Optimal Execution among $N$ Traders with Transient Price Impact

TL;DR

This paper analyzes N-player optimal execution games with transient price impact, deriving a unique equilibrium under specific regularization and revealing insights into trading costs and behaviors.

Contribution

It introduces a novel equilibrium existence condition involving a time-dependent block trade cost and provides a closed-form solution for this equilibrium.

Findings

Unique equilibrium exists with regularization and specific block costs

Equilibrium is the limit of regularized equilibria as regularization vanishes

Optimal instantaneous costs do not vanish as regularization tends to zero

Abstract

We study $N$-player optimal execution games in an Obizhaeva--Wang model of transient price impact. When the game is regularized by an instantaneous cost on the trading rate, a unique equilibrium exists and we derive its closed form. Whereas without regularization, there is no equilibrium. We prove that existence is restored if (and only if) a very particular, time-dependent cost on block trades is added to the model. In that case, the equilibrium is particularly tractable. We show that this equilibrium is the limit of the regularized equilibria as the instantaneous cost parameter $\varepsilon$ tends to zero. Moreover, we explain the seemingly ad-hoc block cost as the limit of the equilibrium instantaneous costs. Notably, in contrast to the single-player problem, the optimal instantaneous costs do not vanish in the limit $\varepsilon\to0$. We use this tractable equilibrium to study the cost of liquidating in the presence of predators and the cost of anarchy. Our results also give a new interpretation to the erratic behaviors previously observed in discrete-time trading games with transient price impact.