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

2025-12-12

AI Summary

This paper studies the discrete-to-continuous limit of optimal execution games among (n) strategic traders in a high-frequency trading environment. Under instantaneous price impact and quadratic transaction costs, we rigorously solve for the Nash equilibrium and prove its convergence to the continuous Obizhaeva–Wang model with boundary block costs at rate (1/N). For the first time, the initial and terminal block cost coefficients—(vartheta_0 = (n-1)/2) and (vartheta_T = 1/2)—are endogenously derived as limits of cumulative discrete instantaneous costs near the endpoints. We establish the equivalence between high-frequency discretization and small instantaneous cost as regularizing mechanisms. Our results unify and generalize prior findings for the two-player case and reveal that, when ( heta = 0) (i.e., vanishing block costs), the equilibrium oscillates and diverges—demonstrating the essential role of boundary block costs in ensuring equilibrium existence.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Blockchains and distributed ledgers

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 $θ(ΔX_t)^2$ on each transaction $ΔX_t$. 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 $vartheta_0(ΔX_0)^2$ and $vartheta_T(ΔX_T)^2$ 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 $θ>0$. By contrast, when $θ=0$ 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.

Problem

• Analyzes high-frequency limit of multi-trader optimal execution game.
• Studies transient price impact with quadratic instantaneous trading costs.
• Identifies unique Nash equilibrium and convergence to continuous-time model.

Innovation

• Discrete-time Nash equilibrium with transient price impact and quadratic costs
• High-frequency limit convergence to continuous-time Obizhaeva-Wang model
• Endogenous emergence of boundary block costs from discrete instantaneous costs