Nash Equilibrium between Brokers and Traders

ArXiv ID: 2407.10561 “View on arXiv”

Authors: Unknown

Abstract

We study the perfect information Nash equilibrium between a broker and her clients – an informed trader and an uniformed trader. In our model, the broker trades in the lit exchange where trades have instantaneous and transient price impact with exponential resilience, while both clients trade with the broker. The informed trader and the broker maximise expected wealth subject to inventory penalties, while the uninformed trader is not strategic and sends the broker random buy and sell orders. We characterise the Nash equilibrium of the trading strategies with the solution to a coupled system of forward-backward stochastic differential equations (FBSDEs). We solve this system explicitly and study the effect of information, profitability, and inventory control in the trading strategies of the broker and the informed trader.

Keywords: Market Microstructure, Nash Equilibrium, Forward-Backward SDEs, Broker-Client Model, Price Impact, Equities

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: The paper heavily employs advanced mathematical concepts including forward-backward stochastic differential equations (FBSDEs) and matrix Riccati differential equations to characterize the Nash equilibrium, resulting in a high math complexity score. In contrast, it lacks code, statistical metrics, or backtesting details, relying instead on theoretical derivations and generic simulations, which gives it a low empirical rigor score.
  flowchart TD
    A["Research Goal:<br>Nash Equilibrium between Broker & Clients"] --> B["Key Methodology:<br>Forward-Backward SDEs (FBSDE)"]
    B --> C["Model Setup:<br>Broker, Informed &<br>Uninformed Traders"]
    C --> D["Market Assumption:<br>Transient Price Impact with<br>Exponential Resilience"]
    C --> E["Client Behavior:<br>Informed (Optimal) vs.<br>Uninformed (Random)"]
    D & E --> F["Computational Process:<br>Derive Coupled FBSDE System"]
    F --> G["Solution Strategy:<br>Explicit Closed-Form<br>Nash Equilibrium"]
    G --> H["Key Findings:<br>Equilibrium Strategies,<br>Impact of Info, Profit, Inventory"]