Optimal Placement in a Limit Order Book

This paper proposes and studies an optimal placement problem in a limit order book. To gain some analytical insights, a simple correlated random walk model with mean-reversion is proposed for the best ask price. Optimal placement strategies for both single-period and multi-period cases are derived. In the single period model, the optimal strategy involves only the market order, the best bid, and the second best bid. In a multi-period model, the optimal strategy is a threshold type. The key component of the analysis for the two-dimensional optimization problem is a generalized reflection principle for the random walk. In the appendix, a model with price impact is analyzed, to illustrate the connection between the optimal execution problem and the optimal placement problem