The optimal placement problem studies how to optimally place orders in a limit order book to purchase/sell a fixed number of shares of a stock. Under a correlated random walk model with mean-reversion for the best ask/bid price, optimal placement strategies for both single-step and multi-step cases are derived in this thesis. In the single-step case, the optimal strategy involves only the market order, the best bid, and the second best bid. In the multi-step case, the optimal strategy is of a threshold type. Critical to the analysis is a generalized reflection principle for correlated random walks.

Furthermore, a simple linear price impact model is also presented, which gives insight into the price impact of executing large orders on placement decisions.

The availability of detailed limit order book information enables more accurate estimation of the order execution probability and price dynamics. Motivated by various optimization problems and models in algorithmic trading, a limiting behavior for order positions and related best bid/ask queues in a limit order book is studied in this thesis. In addition to the fluid and diffusion limits for the processes, fluctuations of the order position and related queues around their fluid limit are analyzed. As a corollary, explicit analytical expressions for various quantities of interests in a limit order book are derived.