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

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.

💡 Research Summary

The paper investigates the high‑frequency (HF) limit of an optimal execution game involving n large traders who trade a single risky asset in discrete time. Each trade generates transient price impact modeled by the Obizhaeva‑Wang kernel G(t)=e^{-ρt} and incurs an additional quadratic instantaneous cost θ(ΔX)² where θ≥0. Traders aim to liquidate their initial inventories x_i by the terminal date T, choosing strategies ξ_i that minimize the expected execution cost, which consists of three parts: (i) the standard transient‑impact term, (ii) a cross‑term that resolves simultaneous orders, and (iii) the instantaneous quadratic penalty.

The authors first formulate the discrete‑time game on a grid T={t₀,…,t_N} and define admissible strategies as bounded, F_{t_k}‑measurable vectors satisfying the liquidation constraint. By introducing the matrices

Γ^θ_{ij}=G(|t_i−t_j|)+2θδ_{ij}, eΓ_{ij}=0 (i<j), ½G(0) (i=j), G(|t_i−t_j|) (i>j),

they construct two deterministic vectors

v=(Γ^θ+(n−1)eΓ)^{-1} 1 ·