Optimal submission problem in a limit order book with VaR constraints

Abstract

We consider an optimal selection problem for bid and ask quotes subject to a value-at-Risk (VaR) constraint when arrivals of the buy and sell orders are governed by a Poisson process. The problem is formulated as a constrained utility maximization problem over a finite time horizon. Using a diffusion approximation to Poisson arrivals of market orders, the dynamic programming principle can be applied here. We propose an efficient procedure to solve this constrained utility maximization problem based on a successive approximation algorithm. Numerical examples with and without the VaR constraint are used to illustrate the effect of the risk constraint on the dealer's choices. We also conduct numerical experiments to analyze the impacts of the risk constraint on dealer's terminal profit. © 2012 IEEE.