Optimal Trade Execution Under Endogenous Pressure to Liquidate: Theory and Numerical Solutions

TL;DR

This paper develops a theoretical model for optimal liquidation of large trading positions considering endogenous market pressure, resulting in strategies aligned with the square-root law and addressing numerical challenges in solution computation.

Contribution

It introduces an endogenous liquidation pressure model with a stopping time horizon and provides a stable numerical method for solving the associated singular differential equations.

Findings

Optimal strategies align with the square-root law of price impact.

Endogenous liquidation pressure influences the optimal stopping time.

Numerical methods address instability in solving the Hamilton-Jacobi-Bellman equation.

Abstract

We study optimal liquidation of a trading position (so-called block order or meta-order) in a market with a linear temporary price impact (Kyle, 1985). We endogenize the pressure to liquidate by introducing a downward drift in the unaffected asset price while simultaneously ruling out short sales. In this setting the liquidation time horizon becomes a stopping time determined endogenously, as part of the optimal strategy. We find that the optimal liquidation strategy is consistent with the square-root law which states that the average price impact per share is proportional to the square root of the size of the meta-order (Bershova and Rakhlin, 2013; Farmer et al., 2013; Donier et al., 2015; T\'oth et al., 2016). Mathematically, the Hamilton-Jacobi-Bellman equation of our optimization leads to a severely singular and numerically unstable ordinary differential equation initial value problem. We provide careful analysis of related singular mixed boundary value problems and devise a numerically stable computation strategy by re-introducing time dimension into an otherwise time-homogeneous task.