A two-player portfolio tracking game

Abstract

We study the competition of two strategic agents for liquidity in the benchmark portfolio tracking setup of Bank et al. (Math Financial Economics 11(2):215–239 2017). Specifically, both agents track their own stochastic running trading targets while interacting through common aggregated temporary and permanent price impact à la Almgren and Chriss (J Risk 3:5–39 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.

Closed‐loop Nash competition for liquidity

“…More specifically, it tracks an “aim portfolio” (a constant multiple $$M_{aim}$$ of the frictionless optimal holdings $$\mu _t/\gamma$$) with a constant (relative) trading rate $$M_{rate}$$. This parallels results for single‐agent models (Gârleanu & Pedersen, 2016), a central planner (Section 4.1), or open‐loop equilibria (Voß, 2019; Casgrain and Jaimungal, 2020 or Section 4.2). However, the coefficients as well as the corresponding optimal value all depend on the form of the agents' strategic interaction.…”

“…To wit, each agent considers the others' feedback controls to be fixed, but takes into account how their own deviations from the equilibrium impact others' trading through their effect on the model's state variables 6. In the present context, single‐agent optimal trading rates (Gârleanu & Pedersen, 2016) and open‐loop Nash equilibria (Voß, 2019) are functions of the (exogenous) trading signal $$\mu _t$$ and the (endogenous) risky positions $$\varphi _t=(\varphi ^1_t,\ldots ,\varphi ^N_t)$$ of the agents. We, therefore, naturally search for a closed‐loop equilibrium in the same class of strategies.…”

“…In dynamic models, different notions of Nash equilibria arise depending on whether agents anticipate that others may react if they deviate from the equilibrium. In the present context, a number of recent papers (Casgrain & Jaimungal, 2020; Neuman & Voß, 2021; Voß, 2019) study open‐loop Nash equilibria, where each agent treats the others' trading rates as fixed stochastic processes. This means that agents do not consider at all how the others may react if they were to deviate from the equilibrium: Definition A collection of admissible trading rates $$\dot{\varphi }=(\dot{\varphi }^1,\ldots ,\dot{\varphi }^N)$$ is an open‐loop Nash equilibrium if no agent $$n=1,\ldots , N$$ has an incentive to deviate from the equilibrium, in that $$$$\begin{equation*} J^n(\dot{\varphi }^n,\dot{\varphi }^{-n}) \ge J^{n}(\dot{\psi }^n,\dot{\varphi }^{-n}), \end{equation*}$$$$for all alternative admissible trading rates $$\dot{\psi }^n \in \mathcal {A}_\rho$$ of agent n .…”

“…(2007), Schied and Schöneborn (2009) as well as many more recent studies 2. More recently, a number of papers also analyze the competition for liquidity between agents that trade to exploit a common trading signal (Drapeau et al., 2021; Evangelista & Thamsten, 2020; Neuman and Voß, 2021; Voß, 2019). To obtain tractable results, these papers focus on open‐loop Nash equilibria.…”

Equilibrium in Functional Stochastic Games with Mean-Field Interaction

“…One of the main challenges in the area of finite population stochastic games is to derive explicitly the Nash equilibrium of the system. Motivating examples from mathematical finance include price impact games with competition for liquidity between agents [36,12,14,35,32], systemic risk games introduced by Carmona et al [9,10], Fouque and Zhang [17], as well as optimal investment problems studied in Lacker and Zariphopoulou [28], Lacker and Soret [27]. In various important extensions of the aforementioned models it turns out that the state variables of the players and their objective functionals naturally depend on the entire trajectory of the controls.…”