Optimal Submission Problem in a Limit Order Book with VaR Constraints

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.