A two-player portfolio tracking game

TL;DR

This paper models a strategic game between two agents competing in portfolio tracking, revealing how their strategies adapt based on price impacts and interaction dynamics, with explicit equilibrium solutions.

Contribution

It introduces a stochastic linear quadratic differential game with terminal constraints for portfolio tracking, providing explicit Nash equilibria and behavioral insights.

Findings

Equilibrium strategies depend on the relation between temporary and permanent price impacts.

Agents may adopt exploitative or liquidity-providing behaviors based on impact parameters.

The model extends predatory trading literature with explicit solutions and behavioral analysis.

Abstract

We study the competition of two strategic agents for liquidity in the benchmark portfolio tracking setup of Bank, Soner, Vo{\ss} (2017). Specifically, both agents track their own stochastic running trading targets while interacting through common aggregated temporary and permanent price impact \`a la Almgren and Chriss (2001). The resulting stochastic linear quadratic differential game with terminal state constraints allows for a unique and explicitly available open-loop Nash equilibrium. Our results reveal how the equilibrium strategies of the two players take into account the other agent's trading targets: either in an exploitative intent or by providing liquidity to the competitor, depending on the relation between temporary and permanent price impact. As a consequence, different behavioral patterns can emerge as optimal in equilibrium. These insights complement and extend existing studies in the literature on predatory trading models examined in the context of optimal portfolio liquidation games.