A Multi-class Queueing Model of Limit Order Book Dynamics

Modern equity markets are computerized technological systems, operating as so-called “electronic limit order books” (LOBs). Market participants, including institutional investors, market makers, and opportunistic investors, are faced with a new set of operational trading challenges as they interact within today’s high-frequency marketplace. This has necessitated the development of electronic trade execution algorithms. At a high level, these algorithms dynamically optimize where, how often, and at what price to trade. They seek to optimize their own best execution objectives while taking into account the state of the exchanges, and real-time market information. Growing evidence for the importance of execution algorithms is evident through formation of specialized groups at investment banks and other organizations that offer algorithmic trading services. This paper describes a multi-class queueing model of a limit order book, drawing a strong connection between the field of high-frequency market microstructure and the ubiquitous service of trade execution to the tools from queueing networks and stochastic control. We subsequently formulate and solve a stylized, yet relevant, short-term optimal execution problem: how to buy or sell a block of shares at the best possible price, taking into account the short term queueing dynamics of the limit order book. Finally, we propose a microstructure model of market impact that captures the expected adverse price movement due to the trader’s own activity, which is an essential ingredient of most trade execution algorithms used in practice. We illustrate our model in an empirically calibrated example. Model. Exchanges typically function as electronic limit order books, operating under a “pricetime” priority rule. They can be modeled as a multi-class queueing system, as follows: Order arrivals. There are two types of orders that arrive to trade in a LOB. (i) Limit orders that seek to buy a quantity of the stock at a price that is less than or equal to an upper price “limit”, where both the quantity and price are defined by the trader; similarly, limit sell orders seek to sell a quantity of the stock at a price that is higher or equal to a lower price limit. The bid is defined to be the highest price at which buyer limit orders are posted and waiting to trade in the LOB, and the ask is the lowest limit prices at which limit orders to sell are posted in the LOB. (ii) Market orders that seek to buy a quantity at the “best” possible price. Since market orders do not impose a constraint on the execution price, they execute instantaneously. In contrast, limit orders may not be able to execute upon their arrival and instead join the queue associated with their price limit. The quantities, price limits, and order inter-arrival times of such orders are stochastic, which we