Portfolio liquidation in dark pools in continuous time

TL;DR

This paper models the optimal liquidation of portfolios in illiquid markets with both primary exchanges and dark pools, deriving a mathematical framework to determine optimal trading strategies considering costs, risks, and uncertain execution.

Contribution

It introduces a continuous-time model combining linear price impact and Poisson-based dark pool execution, providing explicit solutions via Riccati equations and verification theorems.

Findings

Optimal liquidation involves slow primary trading and strategic dark pool orders.

Multi-asset portfolios may require oversizing dark pool orders to reduce risk.

The model yields explicit value functions and optimal strategies under market impact and execution uncertainty.

Abstract

We consider an illiquid financial market where a risk averse investor has to liquidate a portfolio within a finite time horizon [0,T] and can trade continuously at a traditional exchange (the "primary venue") and in a dark pool. At the primary venue, trading yields a linear price impact. In the dark pool, no price impact costs arise but order execution is uncertain, modeled by a multi-dimensional Poisson process. We characterize the costs of trading by a linear-quadratic functional which incorporates both the price impact costs of trading at the primary exchange and the market risk of the position. The liquidation constraint implies a singularity of the value function of the resulting minimization problem at the terminal time T. Via the HJB equation and a quadratic ansatz, we obtain a candidate for the value function which is the limit of a sequence of solutions of initial value problems for a matrix differential equation. We show that this limit exists by using an appropriate matrix inequality and a comparison result for Riccati matrix equations. Additionally, we obtain upper and lower bounds of the solutions of the initial value problems, which allow us to prove a verification theorem. If a single asset position is to be liquidated, the investor slowly trades out of her position at the primary venue, with the remainder being placed in the dark pool at any point in time. For multi-asset liquidations this is generally not the case; it can, e.g., be optimal to oversize orders in the dark pool in order to turn a poorly balanced portfolio into a portfolio bearing less risk.