Broker-Trader Partial Information Nash-Equilibria

ArXiv ID: 2412.17712 “View on arXiv”

Authors: Unknown

Abstract

We study partial information Nash equilibrium between a broker and an informed trader. In this setting, the informed trader, who possesses knowledge of a trading signal, trades multiple assets with the broker in a dealer market. Simultaneously, the broker offloads these assets in a lit exchange where their actions impact the asset prices. The broker, however, only observes aggregate prices and cannot distinguish between underlying trends and volatility. Both the broker and the informed trader aim to maximize their penalized expected wealth. Using convex analysis, we characterize the Nash equilibrium and demonstrate its existence and uniqueness. Furthermore, we establish that this equilibrium corresponds to the solution of a nonstandard system of forward-backward stochastic differential equations (FBSDEs) that involves the two differing filtrations. For short enough time horizons, we prove that a unique solution of this system exists. Finally, under quite general assumptions, we show that the solution to the FBSDE system admits a polynomial approximation in the strength of the transient impact to arbitrary order, and prove that the error is controlled.

Keywords: Nash Equilibrium, Market Microstructure, Forward-Backward SDEs, Adverse Selection, Convex Analysis, Equities

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper is highly theoretical, involving advanced stochastic calculus, convex analysis, and characterization of Nash equilibria via forward-backward stochastic differential equations (FBSDEs) with differing filtrations; it lacks any empirical validation, backtesting, or implementation details, focusing solely on mathematical existence and approximation results.

  flowchart TD
    A["Research Goal<br>Nash Equilibrium in<br>Broker-Trader Partial Info Setting"] --> B
    subgraph B ["Methodology & Inputs"]
        B1["Formulate Two-Person Game<br>Broker vs Informed Trader"]
        B2["Define State Processes<br>Asset Prices & Trading Signal"]
        B3["Convex Analysis &<br>FBSDE Framework"]
    end
    B --> C["Computational Process<br>Characterize Nash Equilibrium via<br>Nonstandard Forward-Backward SDEs"]
    C --> D{"Prove Existence &<br>Uniqueness?"}
    D -- Yes --> E["Key Findings & Outcomes"]
    D -- Limited Time Horizon --> E
    subgraph E ["Outcomes"]
        E1["Equilibrium Existence<br>& Uniqueness Proof"]
        E2["Polynomial Approximation<br>of Solution wrt Impact Strength"]
        E3["Error Control &<br>Explicit Bounds"]
    end