[Paper Review] Optimal Trade Execution in Illiquid Markets

Erhan Bayraktar, Michael Ludkovski|arXiv (Cornell University)|Feb 15, 2009

Economic theories and models8 references4 citations

TL;DR

This paper proposes a novel impulse control framework for optimal trade execution in illiquid markets with discrete order flow, modeling execution opportunities via a Poisson process and minimizing price impact through dynamic strategies. The key contribution is a path-dependent, stochastic control approach that outperforms deterministic strategies, especially under partial information, with information loss in execution costs reaching up to 5% when regime states are unobserved.

ABSTRACT

We study optimal trade execution strategies in financial markets with discrete order flow. The agent has a finite liquidation horizon and must minimize price impact given a random number of incoming trade counterparties. Assuming that the order flow $N$ is given by a Poisson process, we give a full analysis of the properties and computation of the optimal dynamic execution strategy. Extensions, whereby (a) $N$ is a fully-observed regime-switching Poisson process; and (b) $N$ is a Markov-modulated compound Poisson process driven by a hidden Markov chain, are also considered. We derive and compare the properties of the three cases and illustrate our results with computational examples.

Motivation & Objective

• To model optimal trade execution in dark pools and other illiquid markets where liquidity is sparse and information leakage is a concern.
• To address the challenge of minimizing price impact when trades can only occur at random, discrete order arrival times.
• To develop a dynamic execution strategy that adapts to stochastic order flow and time-to-maturity constraints.
• To analyze the impact of partial information on execution costs in regime-switching liquidity environments.
• To extend classical continuous-time execution models to discrete, point-process-based frameworks suitable for dark pool trading.

Proposed method

• Modeling order arrivals as a Poisson process $N$ with intensity $\lambda$, representing discrete execution opportunities.
• Defining the value function $v(k,T)$ as the infimum of expected price impact over all admissible strategies $\xi \in \mathcal{A}_k$.
• Using dynamic programming to derive recursive equations for the value function under different information structures.
• Introducing a regime-switching Poisson process where intensity $\lambda$ depends on a hidden Markov chain $M_t$.
• Applying bounds via Monte Carlo simulation: lower bound $\underline{v}$ via Lemma 2.1, upper bound $\overline{v}$ via Lemma 2.2.
• Comparing fully observed vs. partially observed settings by computing the cost difference $v(k,T,\vec{\pi}) - v(k,T;i)$.

Experimental results

Research questions

• RQ1 How does the optimal execution strategy in a discrete-order-flow market differ from classical continuous-time models?
• RQ2 What is the impact of partial information about liquidity regimes on execution cost?
• RQ3 How do regime-switching dynamics affect the value function and optimal liquidation strategy?
• RQ4 Can tight bounds be computed for the optimal execution cost under stochastic order flow?
• RQ5 How do constraints on trading volume affect the performance gap between full and partial information settings?

Key findings

• The optimal execution strategy is path-dependent and stochastic, in contrast to deterministic strategies in continuous-time models.
• In the unconstrained case, the cost difference between full and partial information is small, around 1-2%.
• When constraints are introduced, the cost difference increases to 4-5%, highlighting the value of observing hidden liquidity states.
• The upper bound $\overline{v}$ computed via Lemma 2.2 provides a close approximation to the true value function, with relative error of about 10-15%.
• In the regime-switching model, the highest cost difference (up to 5%) occurs in regime 2, the transition between high and low liquidity states.
• The constrained value function $\tilde{v}$ is consistently higher than the unconstrained $v$, and exceeds the upper bound $\overline{v}$ in some cases due to additional constraints.