We study a stock dealer's strategy for submitting bid and ask quotes in a limit order book. The agent faces an inventory risk due to the diffusive nature of the stock's mid-price and a transactions risk due to a Poisson arrival of market buy and sell orders. After setting up the agent's problem in a maximal expected utility framework, we derive the solution in a two step procedure. First, the dealer computes a personal indifference valuation for the stock, given his current inventory. Second, he calibrates his bid and ask quotes to the market's limit order book. We compare this "inventory-based" strategy to a "naive" strategy that is symmetric around the mid-price, by simulating stock price paths and displaying the P&L profiles of both strategies. We find that our strategy yields P&L profiles and final inventories that have significantly less variance than the benchmark strategy.

We study the price impact of order book events -limit orders, market orders and cancelations -using the NYSE TAQ data for 50 U.S. stocks. We show that, over short time intervals, price changes are mainly driven by the order flow imbalance, defined as the imbalance between supply and demand at the best bid and ask prices. Our study reveals a linear relation between order flow imbalance and price changes, with a slope inversely proportional to the market depth. These results are shown to be robust to seasonality effects, and stable across time scales and across stocks. We argue that this linear price impact model, together with a scaling argument, implies the empirically observed "square-root" relation between price changes and trading volume. However, the relation between price changes and trade volume is found to be noisy and less robust than the one based on order flow imbalance.

We propose a continuous-time stochastic model for the dynamics of a limit order book. The model strikes a balance between three desirable features: it can be estimated easily from data, it captures key empirical properties of order book dynamics, and its analytical tractability allows for fast computation of various quantities of interest without resorting to simulation. We describe a simple parameter estimation procedure based on high-frequency observations of the order book and illustrate the results on data from the Tokyo Stock Exchange. Using simple matrix computations and Laplace transform methods, we are able to efficiently compute probabilities of various events, conditional on the state of the order book: an increase in the midprice, execution of an order at the bid before the ask quote moves, and execution of both a buy and a sell order at the best quotes before the price moves. Using high-frequency data, we show that our model can effectively capture the short-term dynamics of a limit order book. We also evaluate the performance of a simple trading strategy based on our results.