A Dynamic Model of the Limit Order Book

Transcription

1 A Dynamic Model of the Limit Ode Book Ioanid Roşu Univesity of Chicago This pape pesents a model of an ode-diven maket whee fully stategic, symmetically infomed liquidity tades dynamically choose between limit and maket odes, tading off execution pice and waiting costs. In equilibium, the bid and ask pices depend only on the numbes of buy and sell odes in the book. The model has a numbe of empiical pedictions: (i) highe tading activity and highe tading competition cause smalle speads and lowe pice impact; (ii) maket odes lead to a tempoay pice impact lage than the pemanent pice impact, theefoe to pice oveshooting; (iii) buy and sell odes can cluste away fom the bid-ask spead, geneating a hump-shaped ode book; (iv) bid and ask pices display a comovement effect: afte, e.g., a sell maket ode moves the bid pice down, the ask pice also falls, by a smalle amount, so the bid-ask spead widens; (v) when the ode book is full, tades may submit quick, o fleeting, limit odes. (JEL C7, D4, G1) 1. Intoduction This pape pesents a model of pice fomation in an ode-diven maket, whee agents tade via a limit ode book. 1 Compaed with a quote-diven maket, in which maket makes povide liquidity by setting bid and ask quotes, in an ode-diven maket thee ae no designated maket makes. Instead, liquidity is offeed in a decentalized way, with anonymous tades who place odes in the limit ode book, and wait until the odes get executed. Nowadays, moe than half of the wold s stock exchanges ae ode diven, with a limit ode book at the cente of the tading pocess (see Jain 2003). 2 Yet, despite The autho thanks Rob Battalio, Shane Cowin, Thiey Foucault, Dew Fudenbeg, Xavie Gabaix, Lay Glosten, Buton Hollifield, Segei Izmalkov, Eugene Kandel, Leonid Kogan, Jon Lewellen, Juhani Linnainmaa, Andew Lo, David Musto, Stew Myes, Jun Pan, Chistine Palou, Anna Pavlova, Duane Seppi, Cheste Spatt, Richad Stanton, Dimiti Vayanos, and Jiang Wang fo helpful comments and suggestions. He is also gateful to paticipants at the NBER meeting, May 2004; WFA meeting, June 2005; and to semina audiences at MIT, Bekeley, Note Dame, Toonto, Kellogg, Canegie Mellon, Michigan, Whaton, and Chicago. Send coespondence to Ioanid Roşu, Booth School of Business, Univesity of Chicago, 5807 South Woodlawn Avenue, Chicago, IL 60637; telephone: ; fax: iosu@uchicago.edu. 1 The ode book is the collection of all outstanding limit odes. Limit odes ae pice-contingent odes to buy (sell) if the pice falls below (ises above) a pespecified pice. A sell limit ode is also called an offe (o ask), while a buy limit ode is also called a bid. The lowest offe is called the ask pice, o simply ask, and the highest bid is called the bid pice, o simply bid. 2 Examples of pue ode-diven makets include Euonext, Helsinki, Hong Kong, Swiss, Tokyo, Toonto, and vaious electonic communication netwoks (Island, Instinet, Achipelago). Thee ae also hybid exchanges (NYSE, NASDAQ, London), in which maket makes exist but have to compete with othe tades, who supply liquidity by limit odes. In these makets, the numbe of tansactions that involve a maket make is usually small (see Hasbouck and Sofianos 1993). C The Autho Published by Oxfod Univesity Pess on behalf of The Society fo Financial Studies. All ights eseved. Fo Pemissions, please jounals.pemissions@oxfodjounals.og. doi: /fs/hhp011 Advance Access publication Apil 2, 2009

2 The Review of Financial Studies / v 22 n thei inceasing impotance, the liteatue on ode-diven makets is elatively small. One of the easons fo the scacity of models of ode-diven makets is the shee complexity of the poblem. Unlike in the case of a quote-diven maket, whee only one o pehaps a few maket makes need to be modeled, a satisfactoy model of an ode-diven maket should explain how pices aise fom the inteaction of a lage numbe of anonymous tades, who aive in the maket at diffeent times, choose whethe to tade immediately o to wait, and can behave stategically by changing thei odes at any time. This pape pesents a tactable dynamic model of an ode-diven maket that eflects the featues mentioned above. To the autho s knowledge, this is the fist dynamic model in which agents ae allowed to feely modify and cancel thei limit odes. Supisingly, allowing tades to be fully stategic tuns out to simplify the poblem, athe than complicate it. The model has a numbe of empiical implications about the bid-ask spead, tading volume, the pice impact of tansactions, and the evolution of the limit ode book in time. Some of these implications povide a diffeent intepetation of known empiical facts about ode-diven makets, and some implications ae new, and can be used to test the model. It is inteesting that all these implications ae obtained in the absence of asymmetic infomation among tades. This is an advantage, because asymmetic infomation is had to measue and is usually not obsevable. Instead, this pape elates the shape of the limit ode book to obsevable quantities, such as the numbe of tades and thei aival ates. Moeove, the pape offes a diffeent intepetation of what detemines the shape of the limit ode book. In the maket micostuctue liteatue based on asymmetic infomation, 3 limit odes ae placed at diffeent levels because liquidity povides must potect themselves fom tades with supeio infomation. In paticula, the bid-ask spead is smalle, and the limit ode book is dense when thee is less asymmetic infomation in the maket. By contast, in this pape, limit odes ae placed on diffeent levels because tades have to be compensated fo thei waiting costs. Since the compensation is detemined by the speads between limit odes, when thee is moe tading activity (i.e., tades aive faste in the maket), liquidity povides do not wait that much and so can be compensated with smalle speads. Put diffeently, in this famewok, a maket is consideed liquid if it is fast and/o competitive. In the maket micostuctue based on asymmetic infomation, a maket is liquid if the amount of asymmetic infomation is small. Specifically, this pape consides a continuous-time, infinite-hoizon economy in which thee is only one asset with no dividends. Buyes and selles aive in the maket andomly. They eithe buy o sell one unit of the asset, afte 3 See, e.g., Glosten and Milgom (1985); Kyle (1985); Easley and O Haa (1987); and Glosten (1994). See O Haa (1995) fo a suvey. 4602

3 A Dynamic Model of the Limit Ode Book which they exit the model. It is assumed that all tades ae liquidity tades, in the sense that thei impulse to tade is exogenous to the model. Howeve, they ae discetionay liquidity tades in that they have a choice about when to tade and whethe to place a maket o limit ode. Afte a limit ode is placed, it can be canceled o changed at will. The execution of limit odes is subject to the usual pice pioity ule, and when pices ae equal, to the time pioity ule. All agents incu waiting costs i.e., a loss of utility fom waiting. Depending on whethe they have low o high waiting costs, tades ae patient o impatient. All infomation is common knowledge. To find the equilibium, one notices that all the tades on one side of the limit ode book must have the same expected utility othewise, they would immediately undecut each othe. And, because of the waiting costs, tades place thei limit odes on diffeent levels: fo example, a selle with a limit ode at a highe level gets a bette expected pice than a selle at a lowe level, but has to wait longe in such a way that both selles get the same expected utility. This implies that thee is a (Makov pefect) equilibium whee only the numbe of buyes and selles mattes. Theefoe, thei expected utility follows a ecusive system of diffeence equations. The ecusive system can be solved numeically, and in some cases in closed fom. In equilibium, impatient agents submit maket odes, while patient agents submit limit odes and wait, except fo the states in which the limit ode book is full. 4 In states in which the book is not full, new limit odes ae always placed inside the bid-ask spead. 5 The point whee the book is full coincides with the time when the bid-ask spead is at the minimum. That thee exists a nonzeo minimum bid-ask spead is an inteesting fact since the tick size is zeo in this model. Paticula cases of the model can be solved in closed fom. One impotant example is when thee ae only patient selles and impatient buyes (o buyes can place only maket odes). Section 3 shows how to use these fomulas to deive implications about the aveage bid-ask spead and pice impact and the aveage maximum numbe of tades in the book, and how to use the model to estimate in pactice the aival ates of patient and impatient tades. Fo example, in makets with highe tading activity (measued by the sum of aival ates of all agents) and with highe competition (measued by the atio of aival ates fo patient and impatient tades) both the bid-ask speads and the pice impact will be smalle. 6 An intiguing implication is that moe tading 4 When the book is full, some patient agent eithe places a maket ode o submits a quick (fleeting) limit ode, which some tade fom the othe side of the book immediately accepts. This comes theoetically as a esult of a game of attition among the buyes and selles. 5 In thei analysis of the Pais Bouse maket (now Euonext), Biais, Hillion, and Spatt (1995) obseve that the majoity of limit odes ae spead impoving. 6 These pedictions ae tested fo the Helsinki Exchange by Linnainmaa and Roşu (2008), using weathe as an instument. Tading activity has been shown to explain vaiation in speads, stating with Demsetz (1968). Some evidence that easons othe than infomation may bette explain pices can also be found, e.g., in Huang and 4603

4 The Review of Financial Studies / v 22 n activity leads to lowe asset volatility; moe pecisely, the volatility of the asset must vay in invese popotion to the squae of the tading activity. In ode to discuss pice impact and detemine the distibution of limit odes in the ode book, the model next allows multi-unit maket odes to aive with positive pobability. Then one can define both the tempoay (o instantaneous) pice impact function, which is the actual pice impact suffeed by the maket ode tade, and the pemanent (o subsequent) pice impact, which is the diffeence between the new ask pice and the ask pice befoe the maket ode was submitted. In this setup, the tempoay pice impact is lage than the pemanent pice impact, which is equivalent to pice oveshooting. The intuition is that, befoe a multi-unit maket ode comes, the tades who expect thei limit odes to be executed do not know the exact ode size, so they stay highe in the book. Once the size becomes known, the selles egoup lowe in the book. Also, if multi-unit maket odes aive with pobabilities that do not decease too fast with ode size, then the pice impact function is typically fist concave and then convex. This is the same as saying that the limit odes cluste away fom the bid and the ask, o that the book exhibits a hump shape. 7 This aises because patient tades cluste away fom the bid-ask spead when they expect to take advantage of lage maket odes that ae not too unlikely. The geneal, two-sided case is moe difficult, and the solution must be found numeically. An empiical implication is the comovement effect between bid and ask pices. 8 Fo example, a maket sell ode not only deceases the bid pice due to the mechanical execution of limit odes on the buy side but also subsequently deceases the ask pice. Moeove, the decease in the bid pice is lage than the subsequent decease in the ask pice, which leads to a wide bid-ask spead. The comovement effect is stonge when thee ae moe limit tades on the side of the subsequent pice move, and the competition among them is stonge. Anothe consequence of the geneal case is the existence of quick (o fleeting) limit odes: when the limit ode book becomes full, a buye o selle places a limit ode, and a limit tade on the othe side immediately accepts it by canceling the limit ode and placing a maket ode. In this model, fleeting limit odes appea only when the ode book is full, and ae always placed between the bid and the ask. The comovement effect and the pesence of tempoay and pemanent pice impact aise the issue of compaing the pedictions of the cuent model based on waiting costs with the pedictions of a model based on asymmetic Stoll (1997), who estimate that on aveage appoximately 90% of the bid-ask spead is due to noninfomational fictions ( ode-pocessing costs ). 7 See Bouchaud, Mezad, and Pottes (2002, Figue 2); o Biais, Hillion, and Spatt (1995, p ff.). 8 Biais, Hillion, and Spatt (1995) document a comovement effect in the Pais Bouse (now Euonext), and suggest an explanation based on asymmetic infomation. 4604

5 A Dynamic Model of the Limit Ode Book infomation. Section 6 suggests one way of telling these two stoies apat, based on the behavio of the bid-ask spead. The limit ode book has been analyzed in a vaiety of ways. The static models ae typically based on asymmetic infomation: see Glosten (1994); Chakavaty and Holden (1995); Handa and Schwatz (1996); Rock (1996); and Seppi (1997). The pesent pape is pat of a moe ecent liteatue that analyzes dynamic aspects of the limit ode book, usually in the absence of asymmetic infomation. A pecuso of this liteatue is the gavitational pull model of Cohen et al. (1981), in which tades choose between limit and maket odes based on thei expectations about the evolution of an exogenous pice pocess. Palou (1998) poposes a two-tick model whee tades choose between limit odes and maket odes afte taking into account the effect of thei decision on futue tades stategies; the model can explain vaious pattens in ode placement stategies, including the comovement effect. Foucault (1999) studies the choice between limit and maket odes and focuses on the nonexecution pobability of limit odes and the winne s cuse poblem fo the limit odes when they do execute. Goettle, Palou, and Rajan (2005) solve numeically fo the stationay Makov pefect equilibium in a model in which tades with pivate valuations choose whethe to submit a maket o a limit ode, and also choose the size of the ode. The pape most closely elated to the pesent one is that of Foucault, Kadan, and Kandel (2005), in which tades also face waiting costs, but cannot modify thei odes once submitted; they focus thei analysis on the bid-ask spead, maket esiliency, and time to execution fo limit odes. The pape is oganized as follows. Section 2 descibes the model. Section 3 solves fo the equilibium in a paticula case that epesents the sell side of the book: thee ae only patient selles and impatient buyes. Section 4 discusses the case of multi-unit maket odes and analyzes the pice impact. Section 5 descibes the equilibium in the geneal case with all types of selles and buyes and deives implications about the comovement effect and fleeting odes. Section 6 contasts the pesent model based on waiting costs to a model based on infomation. Section 7 concludes. 2. The Model 2.1 The maket This section descibes the assumptions of the model. Conside a maket fo an asset that pays no dividends. The buy and sell pices fo this asset ae detemined as the bid and ask pices esulting fom tading based on the ules given below. Thee is a constant ange A > B in which the pices lie at all times. 9 Moe specifically, thee is an infinite supply when pice is A, povided 9 One can think about A and B as summaizing the infomation about the asset: (A + B)/2 epesents the aveage value of the asset, while (A B) epesents diffeences of opinion among tades. See Roşu (2008) fo a theoetical model in which the bounds A and B vay ove time along with the efficient pice, and thei diffeence depends on agents pivate costs of tading. 4605

6 The Review of Financial Studies / v 22 n by agents outside the model. Similaly, thee is an infinite demand fo the asset when pice is B. Pices can take any value in this ange i.e., the tick size is zeo. Tading. The time hoizon is infinite, and tading in the asset takes place in continuous time. The only types of tades allowed ae maket odes and limit odes. The limit odes ae subject to the usual pice pioity ule, and, when pices ae equal, the time pioity ule is applied. If seveal maket odes ae submitted at the same time, only one of them is executed, at andom, while the othe odes ae canceled. 10 Limit odes can be canceled fo no cost at any time. 11 Thee is also no delay in tading, both types of odes being posted o executed instantaneously. Tading is based on a publicly obsevable limit ode book. Agents. Tades aive andomly in the maket, due to liquidity needs that ae not modeled hee. The aival pocess is assumed to be exogenous and is descibed in moe detail below. Once tades aive, they choose stategically between maket and limit odes. The tades ae eithe buyes o selles; thei type is fixed fom the beginning and cannot change. Buyes and selles tade at most one unit, afte which they exit the model foeve. 12 Tades ae isk neutal, so thei instantaneous utility function equals the pice fo selles, minus the pice fo buyes. As in Foucault, Kadan, and Kandel (2005), tades lose utility popotionally to thei expected waiting time. 13 If τ is the andom execution time and P τ is the pice obtained at τ, the expected utility of a selle with patience coefficient is f t = E t {P τ (τ t)}. (The expectation opeato takes as given the stategies of all the playes. See the desciption of the stategies below.) Similaly, the expected utility of a buye is g t = E t { P τ (τ t)}, whee we denote g t = E t {P τ + (τ t)}. Notice that g t equals minus the expected utility of a buye; this is done in ode to compae buyes and selles moe easily. The discount coefficient is a constant that can take only two values <. If =, the tade is called patient; and if =, the tade is called impatient. 10 To justify this assumption, think of a maket buy/sell ode as a (maketable) limit ode at a pice equal to the ask/bid. Then, if seveal maket odes ae submitted at the same time, one of them is andomly executed, while the othes emain as limit odes, which can be feely canceled. 11 In most financial makets, cancellation of a limit ode is fee, although one may ague that thee ae still monitoing costs. Foucault, Kadan, and Kandel (2005) conside a model with infinite monitoing costs, in which agents neve change thei odes once submitted. 12 This is a stong assumption: some agents might want to tade lage quantities o decide to emain in the maket to buy and sell secuities, thus in effect becoming maket makes. (Bloomfield, O Haa, and Saa 2005 show expeimentally that maket making aises endogenously in pue limit ode makets.) But, as long as liquidity supplies have some constaints due to inventoy easons o isk avesion, one could adopt a model simila to the pesent one but eplace the one-unit limit with an n-unit limit. 13 The pape is delibeately vague about the exact natue of the waiting costs. Besides the standad time discounting stoy, one can also think of the oppotunity cost of tading in the given asset. Futhemoe, one can intepet the waiting costs as uncetainty avesion, which inceases with the time hoizon. 4606

7 A Dynamic Model of the Limit Ode Book Fo simplicity, it is assumed that is much lage than, which implies that impatient tades always submit maket odes. 14 Aivals. The fou types of tades (patient buyes, patient selles, impatient buyes, and impatient selles) aive in the maket accoding to independent Poisson pocesses with constant, exogenous intensity ates λ PB, λ PS, λ IB, λ IS. 15 By definition, a Poisson aival with intensity λ implies that the numbe of aivals in any inteval of length T has a Poisson distibution with paamete λt. The inte-aival times of a Poisson pocess ae distibuted as an exponential vaiable with the same paamete λ. The mean time until the next aival is then 1/λ. In the est of the pape, to say that an event happens afte Poisson (ν) means that the event time coincides with the fist aival in a Poisson pocess with intensity ν. Stategies. Since this is a model of continuous tading, it is desiable to set the game in continuous time. Thee ae also technical easons why that would be useful: in continuous time, with Poisson aivals the pobability that two agents aive at the same time is zeo. This simplifies the analysis of the game. Anothe impotant benefit of setting the game in continuous time is that agents can espond immediately. Moe pecisely, one can use stategies that specify: Keep the limit ode at a 1 as long as the othe agent stays at a 2 o below. If at some time t the othe agent places an ode above a 2, then immediately afte t undecut at a 2. Immediate punishment allows simple solutions, wheeby existing tades do not need to change thei stategy until the aival of the next tade. Setting the game in continuous time, nevetheless, equies exta cae. We use the famewok developed in Roşu (2006), which allows fo multi-stage games and mixed stategies. 16 The types of equilibium used ae subgame pefect 14 This can be shown using Poposition 12 in the Appendix: in a game of attition with both patient and impatient tades, a significantly moe impatient tade has no eason to wait. 15 Biais, Hillion, and Spatt (1995) show empiically that aivals ae positively coelated: the diagonal effect ; and that they depend on the state of the ode book: the lage the bid-ask spead, the faste limit odes aive to supply liquidity (Hollifield et al ague that the opposite is tue on the Vancouve stock exchange). See Roşu (2008) fo a theoetical model that endogenizes enty decisions, and poduces positively coelated aivals that depend on the state of the ode book. 16 The main difficulty in continuous time game theoy comes fom the fact that given a time t, thee is no last time befoe t and no fist time afte t. The solution is to allow stategies with infinitesimal inetia (as in Begin and MacLeod 1993) and a unifomly bounded numbe of jumps (as in Simon and Stinchcombe 1989). Infinitesimal inetia means that agents do not change thei stategies in the infinitesimal time inteval [t, t + dt] (this is simila to continuous time finance, whee agents do not tade duing [t, t + dt]). The extension to multistage game theoy is needed because of maket odes: when a maket ode aives at time t, an existing limit tade exits the model, and the next stage of the game must take place with fewe tades, at the same time t. This can be done by stopping the clock, so that the next game is also played at t. Also, in continuous time thee can be both mixing ove actions, by choosing andomly an action in the stage game at time t;andmixing ove time, by stating with a deteministic choice at time t but changing the action andomly in the inteval (t, t + dt). Since in this pape natue mixes ove time by binging agents accoding to a Poisson pocess, it is most natual to conside stategies mixed ove time. 4607

8 The Review of Financial Studies / v 22 n equilibium, and Makov pefect equilibium (see Fudenbeg and Tiole 1991, chap. 13). Anothe impotant notion in this famewok is that of competitive Makov equilibium, which is a Makov pefect equilibium fom which local deviations can be stopped by local punishments assuming that the behavio in the est of the game does not change. In othe wods, if a local deviation impoves a tade s expected payoff ignoing what happens in the othe states of the game the tade would deviate. Also, fo the puposes of this pape, one intoduces the notion of igid equilibium, which is a competitive stationay Makov equilibium in which, if some agents have mixed stategies, mixing is done only by the agents with the most competitive limit odes (highest bid o lowest offe). 17 Finally, in this pape all infomation, including agents stategies and beliefs, is common knowledge. 3. Equilibium: One Side of the Book This section analyzes the paticula case in which all the selles ae patient and all the buyes ae impatient. (By symmety, one can deive simila esults fo the maket with only patient buyes and impatient selles.) So assume that the aival ates of patient buyes and impatient selles ae zeo: λ PB = λ IS = 0. To simplify notation, denote the aival ates of patient selles and impatient buyes, espectively, by λ 1 = λ PS and λ 2 = λ IB. This case is vey tactable and has closed-fom solutions. Moeove, while potentially useful in its own ight, it is vey impotant in getting intuition about the geneal case. Indeed, one can egad the sell side of a geneal limit ode book as being diven only by patient selles and impatient buyes, except that the lowe bound B of the sell side is not fixed, but equals the bid pice i.e., the highest bid on the buy side. 3.1 Main intuition Suppose that the limit ode book is empty, and a patient selle labeled 1 aives fist in the maket. Then tade 1 submits a limit sell ode at the maximum level a 1 = A and emains a monopolist until some othe tade aives. 18 Suppose that a second patient selle labeled 2 aives. Now both selles compete fo maket odes fom the incoming impatient buyes. If tade 1 could not cancel his limit ode at A, then tade 2 would undecut by placing 17 In the language of Coollay 2 in the Appendix, in case 4 only equilibia of type c occu. 18 It is assumed implicitly that if the only limit sell odes in the book ae at A, a maket ode fist cleas the odes in the book, and only then elies on the infinite supply at A. 4608

9 A Dynamic Model of the Limit Ode Book a limit ode at a 2 = A δ fo some vey small δ. 19 He expected utility would then be stictly lage than that of tade 1. But tade 1 can change his limit ode, so a pice wa would follow. Undecutting happens instantaneously in this model, because the game is set in continuous time (see the discussion about stategies in Section 2). As a esult, tade 1 does not need to change his limit ode as long as tade 2 places he limit ode at some level a 2 < a 1 = A that is low enough. To find the coect level fo a 2, note that both tades must have the same expected utility in equilibium. If tade 2 placed he ode above a 2 whee she had highe expected utility than tade 1, then tade 1 would immediately undecut by a penny, and so on. So, in the equilibium with two selles, tade 1 has a limit ode at a 1 = A, and tade 2 has a limit ode at a 2 < A. Both tades have the same expected utility: tade 1 obtains in expectation a highe pice than tade 1, but waits longe to get his ode executed. 20 This leads to an equilibium in which patient selles compete with offes at diffeent pices in ode to extact ents fom the impatient buyes. 21 In solving fo the equilibium, it is supising that allowing agents to feely cancel o modify thei limit odes, instead of complicating the solution, actually simplifies it. This happens because the theat of undecutting makes all patient selles have the same expected utility in equilibium. This makes the equilibium Makov, with the numbe of selles as a state vaiable. An impotant popety of this equilibium is that it is competitive, in the sense that a local deviation fom one of the tades can be stopped by anothe tade s immediate undecutting, assuming that the est of the equilibium behavio does not change. One can also imagine a noncompetitive equilibium. Fo example, suppose that all patient selles queue thei limit odes at A until the expected utility of the last tade equals the esevation value B. This equilibium is sustained by Nash theats: tade 1 theatens with competitive behavio if tade 2 does not queue behind him at A. Tade 2 is bette off complying as long as she expects tade 3 to do the same and queue behind he. This equilibium is noncompetitive because punishment implies that behavio in the est of the game will be changed (to the competitive equilibium). Noncompetitive 19 In Foucault, Kadan, and Kandel (2005), even though tade 1 cannot cancel his limit ode, tade 2 would still undecut by moe than a penny in equilibium. This is because the stategy of futue aiving buyes depends on the level of tade 2 s limit ode: the highe it is, the less likely it is that a buye will place a maket ode. Theefoe, in thei model it is impotant that tades on the othe side of the limit ode book ae able to place limit odes. In this model, the main intuition is obust egadless whethe it is a one-sided o a two-sided limit ode book. 20 The intuition is suppoted by empiical wok. Lo, Mackinlay, and Zhang (2002) document that execution times ae vey sensitive to the limit pice (but not to the limit ode size). See also Hollifield, Mille, and Sandas (2004). 21 Unlike the Betand pice competition, in this model tades make positive expected pofits: waiting costs geneate an equilibium with offes at diffeent pices, leading to an expected utility above thei esevation value B. This is consistent with the empiical liteatue, which shows that limit ode tades make positive pofits: see Sandås (2001) fo the case of Stockholm Stock Exchange and Hais and Hasbouck (1996) fo NYSE s SupeDOT system. See also Biais, Matimot, and Rochet (2000) fo a theoetical model in which maket makes competing with supply schedules make positive pofits. 4609

10 The Review of Financial Studies / v 22 n equilibia may be impotant, fo example, in undestanding deale makets. 22 The pesent pape focuses on competitive equilibia, since they ae the moe likely outcome of lage, anonymous ode-diven makets. 3.2 Desciption of the equilibium Conside a limit ode book with uppe bound A; the lowe bound B (which is the esevation value of the selles, as they can always submit a maket ode fo B and exit the model); the selles patience coefficient ; the patient selle aival ate λ 1 ; and the impatient buye aival ate λ 2. Befoe a moe fomal discussion of the esults, some intuition is given about how the equilibium woks (poofs will be given late). As discussed above, in equilibium, all patient selles in the ode book have the same expected utility. Denote the numbe of selles by m, and thei expected utility by f m. This means that, in a Makov equilibium, the numbe of selles m is a state vaiable. The numbe of m evolves accoding to a Makov pocess, and so the selles utility f m satisfies a system of equations, called the ecusive system. The numbe of states must be finite: othewise, the expected execution time of the top-limit selle would be infinite, hence his utility would be negative infinity; but then he would athe submit a maket ode at the lowe bound B, which is the esevation value of the selles. As the numbe of selles m inceases, each selle is stictly wose off, and the ask pice will decease. Denote by M the lagest numbe of limit odes the equilibium book can accommodate. A limit ode book with the maximum numbe of odes M is called full. In that case, it must be that the expected utility of each of the existing M selles equals f M = B: othewise, if f M > B an incoming patient selle would want to join in to get moe than the esevation value B. Now,if the selles utility f M exactly equals the esevation value B, it must be that one of the selles (by choice the bottom selle i.e., the one with the lowest offe) has a mixed stategy: afte Poisson(ν) the bottom selle places a maket ode at B and exits. Obseve that fom a limit ode book with m selles (m = 1,...,M 1), the maket can go eithe to state m + 1 if a patient selle aives afte andom time T 1 exp(λ 1 ); o to state m 1 if an impatient buye aives afte andom time T 2 exp(λ 2 ). Inte-aival times of Poisson pocesses ae exponentially distibuted, so the aival of the fist of the two states happens at T = min(t 1, T 2 ), which is exponential with intensity λ 1 + λ 2 (hence the 1 λ 1 λ 1 +λ 2, expected value of T is λ 1 +λ 2 ). The fist event happens with pobability while the second event happens with pobability λ 2 λ 1 +λ 2. One obtains the fomula f m = λ 1 λ 1 + λ 2 f m+1 + λ 2 λ 1 + λ 2 f m 1 1 λ 1 + λ Chistie and Schultz (1994) document collusion among NASDAQ deales in the ealy 1990s. Thei pape contibuted to the 1997 intoduction in NASDAQ of a public limit ode book. 4610

11 A Dynamic Model of the Limit Ode Book Notice also that thee ae two types of selles: the bottom selle, who has a limit sell ode at the ask pice a m, and the othe selles, who have thei limit odes above a m. If an impatient buye aives and places a maket ode, the bottom selle eceives a m, while the othe selles get f m 1, which is the expected utility of selles in an ode book with m 1 selles. Since all selles must have the same expected utility in equilibium, it must be that the ask pice a m equals the expected utility whee thee is one less selle: a m = f m 1. Fom the state with the maximum numbe of selles M, the system can go only to state M 1, eithe if an impatient buye aives afte andom time T 1 exp(λ 2 ) o if the selle with the cuent bottom limit ode places a maket ode at B and exits afte andom time T 2 exp(ν). 23 Then one obtains the fomula f M = f M 1 1 λ 2 + ν. Define also f 0 = A. 24 In conclusion, f m satisfies a system of diffeence equations, called the ecusive system. Definition 1. Stat with the limit ode book uppe bound A and lowe bound B, the aival ates of patient selles λ 1 and impatient buyes λ 2, and the selles patience coefficient. Then the ecusive system is a collection ( f m, M, ν) of the selles expected utility f m, the maximum numbe of selles in the book M > 0, and the mixed stategy Poisson ate fo the bottom selle ν 0,which satisfy f 0 = A, f m = λ 1 f m+1 + λ 2 1 f m 1, m = 1,...,M 1, λ 1 + λ 2 λ 1 + λ 2 λ 1 + λ 2 f M = f M 1 f M = B. 1 λ 2 + ν, The next theoem descibes the equilibium ode book esulting fom the solution to the ecusive system. Recall that a igid equilibium is a competitive stationay Makov pefect equilibium in which, if some agents have mixed (1) 23 One can ignoe the aival of a new patient selle, because, in equilibium, he will immediately place a maket ode at B and exit, without affecting the state. 24 This is justified by Theoem 1, in the poof of which one can see that a m = f m 1 fo all 1 < m < M. Since the sole tade at m = 1 places an ode at a 1 = A, one has by extension A = a 1 = f

12 The Review of Financial Studies / v 22 n stategies, mixing is done only by the agents with the most competitive limit odes (in this case, only by the selle with the lowest offe). Theoem 1. Conside the limit ode book with only patient selles and impatient buyes (λ PB = λ IS = 0), with uppe bound A, lowe bound B, the aival ates of patient selles λ 1 and impatient buyes λ 2, and the selles patience coefficient. Then thee exists a Makov pefect equilibium of the game, with a maximum numbe M of selles, such that in the state with m M patient selles, the selles expected utility f m is given by (( ) m λ2 f m = A + C 1) + m, if λ 1 λ 2, (2) λ 1 λ 1 λ 2 f m = A bm + m 2, if λ 1 = λ 2, (3) λ 1 + λ 2 whee C > 0,b> 0 and M > 0 ae defined in Poposition 11 in the Appendix. The ask pice a m in the state with m selles is given by a m = f m 1, if m < M; (4) a M = B + λ 2. (5) The stategy of each agent in the state with m selles is the following: If m = 1, then place a limit ode at a 1 = A. If m = 2,...,M 1, place a limit ode at any level above a m, as long as someone has stayed at a m o below; othewise place an ode at a m.ifm= M, the stategy is the same as fo m = 2,...,M 1, except fo the bottom selle at a M, who exits at the fist aival in a Poisson pocess with ate ν 0 by placing a maket ode at B (the numbe ν 0 is defined in Poposition 11 in the Appendix). Ifm> M, then immediately place a maket ode at B. The equilibium descibed above is Makov, with state vaiables: the numbe of existing selles and the ask pice. 25 This equilibium is unique in the class of igid equilibia, in the sense that any othe igid equilibium leads to the same evolution of the state vaiables. Poof. See the Appendix. It should be pointed out that thee is some ambiguity in the way stategies ae fomulated: in the state with m selles, as long as some selle has a limit ode at the ask pice a m (o below), the othe selles can place thei limit odes anywhee above a m. Fom now on, by an abuse of notation, all the equilibia in this class ae consideed to be the same equilibium. Moeove, 25 Since the equilibium ask a m is a function of m, it may seem that m is the only state vaiable in this Makov equilibium. In fact, the ask pice is also a state vaiable, since it descibes what happens out of equilibium: if the selle at the ask a m suddenly inceases his ode, then some othe selle would immediately lowe he ode exactly to the level a m. 4612

13 A Dynamic Model of the Limit Ode Book one can choose in each class a paticulaly impotant epesentative, called the canonical equilibium. Definition 2. To define the canonical equilibium, suppose that a new selle aives when thee ae aleady m 1 selles in the book. Then the equilibium stategies equie that the new selle place an ode at a m, while the othes stay on thei pevious levels. The outcome of this equilibium is that, in state m, selles have thei offes placed at a 1,...a m, and they neve change them. The canonical equilibium aises natually if one intoduces an infinitesimal cancellation cost fo limit odes: selles have no eason to modify thei limit odes if it is costly to do so. The canonical equilibium also appeas as a limiting case of the equilibium when multi-unit maket odes ae allowed but vey unlikely (see Poposition 6 in Section 4). Now we move to deiving a few implications of Theoem 1 that will be useful late. The fist one shows that when the limit ode book is full (in the state M with the maximum numbe of selles), thee is a minimum bid-ask spead. This is inteesting, given that pices can take any value (i.e., the tick size is zeo). Coollay 1. Thee exists a minimum bid-ask spead S min given by S min = a M B = λ 2. (6) One can also compute the dependence of the selles expected utility f m on the lowe bound B of the ode book. This is impotant fo the geneal (twosided) limit ode book, since thee the sell side can be egaded as a one-sided ode book in which the lowe bound B is not constant but equals the bid pice. Poposition 1. If patient selles and impatient buyes aive at diffeent ates (i.e., λ 1 λ 2 ), then the expected utility of m selles f m depends on lowe bound B in the following way: d f m / db = 1 ( λ 2 λ 1 ) m /1 ( λ 2 λ 1 ) M. The deivative is computed by holding the numbe of states M constant. Poof. Denote α = λ 2 λ 1. Diffeentiate the fomula f m = A + C(α m 1) + λ 1 λ 2 m with espect to B: d f m db = dc db (αm 1). Fo m = M the fomula fo f m becomes B = A + C(α M 1) + λ 1 λ 2 M. Diffeentiating this with espect to B and holding M constant, one gets 1 = dc db (αm 1), so dc db = 1 α M 1. Finally, d f m db = 1 αm. 1 α M 3.3 Empiical implications Having descibed a closed-fom solution fo the one-sided limit ode book (with only patient selles and impatient buyes), we now deive empiical implications about the mean and standad deviation of the bid-ask spead and 4613

14 The Review of Financial Studies / v 22 n pice impact, and fo the maximum possible numbe of limit odes in the book M. Some implications povide a diffeent intepetation of known empiical facts about ode-diven makets, and some implications ae new. Given the aival ate of patient selles λ 1, thei patience coefficient, and the aival ate of impatient buyes λ 2, define thee moe numbes: the activity paamete λ: the aival ate of all types of agents; the competition paamete c: the atio of aival ates of patient to impatient tades; 26 and the ganulaity paamete ε: the atio of the selles patience coefficient to tading activity: 27 λ = λ 1 + λ 2 = activity, (7) c = λ 1 λ 2 = competition, (8) ε = λ = ganulaity. (9) Fo now it is assumed that both tading activity λ and competition c ae diectly obsevable. Late in this section, it is shown how to use the model to estimate them. It tuns out that the limit ode book behaves vey diffeently depending on the competition paamete c. Fo example, when the patient selles aive faste than the impatient buyes (c > 1), the limit ode book is esilient i.e., the bid-ask spead on aveage evets to smalle values. By contast, if the patient selles aive at the same speed as the impatient buyes (c = 1) o slowe (c < 1), the bid-ask spead can be quite wide (of the ode of A B, the diffeence between the uppe bound and lowe bound of the book). This is uneasonable, except pehaps in the case of vey illiquid makets. Fo the est of this section, only the case c > 1 will be used. The next esult gives the dependence of the mean and standad deviation of the bid-ask spead on the ganulaity ε and competition c. 28 Since the exact fomulas ae moe complicated, it is moe instuctive to epot the appoximate fomulas 29 only when tading activity λ is lage (o equivalently, the ganulaity ε = /λ is small). Poposition 2. Conside a limit ode book with only patient selles and impatient buyes, with uppe bound A, lowe bound B, selles patience 26 The competition paamete c is also pesent in Foucault, Kadan, and Kandel (2005) (whee it is called ρ) and is agued to be a key deteminant of the esilience of the bid-ask spead i.e., the tendency of the spead to etun to lowe levels. 27 The tem ganulaity is boowed fom the econophysics liteatue (see Fame, Patelli, and Zovko 2005), whee it is elated to the size of gaps in the limit ode book. Similaly, in ou case, the ganulaity ε is of the ode of the minimum bid-ask spead, as seen fom Coollay The coesponding fomulas when c 1 ae in the Appendix, in the poof of Poposition To be pecise about what appoximate means in this context, one says that a vaiable X is asymptotically g(ε) equal to f (ε) and wites X f (ε) ifx can be witten as X = f (ε) + g(ε), with lim ε 0 f (ε) = 0 (in standad mathematical notation, this is witten as X = f (ε) + o( f (ε))). 4614

15 A Dynamic Model of the Limit Ode Book coefficient, tading activity λ, and competition c. Assume that selles aive faste than buyes i.e., c > 1. Let S m = a m B be the equilibium bid-ask spead in the state with m selles. Then, when ganulaity ε = is small, the λ mean spead S and the standad deviation σ(s) can be appoximated by c(c + 1) S ε ln(1/ε) (c 1) ln(c), σ(s) ( (c + 1)(c 3 + c 2 ) 1/2 c) ε(a B). (c 1) 3 (10) Both S and σ(s) decease with c as long as c < 5, and incease with ε; theefoe, S and σ(s) incease with selles patience and decease with tading activity λ. Poof. See the Appendix. Notice that the aveage bid-ask spead is of the ode of ε ln(1/ε), whee ε = /λ is the ganulaity paamete. This should be compaed with Fame, Patelli, and Zovko (2005), who in thei coss-sectional empiical analysis of the London Stock Exchange show that with a high R 2 the aveage bid-ask spead vaies popotionally to ε 3/4, which is close to the theoetical tem ε ln(1/ε). It should be mentioned that thei ganulaity paamete does not include the patience coefficient, which is not diectly obsevable. An impotant implication of Poposition 2 is that the aveage bid-ask spead S is smalle when (i) selles ae moe patient ( is smalle); (ii) tades aive faste in the maket (tading activity λ is highe). 30 Tading activity has been known to explain vaiation in speads (see, e.g., Demsetz 1968), but it is impotant to point out that in this model thee is a causal connection fom tading activity to speads. Linnainmaa and Roşu (2008) test this causal connection fo the Helsinki Exchange by using weathe in Finland as an instument, and find that, indeed, highe tading activity causes smalle speads. Poposition 2 also implies that the aveage bid-ask spead is smalle when competition c = λ 1 /λ 2 is highe i.e., when patient selles aive at a faste ate elative to the impatient buyes. This esult eveses when c is vey high. This comes fom the fact that the aveage bid-ask spead esponds diffeently to selles and buyes aival ates λ 1 and λ 2. Competition c is highe eithe when λ 1 is highe o when λ 2 is lowe. When patient selles aive moe quickly (λ 1 is highe), the bid-ask spead is indeed smalle, due to inceased competition among patient selles. But when impatient buyes aive moe slowly (λ 2 is lowe), the bid-ask spead is actually lage, indicating that patient selles need a highe spead as compensation fo waiting moe. Nomally, the fist effect dominates, but when competition c is vey high (c > 5), the second effect dominates. 30 These esults ae also tue in the model of Foucault, Kadan, and Kandel (2005). 4615

16 The Review of Financial Studies / v 22 n Anothe implication about speads can be deived fom Coollay 1 in Section 3.2: S min = = (c + 1) = ε (c + 1). (11) λ 2 λ Notice that, unlike the aveage bid-ask spead S, when competition c is highe, the minimum spead S min is also highe. 31 This is because the minimum bid-ask spead depends only on the aival ate of impatient buyes λ 2, and as befoe speads ae lage when impatient buyes aive moe slowly. Fo the next esult, one identifies the volatility of the asset as the standad deviation of the ask pice (since the bid pice B is constant), o equivalently the volatility of the spead σ(s). Poposition 3. In the context of Poposition 2, the volatility of the asset σ(s) vaies in invese popotion to λ, the squae oot of tading activity. Also, the aveage spead S vaies appoximately popotionally to the atio σ(s)/ λ. Poof. Equation (10) fom Poposition 2 implies that the volatility σ(s) vaies in popotion to the squae oot of ganulaity ε = /λ i.e., in invese popotion to λ. Fo the second esult, use again Equation (10) to compute the atio S/σ(S). This is popotional to ε ln(1/ε). If one omits the tem ln(1/ε), which goes to infinity at a much slowe ate than the tem ε goes to zeo, then indeed S/σ(S) is popotional to ε, and hence vaies in invese popotion to λ. Empiical evidence fo both esults can be found in Wyat et al. (2008, Equations (32) and (33)), fo high-fequency data. Inteestingly, the fist esult is in contadiction with a lage liteatue that documents a positive elation between tading volume and volatility when dealing with lowe-fequency data (see, e.g., Jones, Kaul, and Lipson 1994). The explanation is that Poposition 3 is a high-fequency esult: it assumes the bounds A and B of the limit ode book to be constant, which is easonable only fo inta-day time intevals. Next, we define pice impact in this context. Conside the canonical equilibium of Definition 2. Fo simplicity, pice impact is defined only fo one-unit maket odes, leaving the case of multi-unit maket odes fo the next section. Then the pice impact of one unit is a m 1 a m, which accoding to Theoem 1 equals f m 2 f m 1 (except fo the case when m = M). The following esult gives an asymptotic fomula fo the aveage pice impact when c > 1. Poposition 4. In the context of Poposition 2, define I m = d f m 1 the pice dm impact of a one-unit maket ode in state m. Then when competition c > 1, 31 One should not expect this conclusion to hold empiically, because it is not obust: the fomula was deived in the one-sided model. In the two-sided model, the minimum bid-ask spead has a much moe complicated dependence on the model paametes, and thee is no eason why the dependence of the minimum bid-ask spead on competition c should even have the same sign. 4616

17 A Dynamic Model of the Limit Ode Book the mean pice impact Ī and the standad deviation σ(i ) can be appoximated by (ε = λ ): Ī ε ln(1/ε) c + 1 ln(c), σ(i ) c + 1 ε(a B) c 1. (12) The aveage pice impact deceases in λ, and deceases in c as long as c < 3.5. Poof. See the Appendix. Notice that pice impact and the aveage bid-ask spead depend on ganulaity ε and competition c in the same way. This should not be supising because both pice impact and the bid-ask spead ae foms of maket depth (in the sense of Kyle 1985). When the maket is deep (fast aivals λ, lage competition c), pice impact is small, because all the inte-limit-ode speads ae small as well. The next esult analyzes M, the maximum possible numbe of (sell) limit odes in the book. This numbe is endogenous, and it depends on the ganulaity ε, competition c, and the ode book bounds A and B. It is impotant to detemine M because it is elated to the aveage density of the limit ode book: a highe M descibes a dense book, while a lowe M descibes a aefied book. Poposition 5. In the context of Poposition 2, the maximum possible numbe M of selles in the book is given by ( ) A B (c 1) ln 2 ( ε c+1 M = + s, with s 1, ln(2) ). (13) ln c ln(c) Fomula (13) is only tue if ganulaity ε is sufficiently small e.g., if ε (A B) c 1 ln(c). The numbe M is inceasing in tading activity λ and deceasing c+1 ln(4c) in competition c. Poof. See the Appendix. It is supising that the maximum numbe of selles in the ode book actually deceases with competition c (holding total activity λ constant). This is because of how the limit odes ae placed in the book. When c > 1 is elatively lage, the speads between the diffeent limit odes ae smalle, but limit odes become moe aefied as one gets futhe away fom the ask pice. On balance, the maximum numbe of odes M is actually smalle when competition c is lage One may think that by a simila agument, the aveage bid-ask spead S should also be lage when c is lage. This would be tue if one used the aithmetic aveage acoss all states. Howeve, hee one takes a weighted aveage with weights given by stationay pobabilities, and these ae popotional to c m. Because competition c is lage than one, smalle speads ae moe likely than highe speads, theefoe the aveage spead deceases in c when c is not too high. 4617

18 The Review of Financial Studies / v 22 n One may wonde which empiical implications of the one-sided model ae obust i.e., cay though to the geneal two-sided model. We think that the asymptotic fomulas fo the mean and standad deviation of the bid-ask spead, fo the pice impact, and fo the maximum numbe of odes in the book M ae tue, except that one may have to multiply the esults by a constant. Such a statement seems difficult to pove fomally. Finally, we discuss how to estimate the model paametes, A, B, λ, and c. The patience coefficient is not obseved, so one may assume that it is constant fo at least a shot peiod of time and deive it fom the implications of the model. One can ague also that should be constant acoss stocks, as it must depend only on agents type. The ode book bounds A and B can be estimated simply by looking at the limits of a (pehaps winsoized) limit ode book. Tading activity λ = λ 1 + λ 2 is also obsevable, as the sum of aival ates of maket and limit odes. A poblem aises if one attempts to estimate competition c = λ 1 /λ 2 as the atio of aival ates of sell limit odes to buy maket odes. This is because, fo example, λ 1 is the aival ate of patient selles, not of sell limit odes. 33 Instead, the model suggests estimating c as the aveage competition conditional on the vaious obseved speads. This is the same as the aithmetic aveage of the atio λ 1 /λ 2 ove all states of the book, which can be shown to equal cm/(m + 1) and theefoe is appoximately equal to the theoetical value c. 4. Multi-Unit Maket Odes and Pice Impact This section studies in moe detail the pice impact of a tansaction, and the shape of the limit ode book. The attention is focused on two questions: (i) If one submits a buy maket ode fo i units, how much does the pice change instantaneously? (ii) How does the limit ode book change subsequently? The answe to the fist question is given by studying the instantaneous (o tempoay) pice impact function. The answe to the second comes fom analyzing the subsequent (o pemanent) pice impact. Simila to the pevious section, it is assumed that thee ae only patient selles and impatient buyes. As in the discussion following Theoem 1, if only oneunit maket odes ae allowed, no limit odes above the ask ae necessaily fixed in equilibium, and so the pice impact function is not uniquely defined. Theefoe, to fix the othe limit odes above the ask, the selles should expect that thei odes might be executed at any time with positive pobability. This means that multi-unit maket odes must be allowed. 33 It is tue that patient selles typically submit limit odes, but thee is an exception when the ode book is full, in which case incoming patient selles submit maket odes. So one cannot estimate c = λ 1 /λ 2 as the total numbe of sell limit odes divided by the total numbe of buy maket odes, because that can be shown to theoetically equal one. In pactice, howeve, this atio is not equal to one due to cancellations, and to the extent that the numbe of cancellations is popotional to λ 1, one may ague that in fact this atio is a good poxy fo competition c. 4618