On regularized optimal execution problems and their singular limits

TL;DR

This paper studies a stochastic control model for optimal portfolio execution considering uncertain volatility and liquidity, providing existence, uniqueness, and a singular limit analysis of optimal strategies.

Contribution

It introduces a new framework modeling liquidity costs with a power law and characterizes optimal strategies via viscosity solutions and singular limits.

Findings

Existence and uniqueness of viscosity solutions for the HJB equation.

Numerical algorithm for computing optimal trading strategies.

Optimal strategies as singular limits of regularized solutions.

Abstract

We investigate the portfolio execution problem under a framework in which volatility and liquidity are both uncertain. In our model, we assume that a multidimensional Markovian stochastic factor drives both of them. Moreover, we model indirect liquidity costs as temporary price impact, stipulating a power law to relate it to the agent's turnover rate. We first analyze the regularized setting, in which the admissible strategies do not ensure complete execution of the initial inventory. We prove the existence and uniqueness of a continuous and bounded viscosity solution of the Hamilton-Jacobi-Bellman equation, whence we obtain a characterization of the optimal trading rate. As a byproduct of our proof, we obtain a numerical algorithm. Then, we analyze the constrained problem, in which admissible strategies must guarantee complete execution to the trader. We solve it through a monotonicity argument, obtaining the optimal strategy as a singular limit of the regularized counterparts.