High-Frequency Analysis of a Trading Game with Transient Price Impact

TL;DR

This paper analyzes the high-frequency limit of a multi-trader optimal execution game with transient price impact and quadratic costs, deriving the continuous-time equilibrium and revealing how discrete costs influence boundary conditions.

Contribution

It extends previous models by deriving the high-frequency limit for multiple traders, showing how boundary costs emerge from discrete costs, and clarifying the effects of trading frictions.

Findings

Discrete equilibrium inventories converge to continuous-time equilibrium at rate 1/N.

Boundary block costs are derived as limits of discrete instantaneous costs near boundaries.

Without boundary costs, no high-frequency limit exists due to persistent oscillations.

Abstract

We study the high-frequency limit of an $n$-trader optimal execution game in discrete time. Traders face transient price impact of Obizhaeva--Wang type in addition to quadratic instantaneous trading costs $\theta(\Delta X_t)^2$ on each transaction $\Delta X_t$. There is a unique Nash equilibrium in which traders choose liquidation strategies minimizing expected execution costs. In the high-frequency limit where the grid of trading dates converges to the continuous interval $[0,T]$, the discrete equilibrium inventories converge at rate $1/N$ to the continuous-time equilibrium of an Obizhaeva--Wang model with additional quadratic costs $\vartheta_0(\Delta X_0)^2$ and $\vartheta_T(\Delta X_T)^2$ on initial and terminal block trades, where $\vartheta_0=(n-1)/2$ and $\vartheta_T=1/2$. The latter model was introduced by Campbell and Nutz as the limit of continuous-time equilibria with vanishing instantaneous costs. Our results extend and refine previous results of Schied, Strehle, and Zhang for the particular case $n=2$ where $\vartheta_0=\vartheta_T=1/2$. In particular, we show how the coefficients $\vartheta_0=(n-1)/2$ and $\vartheta_T=1/2$ arise endogenously in the high-frequency limit: the initial and terminal block costs of the continuous-time model are identified as the limits of the cumulative discrete instantaneous costs incurred over small neighborhoods of $0$ and $T$, respectively, and these limits are independent of $\theta>0$. By contrast, when $\theta=0$ the discrete-time equilibrium strategies and costs exhibit persistent oscillations and admit no high-frequency limit, mirroring the non-existence of continuous-time equilibria without boundary block costs. Our results show that two different types of trading frictions -- a fine time discretization and small instantaneous costs in continuous time -- have similar regularizing effects and select a canonical model in the limit.