We propose a dynamics model of the Limit Order Book (LOB) in which the shape of the LOB, although not observable, is determined endogenously by an expected utility function via a competitive equilibrium argument. Assuming zero resilience, the resulting equilibrium density is random, nonlinear, and time inhomogeneous, and it defines the liquidity cost dynamically in a natural way.
We will then consider an optimal execution problem based on our LOB model. We verify that the value function satisfies the Dynamic Programming Principle and prove that it 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 then construct the optimal purchase strategy via a verification theorem argument, assuming that the PDE has a classical solution.
This is a joint work with Xinyang Wang and Jianfeng Zhang.
Back to Workshop II: The Mathematics of High Frequency Financial Markets: Limit Order Books, Frictions, Optimal Execution and Program Trading