Trading with the Crowd

Abstract

We formulate and solve a multi-player stochastic differential game between financial agents who seek to cost-efficiently liquidate their position in a risky asset in the presence of jointly aggregated transient price impact, along with taking into account a common general price predicting signal. The unique Nash-equilibrium strategies reveal how each agent's liquidation policy adjusts the predictive trading signal to the aggregated transient price impact induced by all other agents. This unfolds a quantitative relation between trading signals and the order flow in crowded markets. We also formulate and solve the corresponding mean field game in the limit of infinitely many agents. We prove that the equilibrium trading speed and the value function of an agent in the finite N -player game converges to the corresponding trading speed and value function in the mean field game at rate O(N −2 ). In addition, we prove that the mean field optimal strategy provides an approximate Nash-equilibrium for the finite-player game.

Statistical Learning with Sublinear Regret of Propagator Models

“…We start with characterising the optimal strategy by using a variational approach. Note that for any u ∈ A the map u → J(u) in (2.7) is strictly concave, which can be easily shown by repeating the same lines as in the proof of Lemma 9.1 of [36] . Therefore, it admits a unique maximizer characterized by the critical point at which the Gâteaux derivative…”

“…In addition, many traders and trading algorithms also strive for using short term price predictors in their dynamic order execution schedules, which are often related to order book dynamics as discussed in [31,32,34,40]. From the modelling point of view, incorporating signals into execution problems translates into taking a non-martingale price process P , in contrast to a martingale price in the classical setting (see [10,5,37,36,1]). This changes the problem significantly as the resulting optimal strategies are often random and in particular signal-adaptive, in contrast to deterministic strategies, which are typically obtained in the martingale price case [6].…”

“…Specifically we decompose the semimatingale price process P in (1.1) to a martingale and to a finite variation process A which has the interpretation of a trading signal (see e.g. [36]). We observe that the optimal trading strategy which is used in the algorithm (see (2.8)) involves the conditional process…”

A two-player portfolio tracking game

“…Other recent work on both finite-player as well as infinite-player mean field price impact games with Almgren-Chriss type price impact include, e.g., Cardaliaguet and Lehalle [8], Huang et al [23], Casgrain and Jaimungal [12,13], Fu et al [21], Fu and Horst [19], Evangelista and Thamsten [18], and Drapeau et al [15], where finitely and infinitely many agents pursue optimal liquidation of their initial positions and interact through common aggregated permanent and temporary price impact. Price impact games of liquidating agents in a market model with transient price impact are analyzed, e.g., in Luo and Schied [24], Schied and Zhang [30], Schied et al [31], Strehle [34]; and very recently in Fu et al [20] and Neuman and Voß [27]. However, these works are all portfolio liquidation games where the agents steer their initial portfolio positions towards zero (with strict liquidation constraints enforced in [18][19][20][21]).…”