Portfolio liquidation games with self‐exciting order flow

Abstract

We analyze novel portfolio liquidation games with self-exciting order flow. Both the 𝑁-player game and the mean-field game (MFG) are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each player solves a mean-field control problem. We characterize open-loop Nash equilibria in both games in terms of a novel mean-field FBSDE system with unknown terminal condition. Under a weak interaction condition, we prove that the FBSDE systems have unique solutions. Using a novel sufficient maximum principle that does not require convexity of the cost function we finally prove that the solution of the FBSDE systems do indeed provide open-loop Nash equilibria.

A Mean-Field Control Problem of Optimal Portfolio Liquidation with Semimartingale Strategies

“…Instead, the large investor rationally anticipates his/her impact on future order flow when making trading decisions and uses it to his/her advantage. Similar effects have previously been observed in the literature; see [23] and references therein for a more detailed discussion of different manipulation strategies in portfolio liquidation models.…”

“…Single and multi-player liquidation models with self-exciting order flow and absolutely continuous controls where dZ t = ξ t dt have recently been considered in Chen et al [15] and Fu et al [23], respectively. In both models, the market order dynamics follows a Hawkes process with exponential kernel, and the large investor's trading triggers an additional flow of child orders whose rate increases linearly in the investor's traded volume.…”

Self-exciting price impact via negative resilience in stochastic order books

“…On the other hand, there recently appeared many papers on trade execution that model selfexcitement of price impact in different ways, while, as explained above, negative resilience is an alternative way of modeling this effect. As in Cayé and Muhle-Karbe (2016) and in Fu et al (2022a), we motivate self-exciting price impact by the following reasons. Imagine, for instance, a large trader performing extensive selling.…”

“…For existing approaches to self-exciting price impact, see Alfonsi and Blanc (2016), Cartea et al (2018), Cayé and Muhle-Karbe (2016), Fu et al (2022a) and references therein. We now explain that, mathematically, all these approaches and ours are pairwise substantially different.…”

“…Cartea et al (2018) again use Hawkes processes but in a quite different way: they consider an execution model where the large trader places limit orders whose fill rates depend on mutually exciting "influential" market order flows. Fu et al (2022a) consider liquidation games between several large traders (and the corresponding mean-field limit as well as the single player subcase) with a self-exciting order flow. In a sense, self-excitement in Fu et al (2022a) is "more endogenous" than in the other mentioned approaches (including ours, where the resilience process is exogenously given), as in Fu et al (2022a) there appear "child orders" triggered by the large traders' trading activity, and as the strategies in Fu et al (2022a) come out as Nash equilibria in the game.…”