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A Mean Field Game between Informed Traders and a Broker

Philippe Bergault, Leandro Sánchez-Betancourt

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.

A Mean Field Game between Informed Traders and a Broker

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.
Paper Structure (18 sections, 6 theorems, 121 equations, 4 figures)

This paper contains 18 sections, 6 theorems, 121 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The game with $N$ informed traders
  3. Framework
  4. The problem of the informed traders
  5. The problem of the broker
  6. Limitations of the finite game model and mean field limit
  7. Facing many informed traders: the mean field limit
  8. Probabilistic framework
  9. Admissible controls
  10. Optimisation problems and definition of equilibria
  11. The informed trader's optimality condition
  12. The broker's optimality condition
  13. The mean field FBSDE system
  14. The closed-form solution to the FBSDE
  15. The optimal strategy of the broker
  16. ...and 3 more sections

Key Result

Lemma 3.2

Let $\nu^B\in\mathcal{A}$. The functional $H^{I,\nu^B}(\cdot):\mathcal{A}\to \mathbb{R}$ defined in representative_obj is strictly concave up to a $\mathbb P \otimes \mathrm{d} t-$null set.

Figures (4)

  • Figure 1: Sample paths for $S_t$$\alpha_t$, $\nu^I_t$, $\nu^B_t$, $Q^I_t$, and $Q^B_t$.
  • Figure 2: Functions $g^{a,b,c}, h^{a,b,c}:[0,T]\to\mathbb{R}$ as time approaches $T$.
  • Figure 3: Functions $f^{a,b,c}, f^{a,I}, f^{b,I}:[0,T]\to\mathbb{R}$ as time approaches $T$.
  • Figure 4: Sample paths for the common signal $\alpha_t$ and the private signal $\alpha^n_t$, together with $S_t$, $\bar{\nu}^*_t$, $\nu^{n*}_t$, $\nu^{B,*}_t$, $\bar{Q}^*_t$, $Q^{n*}_t$, and $Q^{B,*}_t$.

Theorems & Definitions (14)

  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • ...and 4 more