Optimal Execution and Macroscopic Market Making

Abstract

We propose a stochastic game modelling the strategic interaction between market makers and traders. From the trader's perspective, the conventional exogenous permanent price impact is replaced by the endogenous quoting strategies of the market makers. Conversely, from the market maker's perspective, order flows are no longer assumed to be exogenous, but are driven endogenously by the strategic traders. Characterizing the Nash equilibria via forward-backward stochastic differential equations (FBSDEs), we establish a local well-posedness result for the general game. For the specific `Almgren-Chriss-Avellaneda-Stoikov' model, the decoupling approach guarantees the global well-posedness of the FBSDEs by reducing it to a backward stochastic Riccati equation with $M_+$-matrix coefficients. Finally, by introducing small diffusion terms into the inventory processes as an approximation to the general game, we establish its global well-posedness. Simulations reveal a negative correlation between quotes and strategic orders, in contrast to the positive correlation observed between quotes and noise orders.