A Mean Field Game between Informed Traders and a Broker

ArXiv ID: 2401.05257 “View on arXiv”

Authors: Unknown

Abstract

We find closed-form solutions to the stochastic game between a broker and a mean-field of informed traders. In the finite player game, the informed traders observe a common signal and a private signal. The broker, on the other hand, observes the trading speed of each of his clients and provides liquidity to the informed traders. Each player in the game optimises wealth adjusted by inventory penalties. In the mean field version of the game, using a Gâteaux derivative approach, we characterise the solution to the game with a system of forward-backward stochastic differential equations that we solve explicitly. We find that the optimal trading strategy of the broker is linear on his own inventory, on the average inventory among informed traders, and on the common signal or the average trading speed of the informed traders. The Nash equilibrium we find helps informed traders decide how to use private information, and helps brokers decide how much of the order flow they should externalise or internalise when facing a large number of clients.

Keywords: Mean Field Games, Stochastic Differential Equations, Optimal Trading Strategy, Liquidity Provision, Nash Equilibrium, Equity

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats
  • Why: The paper is mathematically dense, featuring a mean-field game framework, Gâteaux derivatives, and explicit solutions to forward-backward stochastic differential equations (FBSDEs), justifying a high math complexity score. However, it is entirely theoretical with no backtesting, datasets, or implementation details provided, placing it firmly in the Lab Rats quadrant (high math, low empirical rigor).
  flowchart TD
    A["Research Goal: <br>Model & Solve Stochastic Game<br>Broker vs. Mean-Field of Informed Traders"] --> B["Data/Inputs"]
    
    B --> C["Common & Private Signals<br>Trading Speeds<br>Wealth & Inventory"]
    
    C --> D["Methodology: <br>Finite Player Game<br>Optimization w/ Inventory Penalties"]
    
    D --> E["Computational Process: <br>Gâteaux Derivative +<br>Forward-Backward SDEs"]
    
    E --> F["Key Findings: <br>Explicit Nash Equilibrium<br>Linear Optimal Strategies<br>Liquidity Provision Logic"]