High-frequency limit of Nash equilibria in a market impact game with transient price impact

TL;DR

This paper analyzes the high-frequency behavior of Nash equilibria in a market impact model with transient price impact, revealing convergence properties and the effects of transaction costs on agents' costs.

Contribution

It establishes the convergence of strategies and costs in high-frequency limits and characterizes the existence of continuous-time Nash equilibria at a critical transaction cost value.

Findings

Strategies oscillate for zero transaction costs.

Strategies converge to a limit for positive transaction costs.

Higher transaction costs can reduce agents' expected costs at high trading speeds.

Abstract

We study the high-frequency limits of strategies and costs in a Nash equilibrium for two agents that are competing to minimize liquidation costs in a discrete-time market impact model with exponentially decaying price impact and quadratic transaction costs of size $\theta\ge0$. We show that, for $\theta=0$, equilibrium strategies and costs will oscillate indefinitely between two accumulation points. For $\theta>0$, however, strategies, costs, and total transaction costs will converge towards limits that are independent of $\theta$. We then show that the limiting strategies form a Nash equilibrium for a continuous-time version of the model with $\theta$ equal to a certain critical value $\theta^*>0$, and that the corresponding expected costs coincide with the high-frequency limits of the discrete-time equilibrium costs. For $\theta\neq\theta^*$, however, continuous-time Nash equilibria will typically not exist. Our results permit us to give mathematically rigorous proofs of numerical observations made in Schied and Zhang (2013). In particular, we provide a range of model parameters for which the limiting expected costs of both agents are decreasing functions of $\theta$. That is, for sufficiently high trading speed, raising additional transaction costs can reduce the expected costs of all agents.