Optimal Execution among $N$ Traders with Transient Price Impact

ArXiv ID: 2501.09638 “View on arXiv”

Authors: Unknown

Abstract

We study $N$-player optimal execution games in an Obizhaeva–Wang model of transient price impact. When the game is regularized by an instantaneous cost on the trading rate, a unique equilibrium exists and we derive its closed form. Whereas without regularization, there is no equilibrium. We prove that existence is restored if (and only if) a very particular, time-dependent cost on block trades is added to the model. In that case, the equilibrium is particularly tractable. We show that this equilibrium is the limit of the regularized equilibria as the instantaneous cost parameter $\varepsilon$ tends to zero. Moreover, we explain the seemingly ad-hoc block cost as the limit of the equilibrium instantaneous costs. Notably, in contrast to the single-player problem, the optimal instantaneous costs do not vanish in the limit $\varepsilon\to0$. We use this tractable equilibrium to study the cost of liquidating in the presence of predators and the cost of anarchy. Our results also give a new interpretation to the erratic behaviors previously observed in discrete-time trading games with transient price impact.

Keywords: Optimal Execution Games, Transient Price Impact, Obizhaeva–Wang Model, Equilibrium Analysis, Cost of Anarchy, Equity Execution

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, involving proofs of existence and uniqueness of equilibria in a continuous-time game with transient price impact, using advanced stochastic control and functional analysis (e.g., Fredholm equations). It lacks any empirical validation, backtests, or real-world data, focusing purely on mathematical modeling and analytical derivations.

  flowchart TD
    A["Research Goal: Find equilibrium in N-player<br>optimal execution game with<br>transient price impact O-W model"] --> B{"Input: Problem Formulation"}
    B --> B1["N-player game<br>Transient impact structure"]
    B --> B2["Regularization: Instantaneous cost ε"]

    subgraph "Methodology: Analysis & Computation"
        B --> C{"Analytical Investigation"}
        C --> C1["With ε > 0<br>Unique equilibrium derived"]
        C --> C2["With ε = 0<br>No equilibrium exists"]
        
        C1 --> D["Limits & Interpretation"]
        C2 --> D
        
        D --> D1["Limits of equilibria as ε→0"]
        D --> D2["Identify necessary cost<br>structure for ε=0"]
    end

    D1 --> E{"Computational Process: Case ε=0"}
    E --> F["Solve for Tractable Equilibrium<br>with Block Trade Cost"]
    F --> G["Compute: Liquidation Cost<br>Cost of Anarchy<br>Predator impact"]

    G --> H["Key Findings & Outcomes"]
    
    H --> H1["Existence restored<br>iff specific time-dependent<br>block cost added"]
    H --> H2["Equilibrium limit ε→0<br>matches block cost solution"]
    H --> H3["Optimal costs do NOT<br>vanish in limit (N-player effect)"]
    H --> H4["Explains erratic behaviors<br>in discrete-time games"]