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

ArXiv ID: 2512.11765 “View on arXiv”

Authors: Marcel Nutz, Alessandro Prosperi

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.

Keywords: Optimal Execution, Obizhaeva–Wang Model, High-Frequency Trading, Nash Equilibrium, Price Impact

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, focusing on discrete-time Nash equilibria, high-frequency limits, and convergence rates (1/N), requiring advanced stochastic calculus, PDEs, and game theory. Empirical rigor is low as it presents no backtests, datasets, or implementation details, relying solely on mathematical derivations and asymptotic analysis.

  flowchart TD
    A["Research Goal: Analyze high-frequency limit<br>of n-trader optimal execution game<br>with transient impact & quadratic costs"] --> B["Model Setup"]
    B --> C["Methodology:<br>Solve discrete Nash equilibrium<br>in high-frequency limit"]
    C --> D["Key Mechanism:<br>Cumulative discrete costs over<br>small neighborhoods converge"]
    D --> E{"Continuum of Trading?"}
    E -- Yes: θ > 0 --> F["Key Findings:<br>1. Converges to CW with θ0=(n-1)/2, θT=1/2<br>2. Endogenous boundary costs<br>3. θ-independent limit"]
    E -- No: θ = 0 --> G["No Limit: Oscillations &<br>non-existence without block costs"]