OPERATIONS RESEARCH
Vol. 58, No. 3, May–June 2010, pp. 549–563 issn 0030-364X eissn 1526-5463 10 5803 0549

informs

®

doi 10.1287/opre.1090.0780 © 2010 INFORMS

A Stochastic Model for Order Book Dynamics
Rama Cont
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027, rama.cont@columbia.edu

Sasha Stoikov
Cornell Financial Engineering Manhattan, New York, New York 10004, sashastoikov@gmail.com

Rishi Talreja
Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027, rt2146@columbia.edu

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. Subject classifications: limit order book; financial engineering; Laplace transform inversion; queueing systems; simulation. Area of review: Financial Engineering. History: Received September 2008; revision received March 2009; accepted August 2009. Published online in Articles in Advance February 26, 2010.

The evolution of prices in financial markets results from the interaction of buy and sell orders through a rather complex dynamic process. Studies of the mechanisms involved in trading financial assets have traditionally focused on quote-driven markets, where a market maker or dealer centralizes buy and sell orders and provides liquidity by setting bid and ask quotes. The NYSE specialist system 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 the order flow by providing an alternative order-driven trading system. These electronic platforms aggregate all outstanding limit orders in a limit order book that is available to market participants and market orders are executed against the best available prices. As a result of the ECN’s popularity, established exchanges such as the NYSE, NASDAQ, the Tokyo Stock Exchange, and the London Stock Exchange have adopted electronic orderdriven platforms, either fully or partially through “hybrid” systems. The absence of a centralized market maker, the mechanical nature of execution of orders and, last but not least, the availability of data have made order-driven markets interesting candidates for stochastic modelling. At a fundamental level, models of order book dynamics may provide
549

some insight into the interplay between order flow, liquidity, and price dynamics (Bouchaud et al. 2002, Smith et al. 2003, Farmer et al. 2004, Foucault et al. 2005). At the level of applications, such models provide a quantitative framework in which investors and trading desks can optimize trade execution strategies (Alfonsi et al. 2010, Obizhaeva and Wang 2006). An important motivation for modelling high-frequency dynamics of order books, is to use the information on the current state of the order book to predict its short-term behavior. We focus, therefore, on conditional probabilities of events, given the state of the order book. The dynamics of a limit order book resembles in many aspects that of a queuing system. Limit orders wait in a queue to be executed against market orders (or canceled). Drawing inspiration from this analogy, we model a limit order book as a continuous-time Markov process that tracks the number of limit orders at each price level in the book. The model strikes a balance between three desirable features: it can be estimated easily using high-frequency data, it reproduces various empirical features of order books, and it is analytically tractable. In particular, we show that our model is simple enough to allow the use of Laplace transform techniques from the queuing literature to compute various conditional probabilities. These include the probability of the midprice increasing in the next move, the

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics

550 probability of executing an order at the bid before the ask quote moves, and the probability of executing both a buy and a sell order at the best quotes before the price moves, given the state of the order book. Although here we only focus on these events, the methods we introduce allow one to compute conditional probabilities involving much more general events such as those involving latency associated with order processing (see Remark 1). We illustrate our techniques on a model estimated from order book data for a stock on the Tokyo Stock Exchange. Related literature. Various recent studies have focused on limit order books. Given the complexity of the structure and dynamics of order books, it has been difficult to construct models that are both statistically realistic and amenable to rigorous quantitative analysis. Parlour (1998), Foucault et al. (2005), and Rosu (2009) propose equilibrium models of limit order books. These models provide interesting insights into the price formation process, but contain unobservable parameters that govern agent preferences. Thus, they are difficult to estimate and use in applications. Some empirical studies on properties of limit order books are Bouchaud et al. (2002), Farmer et al. (2004), and Hollifield et al. (2004). These studies provide an extensive list of statistical features of order book dynamics that are challenging to incorporate in a single model. Bouchaud et al. (2008), Smith et al. (2003), Bovier et al. (2006), Luckock (2003), and Maslov and Mills (2001) propose stochastic models of order book dynamics in the spirit of the one proposed here, but focus on unconditional/steady– state distributions of various quantities rather than the conditional quantities we focus on here. The model proposed here is admittedly simpler in structure than some others existing in the literature: It does not incorporate strategic interaction of traders as in the gametheoretic approaches of Parlour (1998), Foucault et al. (2005), and Rosu (2009), nor does it account for “long memory” features of the order flow as pointed out by Bouchaud et al. (2002, 2008). However, contrarily to these models, it leads to an analytically tractable framework where parameters can be easily estimated from empirical data and various quantities of interest may be computed efficiently. Outline. The paper is organized as follows. Section 1 describes a stylized model for the dynamics of a limit order book, where the order flow is described by independent Poisson processes. Estimation of model parameters from high-frequency order book time-series data is described in §2 and illustrated using data from the Tokyo Stock Exchange. In §3 we show how this model can be used to compute conditional probabilities of various types of events relevant for trade execution using Laplace transform methods. Section 4 explores steady-state properties of the model using Monte Carlo simulation, compares conditional probabilities computed by simulation to those computed with the Laplace transform methods presented in §3, and analyzes a high-frequency trading strategy based on our results in §4.3. Section 5 concludes.

Operations Research 58(3), pp. 549–563, © 2010 INFORMS

1. A Continuous-Time Model for a Stylized Limit Order Book
1.1. Limit Order Books Consider a financial asset traded in an order-driven market. Market participants can post two types of buy/sell orders. A limit order is an order to trade a certain amount of a security at a given price. Limit orders are posted to a electronic trading system, and the state of outstanding limit orders can be summarized by stating the quantities posted at each price level: this is known as the limit order book. The lowest price for which there is an outstanding limit sell order is called the ask price and the highest buy price is called the bid price. A market order is an order to buy/sell a certain quantity of the asset at the best available price in the limit order book. When a market order arrives it is matched with the best available price in the limit order book, and a trade occurs. The quantities available in the limit order book are updated accordingly. A limit order sits in the order book until it is either executed against a market order or it is canceled. A limit order may be executed very quickly if it corresponds to a price near the bid and the ask, but may take a long time if the market price moves away from the requested price or if the requested price is too far from the bid/ask. Alternatively, a limit order can be canceled at any time. We consider a market where limit orders can be placed n representing multiples of a price on a price grid 1 tick. The upper boundary n is chosen large enough so that it is highly unlikely that orders for the stock in question are placed at prices higher than n within the time frame of our analysis. Because the model is intended to be used on the time scale of hours or days, this finite boundary assumption is reasonable. We track the state of the order book with a continuous-time process X t ≡ X1 t Xn t t 0 , where Xp t is the number of outstanding limit orders at price p, 1 p n. If Xp t < 0, then there are −Xp t bid orders at price p; if Xp t > 0, then there are Xp t ask orders at price p. The ask price pA t at time t is then defined by pA t = inf p = 1 n Xp t > 0 ∧ n + 1

Similarly, the bid price pB t is defined by pB t ≡ sup p = 1 n Xp t < 0 ∨ 0

Notice that when there are no ask orders in the book we force an ask price of n + 1, and when there are no bid orders in the book we force a bid price of 0. The midprice pM t and the bid-ask spread pS t are defined by pM t ≡ pB t + pA t 2 and pS t ≡ pA t − pB t

Because most of the trading activity takes place in the vicinity of the bid and ask prices, it is useful to keep track

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics
Operations Research 58(3), pp. 549–563, © 2010 INFORMS

551 • Limit buy (respectively sell) orders arrive at a distance of i ticks from the opposite best quote at independent, exponential times with rate i , • Market buy (respectively sell) orders arrive at independent, exponential times with rate , • Cancellations of limit orders at a distance of i ticks from the opposite best quote occur at a rate proportional to the number of outstanding orders: If the number of outstanding orders at that level is x, then the cancellation rate is i x. This assumption can be understood as follows: if we have a batch of x outstanding orders, each of which can be canceled at an exponential time with parameter i , then the overall cancellation rate for the batch is i x. • The above events are mutually independent. Order arrival rates depend on the distance to the bid/ask with most orders being placed close to the current price. We 1 n → 0 model the arrival rate as a function of the distance to the bid/ask. Empirical studies (Zovko and Farmer 2002 or Bouchaud et al. 2002) suggest a power law, i = k i

of the number of outstanding orders at a given distance from the bid/ask. To this end, we define ⎧ ⎨XpA t −i t 0 < i < pA t QiB t = (1) ⎩ i pB t decreases the quantity at level p x → xp−1 The evolution of the order book is thus driven by the incoming flow of market orders, limit orders, and cancellations at each price level, each of which can be represented as a counting process. It is empirically observed (Bouchaud et al. 2002) that incoming orders arrive more frequently in the vicinity of the current bid/ask price and the rate of arrival of these orders depends on the distance to the bid/ask. To capture these empirical features in a model that is analytically tractable and allows computation of quantities of interest in applications, most notably conditional probabilities of various events, we propose a stochastic model where the events outlined above are modelled using independent Poisson processes. More precisely, we assume that, for i 1,

as a plausible specification. Given the above assumptions, X is a continuous-time Markov chain with state space n and transition rates given by: x → xp−1 x→x x→x x→x p+1 with rate with rate with rate with rate

pA t − p for p < pA t p − pB t for p > pB t

x → xpB

t +1

pA t −1

x → xp+1 p−1 with rate with rate

pA t − p xp for p < pA t p − pB t xp for p > pB t

In practice, the ask price is always greater than the bid price. We say a state is admissible if it fulfills this requirement: ≡ x∈ n ∃k l ∈ p

s.t. 1 l xp

k

l

n xp k

0 for p

l (3)

xp = 0 for k

0 for p

If the initial state of the book is admissible, it remains admissible with probability one: Proposition 1. If ∀t 0 = 1. X 0 ∈ , then X t ∈

Proof. It is easily verified that is stable under each of the six transitions defined above, which leads to our assertion. Proposition 2. If ≡ min1 i n i > 0, then X is an ergodic Markov process. In particular, X has a proper stationary distribution.

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics

552 Proof. Let N ≡ N t t 0 , where N t ≡ n Xp t , p=1 and let N be a birth-death process with birth rate given p and death rate in state i, i ≡ 2 + i . by ≡ 2 n p=1 Notice that N increases by one at a rate bounded from above by and decreases by one at a rate bounded from below by i ≡ 2 + i when in state i. Thus, for all t 0, N is stochastically bounded by N . For k 1, let T0k k and T−0 denote the duration of the kth visit to 0 and the duration between the k − 1 th and kth visit to 0 of prok cess N , respectively. Define random variables T0k and T−0 , k 1, for process N similarly. Then the point process with 1 2 and the point process interarrival times T−0 T01 T−0 T02 1 2 are alternating with interarrival times T−0 T01 T−0 T02 renewal processes. By Theorem VI.1.2 of Asmussen (2003) and the fact that N is stochastically dominated by N , we then have for each k 1, Ɛ T0k Ɛ T0k = lim k t→ + Ɛ T−0 t→ Operations Research 58(3), pp. 549–563, © 2010 INFORMS

2. Parameter Estimation
2.1. Description of the Data Set Our data consist of time-stamped sequences of trades (market orders) and quotes (prices and quantities of outstanding limit orders) for the five best price levels on each side of the order book, for stocks traded on the Tokyo stock exchange over a period of 125 days (Aug.–Dec. 2006). This data set, referred to as Level II order book data, provides a more detailed view of price dynamics than the trade and quotes (TAQ) data often used for high-frequency data analysis, which consist of prices and sizes of trades (market orders) and time-stamped updates in the price and size of the bid and ask quotes. In Table 1, we display a sample of three consecutive trades for Sky Perfect Communications. Each row provides the time, size, and price of a market order. We also display a sample of Level II bid-side quotes. Each row displays the five bid prices (pb1, pb2, pb3, pb4, pb5), as well as the quantity of shares bid at these respective prices (qb1, qb2, qb3, qb4, qb5). 2.2. Estimation Procedure Recall that in our stylized model we assume orders to be of “unit” size. In the data set, we first compute the average sizes of market orders Sm , limit orders Sl , and canceled orders Sc and choose the size unit to be the average size of a limit order Sl . The limit order arrival rate function for 1 i 5 can be estimated by ˆ i = Nl i T∗

N t =0 N t =0 = Ɛ T0k k Ɛ T0k + Ɛ T−0

lim

(4)

Notice that in state 0 both N and N have birth rate . Thus, Ɛ T0k =Ɛ T0k = 1 (5)

Combining (4) and (5) gives us k Ɛ T−0 k Ɛ T−0

(6)

To show N is ergodic, notice the inequalities i i=1 1 ···

< i i=1

1 i!

i

=e

/

−1<

(7)

and
1 ··· i i=1 M i

> i=1 1 ··· i

i

+ i=M+1 2 +M

i

=

(8)

where Nl i is the total number of limit orders that arrived at a distance i from the opposite best quote, and T∗ is the total trading time in the sample (in minutes). Nl i is obtained by enumerating the number of times that a quote increases in size at a distance of 1 i 5 ticks from the opposite best quote. We then extrapolate by fitting a power law function of the form ˆ i = k i (suggested by Zovko and Farmer 2002 or Bouchaud et al. 2002). The power law parameters k and are obtained by a least-squares fit
5

for M > 0 chosen large enough so that 2 +M > . Therefore, by Corollary 2.5 of Asmussen (2003), N is ergodic k so that Ɛ T−0 < . Combining this with the bound (6) and the fact that for each t 0 X t = 0 0 if and only if N t = 0 shows that X is positive recurrent. Because X is clearly also irreducible, it follows that X is ergodic. The ergodicity of X is a desirable feature of theoretical interest: it allows comparison of time averages of various quantities in simulations (average shape of the order book, average price impact, etc.) to unconditional expectations of these quantities computed in the model. The steady-state behavior of X will be further discussed in §4.1. We note, however, that our results involving conditional probabilities in §3 and applications discussed in §4.3 do not rely on this ergodicity result.

min k i=1

ˆ i − k i

2

Estimated arrival rates at distances 0 i 10 from the opposite best quote are displayed in Figure 1(a). The arrival rate of market orders is then estimated by ˆ= N m Sm T ∗ Sl

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics
Operations Research 58(3), pp. 549–563, © 2010 INFORMS

553

Table 1.

A sample of three trades and five quotes for Sky Perfect Communications.
Time 9:11:01 9:11:04 9:11:19 Price 74 300 74 600 74 400 pb3 74 74 74 74 74 000 200 200 200 200 pb4 73 74 74 74 74 900 000 000 000 000 pb5 73 73 73 73 73 800 900 900 900 900 Size 1 2 1 qb1 12 20 21 34 33 qb2 13 12 11 4 4 qb3 1 13 13 13 13 qb4 52 1 1 1 1 qb5 11 52 52 52 52

Time 9:11:01 9:11:03 9:11:04 9:11:05 9:11:19

pb1 74 74 74 74 74 300 400 400 400 400

pb2 74 74 74 74 74 200 300 300 300 300

where T∗ is the total trading time in the sample (in minutes) and Nm is the number of market orders. Note that we ignore market orders that do not affect the best quotes, as is the case when a market order is matched by a hidden order. Because the cancellation rate in our model is proportional to the number of orders at a particular price level, Figure 1. The arrival rates as a function of the distance from the opposite quote.
(a) Limit orders rates
2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 Data Model

in order to estimate the cancellation rates we first need to estimate the steady-state shape of the order book Qi , which is the average number of orders at a distance of i ticks from the opposite best quote, for 1 i 5. If M is the number of quote rows and SiB j the number of shares bid at a distance of i ticks from the ask on the jth row, for 1 j M, we have 1 1 M B QiB = S j Sl M j=1 i The vector QiA is obtained analogously, and Qi is the average of QiA and QiB . An estimator for the cancellation rate function is then given by ˆ i = Nc i Sc for i 5 and T Q i Sl (9) ˆ i =ˆ 5 for i > 5 where Nc i is obtained by counting the number of times that a quote decreases in size at a distance of 1 i 5 ticks from the opposite best quote, excluding decreases due to market orders. The fitted values are displayed in Figure 1(b). Estimated parameter values for Sky Perfect Communications are given in Table 2.

Distance from opposite quote (b) Cancellation rates
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5 6 7 8 9 10

3. Laplace Transform Methods for Computing Conditional Probabilities
As noted above, an important motivation for modelling high-frequency dynamics of order books is to use the information provided by the limit order book for predicting Table 2. Estimated parameters: Sky Perfect Communications. i 1 ˆ i ˆ i ˆ k 1 85 0 71 0 94 1 92 0 52 2 1 51 0 81 3 1 09 0 68 4 0 88 0 56 5 0 77 0 47

Distance from opposite quote

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics

554 short-term behavior of various quantities that are useful in trade execution and algorithmic trading, for instance, the probability of the midprice moving up versus down, the probability of executing a limit order at the bid before the ask quote moves, and the probability of executing both a buy and a sell order at the best quotes before the price moves. These quantities can be expressed in terms of conditional probabilities of events, given the state of the order book. In this section we show that the model proposed in §1 allows such conditional probabilities to be analytically computed using Laplace methods. After presenting some background on Laplace transforms in §3.1, we give various examples of these computations. The probability of an increase in the midprice is discussed in §3.2, the probability that a limit order executes before the price moves is discussed in §3.3, and the probability of executing both a buy and a sell limit order before the price moves is discussed in §3.4. Laplace transform methods allow efficient computation of these quantities, bypassing the need for Monte Carlo simulation. 3.1. Laplace Transforms and First-Passage Times of Birth-Death Processes We first recall some basic facts about two-sided Laplace transforms and discuss the computation of Laplace transforms for first-passage times of birth-death processes (Abate and Whitt 1999). Given a function f → , its two-sided Laplace transform is given by fˆ s = e−st f t dt

Operations Research 58(3), pp. 549–563, © 2010 INFORMS

denominators, which are complex numbers with an = 0 for all n 1, is the sequence wn n 1 , where wn = t1 t2 ··· tn 0 n 1 tk u = ak bk +u k 1

and denotes the composition operator. If w ≡ limn→ wn , then the continued fraction is said to be convergent and the limit w is said to be the value of the continued fraction (Abate and Whitt 1999). In this case, we write w≡ n=1 an bn

Consider now a birth-death process with constant birth rate and death rates i in state i 1, and let b denote the first-passage time of this process to 0 given that it begins in state b. Next, notice that we can write B as the sum b =

b b−1

+

b−1 b−2

+···+

1 0

where i i−1 denotes the first-passage time of the birthdeath process from the state i to the state i − 1, for i=1 b, and all terms on the right-hand side are independent. If fˆb denotes the Laplace transform of b and fˆi i−1 denotes the Laplace transform of i i−1 for i = 1 b, then we have by (10), fˆb s = b i=1

fˆi

i−1

s

(12)

−

where s is a complex numbers. When f is the probability density function (pdf) of some random variable X, we also say that fˆ is the two-sided Laplace transform of the random variable X. We work with two-sided Laplace transforms here because for our purposes the function f will usually correspond to the pdf of a random variable with both positive and negative support. From now on, we drop the prefix “two-sided” when referring to two-sided Laplace transforms. When we say conditional Laplace transform of the random variable X conditional on the event A, we mean the Laplace transform of the conditional pdf of X given A. Recall that if X and Y are independent random variables with well-defined Laplace transforms, then fˆX+Y s = Ɛ e−s X+Y = Ɛ e−sX Ɛ e−sY = fˆX s fˆY s (10)

Therefore, in order to compute fˆb , it suffices to compute the simpler Laplace transforms fˆi i−1 , for i = 1 b. By Equation (4.9) of Abate and Whitt (1999), we see that the Laplace transform of fˆi i−1 is given by fˆi i−1 s =−

1 k=i − +

k k

+s

(13)

The computation there is based on a recursive relationship between the fˆi i−1 , i = 1 b, which is derived by considering the first transition of the birth-death process. Combining (12) and (13), we obtain 1 fˆb s = − b b k=i i=1

− +

k k

+s

(14)

We will use this result in all our computations below. 3.2. Direction of Price Moves We now compute the probability that the midprice increases at its next move. The first move in the midprice occurs at the first-passage time of the bid or ask queue to zero or, if the bid/ask spread is greater than one, the first time a limit order arrives inside the spread. Throughout this section, let XA ≡ XpA · · and XB ≡ XpB · · . Furthermore, let WB ≡ WB t t 0 (WA ≡ WA t t 0 ), where WB t (WA t ) denotes the number of orders remaining at the bid

If for some ∈ we have − fˆ + i d < and f t is continuous at t, then the inverse transform is given by the Bromwich contour integral f t = 1 2 i
+i −i

ets fˆ s ds

(11)

The continued fraction associated with a sequence an n 1 of partial numerators and bn n 1 of partial

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics
Operations Research 58(3), pp. 549–563, © 2010 INFORMS

555 for j 1, and let S ≡ S−1 i . Then (15) is given by i=1 the inverse Laplace transform of FaS b s = 1 ˆS f s a · fˆbS
S S

(ask) at time t of the initial XB 0 (XA 0 ) orders and let B ( A ) be the first-passage time of WB (WA ) to 0. Furthermore, let T be the time of the first change in midprice: T ≡ inf t 0 pM t = pM 0

+s +
S

S

+s

1 − fˆaS 1 − fˆbS
S

S

+s (19)

Given an initial configuration of the book, the probability that the next change in midprice is an increase can then be written as pM T > pM 0 XA 0 = a XB 0 = b pS 0 = S (15)

−s +

S S −s

−s

evaluated at 0. When S = 1, (19) reduces to 1 Fa1 b s = fˆa1 s fˆb1 −s s (20)

where S > 0. For ease of notation, we will omit the condition in (15) in all proofs below. The idea for computing (15) is to use a coupling argument. Lemma 3. Let pS 0 = S. Then 1. There exist independent birth-death processes S and death XA and XB with constant birth rates rates +i S , i 1, such that for all 0 t T, XA t = XA t , and XB t = XB t . 2. There exist independent pure death processes WA and WB with death rate + i S in state i 1, such that for all 0 t T , WA t = WA t and WB t = WB t . Furthermore, WA is independent of XB , WB is independent of XA , WA XA , and WB XB . Proof. We prove Part 1. Part 2 can be proven analogously. X is a continuous-time Markov chain, with transition rates given by (1.2). For 0 t T , pA t = pA 0 and pB t = pB 0 , so substituting in (1.2) yields that XA t and XB t have the following (identical) transition rates for 0 t T n → n + 1 with rate n→n−1 with rate S +n S (16) (17)

Proof. We will first focus on the special case when S = 1 and then extend the analysis to the case S > 1, using Lemma 5 below. Construct the independent birth-death processes XA and XB as in Lemma 3. When S = 1, the price changes for the first time exactly when one of the two processes XA and XB reaches the state 0 for the first time. Thus, given our initial conditions, the distribution of T is given by the minimum of the independent first-passage times A and B . Furthermore, the quantity (15) is given by A < B . By (14), the conditional Laplace transform of A − B given the initial conditions is given by fˆa1 s fˆb1 −s so that the conditional Laplace transform of the cumulative distribution function (cdf) of A − B is given by (20). Thus, our desired probability is given by the inverse Laplace transform of (20) evaluated at 0. i We now move on to the case where S > 1. Let A denote the first time an ask order arrives i ticks away from the i bid and B denote the first time a bid order arrives i ticks away from the ask, for i = 1 S − 1. The time of the first change in midprice is now given by T=
A

∧

B

∧ min

i A

i B

i=1

S −1

Define XA and XB such that • XA t = XA t and XB t = XB t for t T and • X A t XB t t T follow independent birth-death processes with rates given by (16) and (17). The above remarks show that in fact XA t t 0 (respectively XB t t 0 ) has the same law as a birth-death process with rates (16)–(17). To show that XA and XB are independent, we note that because the transition rates of XA (respectively XB ) do not depend on Xp t p = pA 0 (respectively Xp t p = pB 0 ) for 0 t T , we have, in particular, conditional independence of XA t and XB t given X 0 and t T . Henceforth, we let A and B denote the first-passage times of XA and XB to 0, respectively. The conditional probability (15) can then be computed as follows: Proposition 4 (Probability of Increase in Midprice). Let fˆjS be given by 1 fˆjS s = − S j b k=i i=1

Notice that XA and XB are independent of the mutually i i S − 1. Also, independent arrival times A , B , for i = 1 i i notice that A and B are exponentially distributed with rates i for i = 1 S − 1. The first change in midprice is an increase if there is an arrival of a limit bid order within S − 1 ticks of the best ask or XA hits zero, before there is an arrival of a limit ask order within S − 1 ticks of the best bid or XB hits zero. Thus, the quantity (15) can be written as
A

∧

1 B

∧···∧
A

S−1 B

< ∧

B A

∧

1 A

∧···∧

S−1 A

=

∧

B

<

B

(21)

− S +k S S + +k S +s

(18)

where A and B are independent exponential random variables, both with rate S . To compute (21), we first need to compute the conditional Laplace transform of the minimum B ∧ A . This is given in Lemma 5, substituting A for Z. The conditional Laplace transform of the random variable B ∧ A − A ∧ B can then be computed using (10), and the probability (15) can be computed by inverting the conditional Laplace transform of the cdf of this random variable and evaluating at 0 as in the case S = 1.

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics

556 Lemma 5. Let Z be an exponentially distributed random variable with parameter . Then the Laplace transform of the random variable B ∧ Z is given by fˆb1 +s + +s 1 − fˆb1 +s

Operations Research 58(3), pp. 549–563, © 2010 INFORMS

where fˆb1 is given in (18). Proof. We first compute the density f B ∧Z of the random variable B ∧ Z in terms of the density fb of the random variable B . Because Z is exponential with rate , we have for all t 0,
B

S = 1 the probability we are interested in is equal to the probability that the order is executed before the midprice moves away from the desired price, given that the order is not canceled. Although we focus here on an order placed at the bid price, because our model is symmetric in bids and asks, our result also holds for orders placed at the ask price. We introduce some new notation that we will use in this subsection as well as the next. Let NCb (NCa ) denote the event that an order that never gets canceled is placed at the bid (ask) at time 0. Then, the probability that an order placed at the bid is executed before the midprice moves is given by
B

∧Z t
B

Z>t t e− t =1− 1−F

< T XB 0 = b XA 0 = a pS 0 = S NCb

(23)

Taking derivatives with respect to t gives f
B ∧Z

t = fb1 t e− t + Fb1 (fb1

1 − Fb1 t e−

Proposition 6 (Probability of Order Execution ˆ Before Midprice Moves). Define fˆaS s as in (18), let gjS be given by (22)
B. j

t

t t ) is the cdf (pdf) of for t 0, where f B ∧Z t = 0 for t < 0. The Laplace transform of thus given by fˆ B ∧Z s = = = e−st f e
0 −st
B∧ B

Also, B ∧ Z is

gjS s = ˆ i=1 + S i−1 + S i−1 +s

(24)

for j 1, and let S ≡ S−1 i . Then the quantity (23) i=1 is given by the inverse Laplace transform of 1 S FaS b s = gb s ˆ s fˆbS 2 + 2 2
S

−

t dt
− t

−s
S

fb1

t e fb1

+

1−Fb1

t e

− t

ds
−t s+

e
0

−t s+

t dt + +s

0

1−Fb1

t e

dt

S −s

1 − fˆbS 2

S

−s

(25)

= fˆb1 s +

+

1− fˆb1 s +

evaluated at 0. When S = 1, (25) reduces to 1 1 Fa1 b s = gb s fˆa1 −s ˆ s (26)

where the last equality follows from integration by parts. Proposition 4 yields a numerical procedure for computing the probability that the next change in the midprice will be an increase. We discuss implementation of the procedure in §4.2.2. 3.3. Executing an Order Before the Mid-Price Moves A trader that submits a limit order at a given time obtains a better price than a trader that submits a market order at that same time, but faces the risk of nonexecution and the “winner’s curse.” Whereas a market order executes with certainty, a limit order stays in the order book until either a matching order is entered or the order is canceled. The probability that a limit order is executed before the price moves is therefore useful in quantifying the choice between placing a limit order and placing a market order. We now compute the probability that an order placed at the bid price is executed before any movement in the midprice, given that the order is not canceled. Our result holds for initial 1, but we remark that in the case where spread S ≡ pS 0

Proof. Construct XA and WB using Lemma 3. Let us first consider the case S = 1. Let T ≡ B ∧ T denote the first time when either the process WB hits 0 or the midprice changes. Conditional on an infinitely patient order being placed at the bid price at time 0, T is the first time when either that order gets executed or the midprice changes. Notice that conditional on our initial conditions, B is given by a sum of b independent exponentially distributed random variables with parameters + i − 1 1 , for i = 1 b, and independent of XA . Thus, the conditional Laplace transform of B given our initial conditions is given by (24). Because in the case S = 1 the midprice can change before time B if and only if A < B , the quantity (23) can be written simply as B < A . Using (10) with the conditional Laplace transforms of B and A , given in (24) and (18), respectively, we obtain (26). This analysis can be extended to the case where S > 1 just as in the proof of Proposition 4. When S > 1, our desired quantity can be written as B < A∧ B ∧ A . Because the conditional distribution of B ∧ A is exponential with parameter 2 S , Lemma 5 then yields the result.

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics
Operations Research 58(3), pp. 549–563, © 2010 INFORMS

557 placed at the best bid and ask prices at time 0, T is the first time when either both the orders get executed or the midprice changes. Furthermore, by Lemma 3, WA and WB are independent pure death processes with death rate + i 1 XA t and WB t XB t . This in state i 1, and WA t implies that A and B are independent of each other and A and B are independent of each other with A A and . Using these properties, we obtain B B max = = + =
A B B B A

3.4. Making the Spread We now compute the probability that two orders, one placed at the bid price and one placed at the ask price, are both executed before the midprice moves, given that the orders are not canceled. If the probability of executing both a buy and a sell limit order before the price moves is high, a statistical arbitrage strategy can be designed by submitting limit orders at the bid and the ask and wait for both orders to execute. If both orders execute before the price moves, the strategy has paid off the bid-ask spread: we refer to this situation as “making the spread.” Otherwise, losses may be minimized by submitting a market order and losing the bid-ask spread. We restrict attention to the case where the initial spread is one tick: S = 1. The probability of making the spread can be expressed as max
A B

< min
A A A A A B

B B B B B

A

< <
B

< <
A

<
A

A

<
B a

<
A

<
B

B A A

<

<

+

<

<

B

< T X B 0 = b XA 0 = a (27)

= ha b + hb

(33)

pS 0 = 1 NCa NCb

The following result allows one to compute this probability using Laplace transform methods: Proposition 7. The probability (27) of making the spread is given by ha b + hb a , where a where we define ha b ≡ B < A < B , the probability that the order placed at the bid is executed before the order placed at the ask, and the order at the ask is executed before the bid quote disappears. We now focus on computing ha b . Conditioning on the value of B gives ha b =
B

<

ha b = i=0 j=1

j

<

i

0

X P0 i

t

W Pa j

t

1 gb

t dt

(28)

0

A

<

B

B

1 = t gb t dt

(34)

where
X P0 i t ≡

Focusing on the first factor in the integrand in (34) and conditioning on the values of XB t and WA t gives us
X

e−

X

t

X

t

i

i!
W

t ≡ k 1 − e−

t

(29) = (30)

B

< a A

<
B

B

B

=t
B B

W Pa j t ≡ eQa t

a

j ≡

⎡

tk W Qa k=0 k! 0 0

<

A

<

= t XB t = i WA t = j
B

i=0 j=0

a j

⎢ ⎢ ⎢ ⎢ ⎢ W Qa ≡ ⎢ 0 ⎢ ⎢ ⎢ ⎣ 0

0

0 − +

··· ··· ···

0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (31)

·

XB t = i WA t = j

=t

(35)

− −

0

···

+ a−1

− − a−1

1 and gb is the inverse Laplace transform of gb , which is ˆ1 given in (24).

The first conditional probability on the right hand of (35) can now be simplified as follows. For i = 0 or j = 0 it is simply 0. For i j 1, under the condition of the probability, at time t there are j orders in the ask queue that have been placed before time 0 that have yet to be executed, and there are a total of i orders in the bid queue. Thus, the probability of interest is simply the probability that the j ask orders get executed before the number of orders in the bid queue hits 0. Thus,
B

< =

A

< j B i

B

= t XB t = i WA t = j (36)

Proof. Because S = 1, T = min (27) can be written as max
B A

A

B

, and the quantity

<

< min

B

A

(32)

Furthermore, the second probability on the right-hand side of (35) can be written as XB t = i WA t = j = = XB t = i XB t = i
B B B

Construct XA , XB , WA , and WB using Lemma 3. Let T = max A B ∧ T denote the first time when either both of the processes WA and WB have hit 0, or the midprice has changed. Conditional on infinitely patient orders being

=t WA t = j WA t = j
B

=t =t

=t (37)

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics

558 Combining Equations (33)–(35) and using Tonelli’s theorem to interchange the integral and the summation gives us a Operations Research 58(3), pp. 549–563, © 2010 INFORMS

4.1. Long-Term Behavior Recent empirical studies on order books (Bouchaud et al. 2002, 2008) have focused mainly on average properties of the order book, which in our context correspond to unconditional expectations of quantities under the stationary measure of X: the steady-state shape of the book and the volatility of the midprice. The ergodicity of the Markov chain X, shown in Proposition 2, implies that such expectations E f X can be computed in the model by simulating the order book over a large horizon T and averaging f X t over the simulated path: 1 T
T 0

ha b = i=0 j=1

j

<

i

0

XB t = i

B

=t

·

1 WA t = j gb t dt

The quantity XB t = i B = t can be computed using an analogy with the M/M/ queue. The number of orders in the bid queue at the time when the bid order placed at time 0 has executed is simply the number of customers at time t in an initially empty M/M/ queue with arrival rate and service rate , which has a Poisson distribution with mean given by X t in (29). This leads to X the expression for P0 i t in (29). The quantity WA t = j is the probability that a pure death process with death rate + k − 1 1 in state k 1 is in state j at time t, given that it begins in state a. The infinitesimal generator of this pure death process is given by (31). Thus, by Corollary II.3.5 of Asmussen (2003), WA t = j is given by (30). Remark 1. We note here that the probabilities computed in this section can also be computed using transition matrices of appropriately defined transient discrete-time Markov chains. In general, for a continuous-time Markov chain the probability of hitting state i before state j can be determined by constructing a corresponding embedded discretetime Markov chain with states i and j absorbing states and computing the fundamental matrices of this Markov chain (see, for example, §4.4 of Ross 1996). However, our Laplace transform approach has the advantage of computing full distributions of random variables such as A , B , A , and B . This could be used, for example, to compute probabilities such as > 0, which are A + < B , for useful when latency in order processing is an issue.

f X t dt → E f X

a.s.

as T →

4.1.1. Steady-State Shape of the Book. We simulate the order book over a long horizon (n = 106 events) and observe the mean number of orders Qi at distances 1 i 30 ticks from the opposite best quote. The results are displayed in Figure 2. The steady-state profile of the order book describes the average market impact of trades (Farmer et al. 2004, Bouchaud et al. 2008). Figure 2 shows that the average profile of the order book displays a hump (in this case, at two ticks from the bid/ask), as observed in empirical studies (Bouchaud et al. 2008). Note that this hump feature does not result from any fine-tuning of model parameters or additional ingredients such as correlation between order flow and past price moves. 4.1.2. Volatility. Define the realized volatility of the asset over a day by n RVn = i=1 log

Pi+1 Pi

2

(38)

4. Numerical Results
Our stochastic model allows one to compute various quantities of interest both by simulating the evolution of the order book and by using the Laplace transform methods presented in §3, based on parameters , , and estimated from the order flow. In this section we compute these quantities—for example, of Sky Perfect Communications— and compare them to empirically observed values in order to assess the precision of the description provided by our model. In §4.1, we compare empirically observed long-term behavior (e.g., unconditional properties) of the order book to simulations of the fitted model. Although these quantities may not be particularly important for traders who are interested in trading in a short time scale, they indicate how well the model reproduces the average properties of the order book. In §4.2, we compare conditional probabilities of various events in our model to frequencies of the events in the data. We also compare results using the Laplace transform methods developed in §3 to our simulation results.

Figure 2.
2.0

Simulation of the steady-state profile of the order book: Sky Perfect Communications.

1.8

Average number of orders

1.6

1.4

1.2

1.0

0.8

0

5

10

15

20

25

30

Distance from opposite best quote in ticks

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics
Operations Research 58(3), pp. 549–563, © 2010 INFORMS

559 where Bup = m QiB Tm = n QiB Tm+1 > n ˆ Aup = m QiA Tm = n QiA Tm+1 > n Bchange = m QiB Tm = n QiB Tm+1 = n ˆ Achange = m QiA Tm = n QiA Tm+1 = n and

where n is the number of quotes in the day and the prices Pi represent the midprice of the stock, for i = 1 n. In the first day of the sample, we compute a realized volatility of 0 0219 after a total of 370 trades. After repeatedly simulating our model for 370 trades, we obtained a 95% confidence interval for realized volatility of 0 0228 ± 0 0003. Interestingly, this estimator yields the correct order of magnitude for realized volatility based solely on intensity parameters for the order flow . 4.2. Conditional Distributions As discussed in the introduction, conditional distributions are the main quantities of interest for applications in highfrequency trading. A good description of conditional distributions of variables characterizing the order book gives one the ability to predict their behavior in the short term, which is of obvious interest in optimal trade execution and the design of trading strategies. 4.2.1. One-Step Transition Probabilities. In order to assess the model’s usefulness for short-term prediction of order book behavior, we compare one-step transition probabilities implied by our model to corresponding empirical frequencies. In particular, we consider the probability that the number of orders at a given price level increases given that it changes. Define Tm as the time of the mth event in the order book: T0 = 0 Tm+1 ≡ inf t Tm X t = X Tm (39)

In Figure 3, Pi n and Pi n for 1 i 5 are shown for Sky Perfect Communications. We see that these probabilities are reasonably close in most cases, indicating that the transition probabilities of the order book are well described by the model. 4.2.2. Direction of Price Moves. This subsection and the next two are devoted to the computation of conditional probabilities using the Laplace transform methods described in §3. These computations require the numerical inversion of Laplace transforms. The inversions are performed by shifting the random variable X under study by a constant c such that X + c 0 ≈ 1, then inverting the corresponding one-sided Laplace transform using the methods proposed in Abate and Whitt (1992, 1995). When computing the probability of an increase in midprice, one can find a good shift c by using the fact that when a = b the probability of an increase in midprice is 0.5. This shift c should also serve well for cases where a = b. Table 3 compares the empirical frequencies of an increase in midprice to model-implied probabilities, given an initial configuration of b orders at the bid price, a orders at the ask price, and a spread of 1, for various values of a and b. We computed these quantities using Monte Carlo simulation (using 30,000 replications) and the Laplace transform methods described in §3. The simulation results, reported as 95% confidence intervals, agree with the Laplace transform computations and show that the probability of an increase in the midprice is well captured by the model. 4.2.3. Executing an Order Before the Midprice Moves. Table 4 gives probabilities computed using both simulation and our Laplace transform method for executing a bid order before a change in midprice for various values of a and b and for S = 1. Because our data set does not allow us to track specific orders, empirical values for these quantities, as well as the quantities in §4.2.4, are not obtainable. 4.2.4. Making the Spread. Table 5 gives probabilities computed using both simulation and our Laplace transform method for executing both a bid and an ask order at the best quotes before the midprice changes. One interesting observation here is that for a fixed value of a, as b is increased, the probability of making the spread is not monotone. Thus, for a fixed number of orders at the ask price the probability of making the spread is maximized for a nontrivial optimal number of orders at the bid price.

The probability that the number of orders at a distance i from the opposite best quote moves from n to n + 1 at the next change is given by Pi n ≡ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ QiA Tm+1 = n + 1 QiA Tm = n QiA Tm+1 = n 1 + 1 +n 1 i=1 (40) i>1

=

i i +n i

To see how the above expression arises, consider the case A i = 1. The next change in Q1 is an increase if an arrival A of a limit order at price Q1 occurs before any of the limit A orders at Q1 cancel or a market buy order occurs. However, A because an arrival of a limit order at price Q1 occurs with rate 1 and a cancellation or market buy order occurs at rate + n 1 , the probability that an arrival of a limit order occurs first is given by 1 / 1 + + n 1 . Denoting empirical quantities with a hat, e.g., QiB t is the empirically observed number of bid orders at a distance of i units from the ask price at time t, an estimator for the above probability is given by Pi n ≡ ˆ Bup + Aup ˆ Bchange + Achange

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics

560 Figure 3.

Operations Research 58(3), pp. 549–563, © 2010 INFORMS

Probability of an increase in the number of orders at distance i from the opposite best quote in the next change, for i = 1 5.
1 tick from opposite quote 2 ticks from opposite quote
1.0

1.0

Probability of increase

Probability of increase
7

Empirical Model 0.5

0.5

0

0

1

2

3

4

5

6

0

0

1

2

3

4

5

6

7

Queue size 3 ticks from opposite quote
1.0 1.0

Queue size 4 ticks from opposite quote Probability of increase

Probability of increase

0.5

0.5

0

0

1

2

3

4

5

6

7

0

0

1

2

3

4

5

6

7

Queue size 5 ticks from opposite quote
1.0

Queue size

Probability of increase

0.5

0

0

1

2

3

4

5

6

7

Queue size

4.3. An Application to High-Frequency Trading The conditional probabilities described in the above section may be used as a building block to construct systematic trading strategies. Such strategies fall into the realm of statistical arbitrage because they do not guarantee a profit, but lead to trades with positive expected returns and bounded losses. As a final exercise, we provide the reader with one such example based on our results in §3.2 on the probability that the midprice increases, conditional on the configuration of the book. In particular, using Equation (19), we can compute the probability that the midprice increases given that the spread is S = 2, the number of orders at the bid is XB 0 = b = 3, and the number of orders at the ask XA 0 = a = 1. A simple application of our Laplace transform results, with our estimated parameters for Sky Perfect Communications given in Table 2, yields a probability 0 62 of the midprice increasing. We use this as the basis for the following strategy, which we test in simulation: Entering the position. If the spread is S = 1, the number of orders at the bid is XB 0 3, the number of orders at the ask is XA 0 = 1 and the number of orders at the second-best ask is XpA 0 +1 0 1, then submit a market buy order. Right after this trade, if XpA 0 +1 0 = 1, the new configuration of the order book will have XB 0+ = XB 0 3, XA 0+ = XpA 0 +1 0 = 1,

and the spread will be S = 2. In this scenario, the probability of the midprice increasing is now 0 62, as stated above, and we have entered the position at the current midprice. Thus, we are in a good position to make a profit. In the case where XpA 0 +1 0 = 0, the order was bought at a price XA 0 , which is strictly lower than the new midprice XB 0+ + XA 0+ /2 1 XA 0 − 1 + XA 0 + 2 /2 = XA 0 + 2 . In order for the trade to be welldefined, we must define an exit strategy. Exiting the position. We submit a market sell order at the first time such that either > pA 0 , in which case we are selling at a price 1. pB that is strictly greater than our buying price, or = pB 0 and XB = 1, which results in a loss 2. pB of one tick. The probability of success of this round-trip transaction need not be recomputed in real time: if an “offline” computation (for example, using Laplace transform methods described in §3) indicates that the probability in (19) is large, this suggests that this strategy would perform well. Comparing this probability across different stocks may be a good indicator of the profitability of this strategy. After running our simulation for 15,788 trades, roughly the equivalent of 30 days of trading, our algorithm does a total of 2,376 round-trip trades, and we display the P&L

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics
Operations Research 58(3), pp. 549–563, © 2010 INFORMS

561

Table 3.

Probability of an increase in mid-price: empirical frequencies (top), simulation results (95% confidence intervals, middle), and Laplace transform method results (bottom). a b 1 2 3 4 5 1 0 512 0 691 0 757 0 806 0 822 2 0 304 0 502 0 601 0 672 0 731 3 0 263 0 444 0 533 0 580 0 640 a 4 0 242 0 376 0 472 0 529 0 714 5 0 226 0 359 0 409 0 484 0 606

b 1 2 3 4 5

1 0.499 ± 0.006 0.663 ± 0.005 0.743 ± 0.006 0.788 ± 0.005 0.811 ± 0.004

2 0.333 ± 0.005 0.495 ± 0.006 0.589 ± 0.006 0.652 ± 0.006 0.693 ± 0.005

3 0.258 ± 0.005 0.411 ± 0.006 0.506 ± 0.006 0.564 ± 0.006 0.615 ± 0.006 a

4 0.213 ± 0.005 0.346 ± 0.005 0.434 ± 0.006 0.503 ± 0.006 0.547 ± 0.006

5 0.187 ± 0.005 0.307 ± 0.005 0.389 ± 0.006 0.452 ± 0.006 0.504 ± 0.006

b 1 2 3 4 5

1 0 500 0 664 0 741 0 784 0 812

2 0 336 0 500 0 593 0 652 0 693

3 0 259 0 407 0 500 0 563 0 609

4 0 216 0 348 0 437 0 500 0 548

5 0 188 0 307 0 391 0 452 0 500

distribution in Figure 4. Note that the computed probability of 0.62 is not directly linked to the probability of the trade being successful, which may only be computed through simulation. Indeed, the probability of success of each round-trip transaction is less than 0.5, although the average profit of each trade was 0 068 ticks, or 6 8 yen. The analysis of the above trading strategy does not take into

account transaction costs, but these can easily be included in the analysis.

5. Conclusion
We have proposed a stylized stochastic model describing the dynamics of a limit order book, where the occurrences of market events—market orders, limit orders

Table 4.

Probability of executing a bid order before a change in midprice: simulation results (95% confidence intervals, top) and Laplace transform method results (bottom). a b 1 2 3 4 5

1 0.498 ± 0.004 0.299 ± 0.004 0.204 ± 0.004 0.152 ± 0.003 0.117 ± 0.003

2 0.642 ± 0.004 0.451 ± 0.004 0.335 ± 0.004 0.264 ± 0.004 0.213 ± 0.004

3 0.709 ± 0.004 0.536 ± 0.004 0.422 ± 0.004 0.344 ± 0.004 0.291 ± 0.004 a

4 0.748 ± 0.004 0.592 ± 0.004 0.484 ± 0.004 0.403 ± 0.004 0.342 ± 0.004 0.779 0.632 0.532 0.450 0.394

5 ± 0.004 ± 0.004 ± 0.004 ± 0.004 ± 0.004

b 1 2 3 4 5

1 0 497 0 302 0 206 0 152 0 118

2 0 641 0 449 0 336 0 263 0 213

3 0 709 0 535 0 422 0 344 0 287

4 0 749 0 591 0 483 0 404 0 346

5 0 776 0 631 0 528 0 452 0 393

Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics

562 Table 5.

Operations Research 58(3), pp. 549–563, © 2010 INFORMS

Probability of making the spread: simulation results (95% confidence intervals, top) and Laplace transform method results (bottom). a b 1 2 3 4 5

1 0.268 ± 0.004 0.306 ± 0.004 0.312 ± 0.004 0.301 ± 0.004 0.286 ± 0.004

2 0.306 ± 0.004 0.384 ± 0.004 0.406 ± 0.004 0.411 ± 0.004 0.401 ± 0.004

3 0.312 ± 0.004 0.406 ± 0.004 0.441 ± 0.004 0.455 ± 0.004 0.456 ± 0.004 a

4 0.301 ± 0.004 0.411 ± 0.004 0.455 ± 0.004 0.473 ± 0.004 0.485 ± 0.004 0.286 0.401 0.456 0.485 0.491

5 ± 0.004 ± 0.004 ± 0.004 ± 0.004 ± 0.004

b 1 2 3 4 5

1 0 266 0 308 0 309 0 300 0 288

2 0 308 0 386 0 406 0 406 0 400

3 0 309 0 406 0 441 0 452 0 452

4 0 300 0 406 0 452 0 471 0 479

5 0 288 0 400 0 452 0 479 0 491

and cancellations—are governed by independent Poisson processes. The formulation of the model, which can be viewed as a queuing system, is entirely based on observable quantities so that its parameters can be easily estimated from observations of events in an actual order book. The model is simple enough to allow semianalytical computation of various conditional probabilities of order book events via Laplace transform methods, yet rich enough to adequately capture the short-term behavior of the order book: conditional distributions of various quantities of interest show good agreement with the corresponding empirical distributions for parameters estimated from data sets from the Tokyo Stock Exchange. The ability of our model to compute conditional distributions is useful for short-term prediction and design of automated trading strategies. Finally, simulation results illustrate that our model also yields realistic features for long-term (steady-state) average behavior of the order book profile and of price volatility. One by-product of this study is to show how far a stochastic model can go in reproducing the dynamic properties of a limit order book without resorting to Figure 4.
0.6 0.5 0.4 0.3 0.2 0.1 0 –1 0 1 2 3 4 5 6

detailed behavioral assumptions about market participants or introducing unobservable parameters describing agent preferences, as in the market microstructure literature. This model can be extended in various ways to take into account a richer set of empirically observed properties (Bouchaud et al. 2008). Correlation of the order flow with recent price behavior can be modeled by introducing statedependent intensities of order arrivals. The heterogeneity of order sizes, which appears to be an important ingredient in actual order book dynamics, can be incorporated by making order sizes independent and identically distributed random variables. Both of these features would conserve the Markovian nature of the process. A more realistic distribution of interevent times may also be introduced by modelling the event arrivals via renewal processes. It remains to be seen whether the analytical tractability of the model can be preserved when such generalities are introduced. We look forward to exploring some of these extensions in future work.

Acknowledgments
The authors thank Ning Cai, Alexander Cherny, Jim Gatheral, Zongjian Liu, Peter Randolph, and Ward Whitt for useful discussions. References
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Probability distribution of P&L per roundtrip trade, in ticks.

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