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1、A stochastic model for order book dynamicsRama Cont, Sasha Stoikov, Rishi TalrejaIEOR Dept, Columbia University, New Yorkrama.contcolumbia.edu, , rt2146columbia.eduWe propose a stochastic model for the continuous-time dynamics of a limit order book. The model strikesa balance between two desirable f
2、eatures: it captures key empirical properties of order book dynamics andits analytical tractability allows for fast computation of various quantities of interest without resorting tosimulation. We describe a simple parameter estimation procedure based on high-frequency observations ofthe order book
3、and illustrate the results on data from the Tokyo stock exchange. Using Laplace transformmethods, we are able to eciently compute probabilities of various events, conditional on the state of theorder book: an increase in the mid-price, execution of an order at the bid before the ask quote moves, and
4、execution of both a buy and a sell order at the best quotes before the price moves. Comparison with high-frequency data shows that our model can capture accurately the short term dynamics of the limit orderbook.Key words: Limit order book, financial engineering, Laplace transform inversion, queueing
5、 systems,simulation.1Cont, Stoikov and Talreja: A stochastic model for order book dynamics2Contents1 Introduction 32 A continuous-time model for a stylized limit order book 42.1 Limitorderbooks . 42.2 Dynamicsoftheorderbook. 53 Parameter estimation 63.1 Descriptionofthedataset. 63.2 Estimationproced
6、ure . 74 Laplace transform methods for computing conditional probabilities 84.1 Laplacetransformsandfirst-passagetimesofbirth-deathprocesses. 94.2 Directionofpricemoves. 104.3 Executinganorderbeforethemid-pricemoves . 124.4 Makingthespread . 135 Numerical Results 155.1 Longtermbehavior . 155.1.1 Ste
7、adystateshapeofthebook. 155.1.2 Volatility . 155.2 Conditionaldistributions . 165.2.1 One-step transition probabilities . . . 165.2.2 Directionofpricemoves. 175.2.3 Executinganorderbeforethemid-pricemoves. 185.2.4 Makingthespread. 186Conclusion 18Cont, Stoikov and Talreja: A stochastic model for ord
8、er book dynamics31. IntroductionThe evolution of prices in financial markets results from the interaction of buy and sell ordersthrough a rather complex dynamic process. Studies of the mechanisms involved in trading financialassets have traditionally focused on quote-driven markets, where a market m
9、aker or dealer central-izes buy and sell orders and provides liquidity by setting bid and ask quotes. The NYSE specialistsystem is an example of this mechanism. In recent years, Electronic Communications Networks(ECNs) such as Archipelago, Instinet, Brut and Tradebook have captured a large share of
10、theorder flow by providing an alternative order-driven trading system. These electronic platformsaggregate all outstanding limit orders in a limit order book that is available to market participantsand market orders are executed against the best available prices. As a result of the ECNs popular-ity,
11、 established exchanges such as the NYSE, Nasdaq, the Tokyo Stock Exchange and the LondonStock Exchange have adopted electronic order-driven platforms, either fully or partially through“hybrid” systems.The absence of a centralized market maker, the mechanical nature of execution of orders andlast but
12、 not least the availability of data have made order-driven markets interesting candidatesfor stochastic modelling . At a fundamental level, models of order book dynamics may providesome insight into the interplay between order flow, liquidity and price dynamics Bouchaud et al.(2002), Smith et al. (2
13、003), Farmer et al. (2004), Foucault et al. (2005). At the level of applications,such models provide a quantitative framework for investors and trading desks to optimize tradeexecution strategies Alfonsi et al. (2007), Obizhaeva and Wang (2006). An important motivationfor modelling high-frequency dy
14、namics of order books is to use the information on the currentstate of the order book to predict its short-term behavior. The focus is therefore on conditionalprobabilities of events, given the state of the order book.The dynamics of a limit order book resembles in many aspects that of a queuing sys
15、tem. Limitorders wait in a queue to be executed against market orders (or canceled). Drawing inspirationfrom this analogy, we model a limit order book as a continuous-time Markov process that tracks thenumber of limit orders at each price level in the book. The model strikes a balance between threedes
