiplicative price impact and unobservable market trends, providing a solution framework involving stochastic control and optimal stopping, with numerical methods to analyze parameter sensitivity.

Contribution

It introduces a novel singular stochastic control model with partial information and derives a two-dimensional optimal stopping problem for optimal execution.

Findings

The optimal selling boundary is characterized by a unique nonlinear integral equation.

Numerical solutions reveal how model parameters influence the optimal strategy.

The value of information significantly affects the optimal liquidation policy.

Abstract

We study an optimal liquidation problem with multiplicative price impact in which the trend of the asset's price is an unobservable Bernoulli random variable. The investor aims at selling over an infinite time-horizon a fixed amount of assets in order to maximize a net expected profit functional, and lump-sum as well as singularly continuous actions are allowed. Our mathematical modelling leads to a singular stochastic control problem featuring a finite-fuel constraint and partial observation. We provide the complete analysis of an equivalent three-dimensional degenerate problem under full information, whose state process is composed of the asset's price dynamics, the amount of available assets in the portfolio, and the investor's belief about the true value of the asset's trend. The optimal execution rule and the problem's value function are expressed in terms of the solution to a truly two-dimensional optimal stopping problem, whose associated belief-dependent free boundary $b$ triggers the investor's optimal selling rule. The curve $b$ is uniquely determined through a nonlinear integral equation, for which we derive a numerical solution through an application of the Monte-Carlo method. This allows us to understand the sensitivity of the problem's solution with respect to the relevant model's parameters as well as the value of information in our model.