Broker–Trader Nash Equilibrium with Price Impact and Inventory Costs

Summary

This paper models a broker interacting with an informed client and a nonstrategic uninformed client. The broker executes trades on a lit exchange where price impact is immediate and then fades exponentially. The informed trader and broker maximize expected wealth while facing inventory penalties; the uninformed trader instead sends random buy and sell orders.

The authors characterize equilibrium trading strategies by solving a coupled system of forward-backward stochastic differential equations, which they solve explicitly. They then examine how information, profitability, and inventory control affect the broker's and informed trader's strategies. The account is a theoretical model of strategic trading and intermediation, rather than an empirical test. Its conclusions depend on the specified information structure, price-impact dynamics, random uninformed flow, and inventory penalties.

Key ideas

• The model has a broker, an informed strategic trader, and a nonstrategic uninformed trader.
• Exchange trades create immediate price impact that decays exponentially.
• The broker and informed trader maximize expected wealth subject to inventory penalties.
• Equilibrium strategies are characterized and solved through coupled forward-backward stochastic equations.
• The analysis explores how information, profitability, and inventory control shape strategy.

Tags

# Nash Equilibrium between Brokers and Traders


# Nash Equilibrium between Brokers and Traders









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.