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

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, both strategies and 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 [arXiv:1305.4013, 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.