A mean field game between informed traders and a broker

Journal article

We find the solution 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 in his own inventory, in the average inventory among informed traders, and in 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.

Authors: Bergault, P; Sánchez-Betancourt, L

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Financial Mathematics
• Volume: 16
• Issue: 2
• Pages: 358-388
• Publication date: 2025-04-23
• Acceptance date: 2025-01-28
• DOI: 10.1137/24M1630414
• EISSN: 1945-497X
• Language: English
• Keywords: market making; algorithmic trading; externalization; internalization; mean field games