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

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.

• The paper establishes that minimal trading frictions, either through time discretization or instantaneous costs, yield continuous-time equilibria with endogenous boundary block costs.
• It quantifies the convergence rate of discrete Nash equilibria, demonstrating that strategies converge at a 1/N rate and clarifying the universal regularization effect.
• The study shows that omitting instantaneous costs leads to persistent oscillations, emphasizing the critical role of regularization in ensuring well-posed trading models.

High-Frequency Nash Equilibria in Trading Games with Transient Price Impact

Introduction and Motivation

This paper studies the high-frequency limit of Nash equilibria in discrete-time $n$-player optimal execution games with transient price impact of Obizhaeva–Wang type and instantaneous quadratic trading costs. The canonical setting involves $n \ge 2$ agents aiming to liquidate inventories over a finite horizon $[0,T]$, where each trade produces exponentially decaying impact on the asset price, combined with quadratic instantaneous cost penalties. Distinct from earlier two-player studies, this work generalizes results to arbitrary $n$, provides explicit convergence rates, and tracks the emergence of endogenous boundary block costs in the continuous-time limit. The principal contribution is the rigorous demonstration that both time discretization and small instantaneous trading frictions have a universal regularizing effect, giving rise to the specific boundary frictions required for well-posedness in continuous time.

Discrete-Time Nash Equilibrium: Model and Structure

The discrete-time model considers trading on an equidistant grid $\mathbb{T}_N = \{0, T/N, 2T/N, \dots, T\}$, with the price impact kernel $G(t) = e^{-\rho t}$, following the Obizhaeva–Wang framework. Each agent $i$ selects an adapted strategy $\bm{\xi}_i$ satisfying deterministic liquidation, facing both the transient impact and quadratic instantaneous cost $\theta (\Delta X_t)^2$. The execution cost functional incorporates impact, instantaneous cost, and a symmetric tie-breaking rule for simultaneous trades.

A Nash equilibrium consists of a deterministic profile $(\bm{\xi}_1^*, ..., \bm{\xi}_n^*)$ minimizing each player's expected cost against the opponents, and its explicit closed form can be constructed using two fundamental inventory processes $\bm{v}$ and $\bm{w}$. The equilibrium inventories for agent $i$ at time $t$ decompose as

$$X_t^{(N),i} = \bar{x} V_t^{(N)} + (x_i - \bar{x}) W_t^{(N)}, \quad \bar{x} = \frac{1}{n}\sum_{j=1}^n x_j~,$$

where $V_t^{(N)}$ and $W_t^{(N)}$ correspond to symmetric and zero-net-supply cases, respectively.

Continuous-Time Limit and Boundary Block Costs

In continuous time, the optimal execution game with solely transient impact generically admits no Nash equilibrium: Schied et al. (for $n=2$) and Campbell–Nutz (for general $n$) established that existence uniquely requires the addition of quadratic boundary block costs at $t=0$ and $t=T$, with coefficients $\vartheta_0 = (n-1)/2$ and $\vartheta_T = 1/2$. The continuous-time Nash strategy for agent $i$ is

$X_t^{*,i} = \mathbbm{g}(t) \bar{x} + \mathbbm{f}(t)(x_i - \bar{x})$

with explicit expressions for $\mathbbm{f}(t)$, $\mathbbm{g}(t)$, and a total cost that decomposes into impact and boundary block cost contributions. The values of $\vartheta_0$, $\vartheta_T$ are canonical: any deviation precludes equilibrium for generic inventories (except for the fully symmetric or net-zero initial configurations).

High-Frequency Limit: Rates and Identification of Block Frictions

The main results provide a detailed description of the limiting behavior of the discrete Nash equilibrium as $N\to\infty$, specifically:

• Convergence of Strategies: For any fixed $\theta > 0$, $X_t^{(N),i} \to X_t^{*,i}$ pointwise on $(0,T)$ with rate $1/N$. The limiting inventory processes universally coincide with the continuous-time limit, independently of the magnitude of $\theta$ as long as $\theta>0$.

  Figure 1: Convergence of $V_t^{(N)}$ as $N$ increases, demonstrating the approach to $\mathbbm{g}(t)$ for $n=10$, $\theta=0.1$, $\rho=1$.

  

  Figure 2: Convergence of $W_t^{(N)}$ as $N$ increases, approaching $\mathbbm{f}(t)$, for the same parameters as Figure 1.

• Emergence of Block Frictions: The initial and terminal block costs of the continuous-time model ($\vartheta_0 = \frac{n-1}{2}$, $\vartheta_T = \frac12$) are shown to arise as weak limits of the cumulative instantaneous costs accrued in shrinking neighborhoods of the endpoints. The limiting procedure is robust: the result does not depend on the particular choice of $\theta>0$. In the high-frequency limit, the only nonvanishing contribution to instantaneous cost comes from oscillations near the endpoints, whose mass converges to the prescribed block cost.
• Impact of Removing Instantaneous Costs: When $\theta=0$, the discrete-time Nash equilibrium oscillates persistently as $N \to \infty$. There is no continuous-time equilibrium, mirroring the non-existence results for the unregularized model. Subsequence analysis shows inventories cycling between explicit cluster points, with the partition into even/odd grid sizes controlling the limit.

  Figure 3: Oscillatory patterns in discrete inventories under block costs charged only on $[T/2,T]$; cluster points from the oscillation theorems are indicated.

• Boundary Sensitivity: Charging instantaneous cost only on part of the interval (e.g., $[0, T/2]$ or $[T/2, T]$) allows for convergence to equilibrium only for inventories in symmetric/zero-sum configurations, respectively, matching the existence results for the continuous-time model with only one correct boundary block cost.

  Figure 4: Analogous oscillatory behavior when instantaneous cost is charged only on $[0, T/2]$, isolating convergence in the symmetric case.

Numerical and Analytical Consequences

The analysis gives strong, explicit convergence rates for the microscopic-to-macroscopic limit in $n$-player Nash equilibria, clarifies the exact mechanism by which boundary frictions become endogenous, and displays the universal nature of regularization via either fine time-discretization or explicit instantaneous cost. The precise oscillatory form for $\theta=0$ further sharpens the connection between discrete grid friction and mathematical ill-posedness of pure Obizhaeva–Wang games.

Implications and Theoretical Significance

The findings have several important implications:

• Canonical Regularization: Any positive trading friction (instantaneous cost or positive discretization step) universally selects the same continuous-time model with endogenous block costs, regardless of how small the regularization is.
• Universality: Various trading frictions (small instantaneous penalty, time discretization, or time-inhomogeneous local friction) play interchangeable roles in the existence theory for strategic execution games with transient impact.
• Block Costs Are Endogenous: The required block cost penalties in continuous models should be viewed as emerging features of the market microstructure (tick size, latency, or other "hidden" trading costs).
• Robust Well-posedness: The methods provide a framework for addressing well-posedness and equilibrium computation even in more complex models with signals, non-exponential decay, or other generalizations.

Conclusion

This paper offers a definitive high-frequency analysis of multi-agent optimal execution games with transient price impact. It rigorously establishes the universal appearance of critical block frictions in continuous-time Nash equilibria, derived from both time discretization and vanishing instantaneous costs, and quantifies the resulting Nash strategies and cost convergence. The results provide a comprehensive bridge between microstructure modeling and continuous-time equilibrium theory, with direct relevance for both theoretical mathematical finance and algorithmic market design.

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Reference: "High-Frequency Analysis of a Trading Game with Transient Price Impact" (2512.11765)