Nash Equilibrium between Brokers and Traders

TL;DR

This paper models the strategic interaction between a broker and two types of traders, deriving explicit Nash equilibria in a complex stochastic environment with implications for market behavior and information asymmetry.

Contribution

It provides an explicit solution to the Nash equilibrium in a model with informed and uninformed traders, incorporating transient price impact and inventory penalties.

Findings

Explicit Nash equilibrium strategies derived

Information asymmetry impacts trading behavior

Inventory penalties influence trader strategies

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.