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operationsresearch informs vol 58 no 3 may june 2010 pp 549 563 issn0030 364x eissn1526 5463 10 5803 0549 doi10 1287 opre 1090 0780 2010 informs astochastic model for order ...

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                OPERATIONSRESEARCH                                                                                                                informs
                                                                                                                                                                ®
                Vol. 58, No. 3, May–June 2010, pp. 549–563
                issn0030-364Xeissn1526-54631058030549                                                                              doi10.1287/opre.1090.0780
                                                                                                                                               ©2010 INFORMS
                            AStochastic Model for Order Book Dynamics
                                                                                RamaCont
                                  Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027,
                                                                             rama.cont@columbia.edu
        .org/.                                                               Sasha Stoikov
        ms                                             Cornell Financial Engineering Manhattan, New York, New York 10004,
     author(s).or                                                            sashastoikov@gmail.com
     the.inf                                                                   Rishi Talreja
     to nals                      Department of Industrial Engineering and Operations Research, Columbia University, New York, New York 10027,
                                                                              rt2146@columbia.edu
     tesy               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
     courhttp://jour    dynamics, and its analytical tractability allows for fast computation of various quantities of interest without resorting to
     a  at              simulation. We describe a simple parameter estimation procedure based on high-frequency observations of the order book
     as le              and illustrate the results on data from the Tokyo Stock Exchange. Using simple matrix computations and Laplace transform
     y                  methods, we are able to ef“ciently compute probabilities of various events, conditional on the state of the order book: an
        ailab           increase in the midprice, execution of an order at the bid before the ask quote moves, and execution of both a buy and a
        v               sell order at the best quotes before the price moves. Using high-frequency data, we show that our model can effectively
     copa
        is              capture the short-term dynamics of a limit order book. We also evaluate the performance of a simple trading strategy based
     this,              on our results.
                        Subject classi“cations: limit order book; “nancial engineering; Laplace transform inversion; queueing systems;
     utedpolicies          simulation.
     ib                 Area of review: Financial Engineering.
     distr              History: Received September 2008; revision received March 2009; accepted August 2009. Published online in Articles in
                           Advance February 26, 2010.
     andmission
        per
     ticle         The evolution of prices in “nancial markets results from                 some insight into the interplay between order ”ow, liquid-
     ar and     the interaction of buy and sell orders through a rather com-                ity, and price dynamics (Bouchaud et al. 2002, Smith et al.
                plex dynamic process. Studies of the mechanisms involved                    2003, Farmer et al. 2004, Foucault et al. 2005). At the level
     thisights  in trading “nancial assets have traditionally focused on                    of applications, such models provide a quantitative frame-
        r
     to         quote-driven markets, where a market maker or dealer cen-                   work in which investors and trading desks can optimize
                tralizes buy and sell orders and provides liquidity by set-                 trade execution strategies (Alfonsi et al. 2010, Obizhaeva
        includingting bid and ask quotes. The NYSE specialist system is                     and Wang 2006). An important motivation for modelling
     yright     an example of this mechanism. In recent years, electronic                   high-frequency dynamics of order books, is to use the infor-
     cop        communications networks (ECNs) such as Archipelago,                         mation on the current state of the order book to predict
        mation, Instinet, Brut, and Tradebook have captured a large share                   its short-term behavior. We focus, therefore, on conditional
        or      of the order ”ow by providing an alternative order-driven                   probabilities of events, given the state of the order book.
     holdsinf   trading system. These electronic platforms aggregate all                       The dynamics of a limit order book resembles in many
                outstanding limit orders in a limit order book that is avail-               aspects that of a queuing system. Limit orders wait in a
                able to market participants and market orders are exe-                      queue to be executed against market orders (or canceled).
                cuted against the best available prices. As a result of                     Drawing inspiration from this analogy, we model a limit
     INFORMSAdditionalthe ECN’s popularity, established exchanges such as the               order book as a continuous-time Markov process that tracks
                NYSE, NASDAQ, the Tokyo Stock Exchange, and the                             the number of limit orders at each price level in the book.
                London Stock Exchange have adopted electronic order-                        The model strikes a balance between three desirable fea-
                driven platforms, either fully or partially through “hybrid”                tures: it can be estimated easily using high-frequency data,
                systems.                                                                    it reproduces various empirical features of order books, and
                   The absence of a centralized market maker, the mechan-                   it is analytically tractable. In particular, we show that our
                ical nature of execution of orders and, last but not least,                 model is simple enough to allow the use of Laplace trans-
                the availability of data have made order-driven markets                     form techniques from the queuing literature to compute
                interesting candidates for stochastic modelling. At a funda-                various conditional probabilities. These include the prob-
                mental level, models of order book dynamics may provide                     ability of the midprice increasing in the next move, the
                                                                                       549
                                                                             Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics
              550                                                                               Operations Research 58(3), pp. 549–563, ©2010 INFORMS
              probability of executing an order at the bid before the ask       1. A Continuous-Time Model for a
              quote moves, and the probability of executing both a buy              Stylized Limit Order Book
              and a sell order at the best quotes before the price moves,
              given the state of the order book. Although here we only          1.1. Limit Order Books
              focus on these events, the methods we introduce allow one         Consider a “nancial asset traded in an order-driven market.
              to compute conditional probabilities involving much more          Market participants can post two types of buy/sell orders. A
              general events such as those involving latency associated         limit order is an order to trade a certain amount of a security
              with order processing (see Remark 1). We illustrate our           at a given price. Limit orders are posted to a electronic
              techniques on a model estimated from order book data for          trading system, and the state of outstanding limit orders can
       .org/. a stock on the Tokyo Stock Exchange.                              be summarized by stating the quantities posted at each price
    author(s).ms Related literature. Various recent studies have focused        level: this is known as the limit order book. The lowest
       or     on limit order books. Given the complexity of the struc-          price for which there is an outstanding limit sell order is
    the.inf   ture and dynamics of order books, it has been dif“cult            called the ask price and the highest buy price is called the
    to nals   to construct models that are both statistically realistic and     bid price.
              amenable to rigorous quantitative analysis. Parlour (1998),         Amarket order is an order to buy/sell a certain quantity
    tesy      Foucault et al. (2005), and Rosu (2009) propose equilib-          of the asset at the best available price in the limit order
              rium models of limit order books. These models provide            book. When a market order arrives it is matched with the
    courhttp://jourinteresting insights into the price formation process, but   best available price in the limit order book, and a trade
    a  at     contain unobservable parameters that govern agent prefer-         occurs. The quantities available in the limit order book are
    as le     ences. Thus, they are dif“cult to estimate and use in appli-      updated accordingly.
    y         cations. Some empirical studies on properties of limit order        Alimit order sits in the order book until it is either exe-
       ailab  books are Bouchaud et al. (2002), Farmer et al. (2004),
       v                                                                        cuted against a market order or it is canceled. A limit order
    copa      and Holli“eld et al. (2004). These studies provide an exten-
       is     sive list of statistical features of order book dynamics that     may be executed very quickly if it corresponds to a price
    this,                                                                       near the bid and the ask, but may take a long time if the
              are challenging to incorporate in a single model. Bouchaud        market price moves away from the requested price or if the
    uted      et al. (2008), Smith et al. (2003), Bovier et al. (2006),         requested price is too far from the bid/ask. Alternatively, a
    ib policiesLuckock (2003), and Maslov and Mills (2001) propose
              stochastic models of order book dynamics in the spirit of         limit order can be canceled at any time.
    distr     the one proposed here, but focus on unconditional/steady–           We consider a market where limit orders can be placed
       missionstate distributions of various quantities rather than the con-    on a price grid 1nrepresenting multiples of a price
    andper    ditional quantities we focus on here.                             tick. The upper boundary n is chosen large enough so that
                 The model proposed here is admittedly simpler in struc-        it is highly unlikely that orders for the stock in question are
    ticleand  ture than some others existing in the literature: It does not     placed at prices higher than n within the time frame of our
    ar        incorporate strategic interaction of traders as in the game-      analysis. Because the model is intended to be used on the
    thisights theoretic approaches of Parlour (1998), Foucault et al.           time scale of hours or days, this “nite boundary assumption
       r                                                                        is reasonable. We track the state of the order book with
    to        (2005), and Rosu (2009), nor does it account for “long
                                                                                a continuous-time process Xt ≡ X tX t        ,
              memory” features of the order ”ow as pointed out by                                                       1          n    t0
                                                                                where X t is the number of outstanding limit orders at
              Bouchaud et al. (2002, 2008). However, contrarily to these                 p
    yrightincludingmodels, it leads to an analytically tractable framework      price p,1pn.IfXpt<0, then there are ŠXpt bid
                                                                                orders at price p;ifX t > 0, then there are X t ask
              where parameters can be easily estimated from empirical                                  p                           p
    cop       data and various quantities of interest may be computed           orders at price p.
                                                                                  The ask price p t at time t is then de“ned by
       mation,ef“ciently.                                                                         A
    holdsor      Outline. The paper is organized as follows. Section 1
       inf                                                                      p t=infp=1nX t>0∧n+1
              describes a stylized model for the dynamics of a limit             A                          p
              order book, where the order ”ow is described by inde-             Similarly, the bid price p t is de“ned by
              pendent Poisson processes. Estimation of model param-                                      B
              eters from high-frequency order book time-series data is          p t≡supp=1nX t<0∨0
    INFORMSAdditionaldescribed in §2 and illustrated using data from the Tokyo   B                          p
              Stock Exchange. In §3 we show how this model can be               Notice that when there are no ask orders in the book we
              used to compute conditional probabilities of various types        force an ask price of n + 1, and when there are no bid
              of events relevant for trade execution using Laplace trans-       orders in the book we force a bid price of 0. The midprice
              form methods. Section 4 explores steady-state properties of       p t and the bid-ask spread p t are de“ned by
                                                                                 M                              S
              the model using Monte Carlo simulation, compares condi-                    p t+p t
              tional probabilities computed by simulation to those com-         p t≡ B          A      and   p t≡p tŠp t
                                                                                 M             2                 S       A       B
              puted with the Laplace transform methods presented in §3,
              and analyzes a high-frequency trading strategy based on             Because most of the trading activity takes place in the
              our results in §4.3. Section 5 concludes.                         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
               of the number of outstanding orders at a given distance                • Limit buy (respectively sell) orders arrive at a dis-
               from the bid/ask. To this end, we de“ne                             tance of i ticks from the opposite best quote at independent,
                        ⎧                                                          exponential times with rate 	i,
                        ⎨X        t   0p t
    and                                                    A                                                      B               B
       per     quantity at level p x →xpŠ1                                        x→xpBt+1      with rate 

                 • a limit sell order at price level p>p t increases the
    ticleand                                               B                              p tŠ1
    ar         quantity at level p x →xp+1                                        x→xA            with rate 

       ights     • a market buy order decreases the quantity at the ask            x→xp+1 with rate p tŠpx  for p

p t B p B p t+1 price: x →x B yrightincluding• a cancellation of an oustanding limit buy order at price In practice, the ask price is always greater than the bid level p

p t decreases the quantity at level p x→xpŠ1 mation, B or The evolution of the order book is thus driven by the ≡x∈n∃kl∈ s.t. 1klnx 0 for pl holdsinf p incoming ”ow of market orders, limit orders, and cancella- x =0 for kplx 0 for pk (3) tions at each price level, each of which can be represented p p as a counting process. It is empirically observed (Bouchaud If the initial state of the book is admissible, it remains et al. 2002) that incoming orders arrive more frequently admissible with probability one: INFORMSAdditionalin the vicinity of the current bid/ask price and the rate of arrival of these orders depends on the distance to the Proposition 1. If X0 ∈ , then  Xt ∈  bid/ask. ∀t 0=1. To capture these empirical features in a model that is Proof. It is easily veri“ed that  is stable under each analytically tractable and allows computation of quantities of the six transitions de“ned above, which leads to our of interest in applications, most notably conditional prob- assertion.  abilities of various events, we propose a stochastic model where the events outlined above are modelled using inde- Proposition 2. If ≡ min1in i > 0, then X is an pendent Poisson processes. More precisely, we assume that, ergodic Markov process. In particular, X has a proper sta- for i 1, tionary distribution. Cont, Stoikov, and Talreja: A Stochastic Model for Order Book Dynamics 552 Operations Research 58(3), pp. 549–563, ©2010 INFORMS  Proof. Let N ≡Ntt0, where Nt≡ n X t, 2. Parameter Estimation p=1 p and let N be a birth-death process with birth rate given by ≡2n p and death rate in state i, i ≡2 +i . 2.1. Description of the Data Set p=1 Notice that N increases by one at a rate bounded from Our data consist of time-stamped sequences of trades (mar- above by and decreases by one at a rate bounded from ket orders) and quotes (prices and quantities of outstanding below by ≡ 2 + i when in state i. Thus, for all limit orders) for the “ve best price levels on each side of the i k t 0, N is stochastically bounded by N.Fork1, let T0 order book, for stocks traded on the Tokyo stock exchange and Tk denote the duration of the kth visit to 0 and the over a period of 125 days (Aug.–Dec. 2006). This data set, Š0 duration between the k Š1th and kth visit to 0 of pro- referred to as Level II order book data, provides a more .org/. k k cess N, respectively. De“ne random variables T and T , detailed view of price dynamics than the trade and quotes ms 0 Š0 author(s).ork1,for process N similarly. Then the point process with (TAQ) data often used for high-frequency data analysis, interarrival times T1 T1T2 T2and the point process which consist of prices and sizes of trades (market orders) .inf Š0 0 1 Š0 1 0 2 2 the with interarrival times T T T T  are alternating and time-stamped updates in the price and size of the bid Š0 0 Š0 0 to nals renewal processes. By Theorem VI.1.2 of Asmussen (2003) and ask quotes. and the fact that N is stochastically dominated by N,we In Table 1, we display a sample of three consecutive tesy then have for each k 1, trades for Sky Perfect Communications. Each row provides k the time, size, and price of a market order. We also display courhttp://jour Ɛ T  0 =lim Nt=0 a sample of Level II bid-side quotes. Each row displays the a at Ɛ Tk+Ɛ Tk  t→ “ve bid prices (pb1, pb2, pb3, pb4, pb5), as well as the le 0 Š0 quantity of shares bid at these respective prices (qb1, qb2, as k Ɛ T  0 y ailab lim Nt=0= k k  (4) qb3, qb4, qb5). t→ v Ɛ T +Ɛ T  copa 0 Š0 is 2.2. Estimation Procedure this, Notice that in state 0 both N and N have birth rate . Thus, Recall that in our stylized model we assume orders to be k k 1 of “unit” size. In the data set, we “rst compute the average Ɛ T =Ɛ T =  (5) utedpolicies 0 0 sizes of market orders S , limit orders S , and canceled ib m l Combining (4) and (5) gives us orders Sc and choose the size unit to be the average size distr of a limit order S . The limit order arrival rate function for mission l Ɛ Tk k 1i5canbe estimated by andper Š0Ɛ TŠ0 (6) ˆ Nli ticleand To show N is ergodic, notice the inequalities i=  T ar    i ∗  i  < 1 =e / Š1< (7) where N i is the total number of limit orders that arrived thisights l r ··· i! i=1 1 i i=1 at a distance i from the opposite best quote, and T is to the total trading time in the sample (in minutes). N ∗ and li is obtained by enumerating the number of times that a quote   including M  i increases in size at a distance of 1  i  5 ticks from the yright  1··· i > 1··· i+  2 +M = (8) opposite best quote. We then extrapolate by “tting a power i i cop i=1 i=1 i=M+1 law function of the form mation, for M>0chosenlargeenoughsothat2 +M > .There- k or ˆ holds i= inf fore, by Corollary 2.5 of Asmussen (2003), N is ergodic i k so that Ɛ T <.Combining this with the bound (6) and Š0 (suggested by Zovko and Farmer 2002 or Bouchaud et al. the fact that for each t  0 Xt=00 if and only if 2002). The power law parameters k and are obtained by Nt=0 shows that X is positive recurrent. Because X is a least-squares “t INFORMSAdditionalclearly also irreducible, it follows that X is ergodic.   The ergodicity of X is a desirable feature of theoretical 5 2 min ˆ k interest: it allows comparison of time averages of various k i=1 iŠ i  quantities in simulations (average shape of the order book, average price impact, etc.) to unconditional expectations of Estimated arrival rates at distances 0  i  10 from the these quantities computed in the model. The steady-state opposite best quote are displayed in Figure 1(a). behavior of X will be further discussed in §4.1. We note, The arrival rate of market orders is then estimated by however, that our results involving conditional probabilities in §3 and applications discussed in §4.3 do not rely on this ˆ = Nm Sm ergodicity result. T S  ∗ l

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...Operationsresearch informs vol no may june pp issn x eissn doi opre astochastic model for order book dynamics ramacont department of industrial engineering and operations research columbia university new york rama cont edu org sasha stoikov ms cornell financial manhattan author s or sashastoikov gmail com the inf rishi talreja to nals rt tesy we propose a continuous time stochastic limit strikes balance between three desirable features it can be estimated easily from data captures key empirical properties courhttp jour its analytical tractability allows fast computation various quantities interest without resorting at simulation describe simple parameter estimation procedure based on high frequency observations as le illustrate results tokyo stock exchange using matrix computations laplace transform y methods are able efciently compute probabilities events conditional state an ailab increase in midprice execution bid before ask quote moves both buy v sell best quotes price show that ou...

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