Dynamic Equilibrium Limit Order Book Model and Optimal Execution Problem

TL;DR

This paper introduces a dynamic, endogenous Limit Order Book model based on equilibrium theory, and investigates an optimal execution problem with a focus on liquidity costs and strategy structure.

Contribution

It develops a novel LOB model where the shape is determined endogenously and analyzes the optimal execution strategy within this framework.

Findings

Equilibrium density of LOB is random, nonlinear, and time inhomogeneous.

Liquidity costs are defined dynamically in the model.

Optimal strategies are characterized via a viscosity solution to a complex HJB equation.

Abstract

In this paper we propose a dynamic model of Limit Order Book (LOB). The main feature of our model is that the shape of the LOB is determined endogenously by an expected utility function via a competitive equilibrium argument. Assuming zero resilience, the resulting equilibrium density of the LOB is random, nonlinear, and time inhomogeneous. Consequently, the liquidity cost can be defined dynamically in a natural way. We next study an optimal execution problem in our model. We verify that the value function satisfies the Dynamic Programming Principle, and is a viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation which is in the form of an integro-partial-differential quasi-variational inequality. We also prove the existence and analyze the structure of the optimal strategy via a verification theorem argument, assuming that the PDE has a classical solution.