A new model for limit order book dynamics

Transcription

1 Anewmodelforlimiorderbookdynmics JeffreyR.Russell UniversiyofChicgo,GrdueSchoolofBusiness TejinKim UniversiyofChicgo,DeprmenofSisics Absrc:Thispperproposesnewmodelforlimiorderbookdynmics.Thelimiorderbookconsiss ofquniiesvilbleforrdedifferenpricessodynmicmodelmusdescribeheevoluionof curves.theproposedmodelcpuresdependenceinhecurveusingnuoregressivesrucureinwo componens of he curve h re simple o inerpre from n economic perspecive.the firs erm describesheverge(weighed)disncehhedephononesideofhemrke(buyorsell)lies wy from he midquoe.the second erm describes how spred ou he deph is cross prices. MximumlikelihoodesimesreconsrucedusingdfromArchipelgoExchnge(currenlyARCA) ndspecificionessredescribed.byincludingddiionlexplnoryvribles,weeshypohesis bouhowmrkeliquidiyrespondsoinformionflow.wefindhdephendsomovewyfrom he midquoe in periods of low voliliy, high volume, nd wide spreds.we find h he deph is morespredoucrosspriceswhenvoliliyishigh,rdingvolumeislrge,ndspredsrewide.

2 I. Inroducion Nerlyhlfheworld ssockexchngesreorgnizedsorderdrivenmrkessuchsecn s. Thesrucureofhelimiorderbooknypoininimedeermineshecosofnyrde.The dynmics of he limi order book deermine how his cos vries over ime.despie he prevlenceoforderdrivenmrkesherereremrkblyfewmodelsforhedeerminnsof he srucure of he limi order book nd is dynmics.this pper proposes new dynmic modelforhedeerminnsofhesrucureofhelimiorderbooksdeerminedbyhese ofhemrkendssechrcerisics. Exising work on liquidiy in order driven mrkes cn be spli ino wo groups.the firs pprochdireclymodelshelimiorderplcemensregiesofindividulrders.exmples includebiis,hillioin,ndsp(1998)coppejnsnddomowiz(2002),rnldo(2004),nd HllndHusch(2004).Thispprochprovidesinsighinohemicrobehviorofdecisions, buprovidesonlyindirecevidencebouheoverllsrucureofhelimiorderbook.the secondpprochfocusesonspecificspecsofhelimiorderbooksuchsbidskspredsor he deph he bes bid or sk.this pproch provides deiled semensbou specific feures of he limi order book bu gin, provides only indirec evidence bou he overll srucureofhelimiorderbook.inheend,hesepprochescnnoprovidedirecnswers oquesionslike whisheexpecedcosofbuying$1000ofibminoneminue? Answers o hese quesions require more complee model of he limi order book h models he eniresrucure,nojuscomponen.asresul,hesepprochesrefocusingonnrrow specsofliquidiy. Thispperkesdifferenpproch.Thelimiorderbookisseofquniiesobebough orsolddifferenprices.weproposedireclymodelingheimevryingcurves.theforecs ofhemodelishereforefuncionproducingexpecedquniiesoverrngeofpricess funcionofhehisoryofhelimiorderbookndmrkendssecondiions.themodel, herefore,cndireclynswerhequesionsregrdingheexpecedcosofpurchse(orsle) inoneminue. The model is prmeerized in wy h llows for esy inerpreion nd herefore he model is useful in ssessing nd inerpreing how mrke condiions ffec he shpe of he limiorderbookndhereforeliquidiy.thedisribuionofdephcrosshelimiorderbookis modeled by ime vrying norml disribuion nd herefore depends on wo ime vrying prmeers.the firs deermines he verge disnce h he deph lies wy from he midquoe. As his prmeer increses, mrke liquidiy ends o decrese. The second prmeerdeermineshowspredouhedephis.lrgervluesofhisprmeerledo

3 flerlimiorderbook.theseprmeersremdeimevryinginnuoregressivemnner sohheshpeofhelimiorderbooknexperioddependsonheshpeofhelimiorder bookinhepreviousperiodndpossiblyohervribleshchrcerizehemrkecondiion. Themodelisppliedoonemonhoflimiorderbookd.ThedcomefromArchipelgo Exchnge. Model esimes re presened for limi order book dynmics one minue incremens.wefindhhelimiorderbookexhibisverysrongpersisencesuggesingh newlimiordersresloworeplenishhebook.welsofindhdephendsomovewy fromhemidquoe,sohhemrkebecomeslessliquid,followinglrgerspreds,smller rdevolume,higherrnscionres,ndhighervoliliy.welsofindhhebookends obecomemoredisperse(fler)whenspredsrelow,rdesizeislrge,rnscionres rehigh,ndvoliliyishigh. II. Themodel This secion presens model for he disribuion of he deph cross muliple prices.our pprochdecomposeshelimiorderbookinowocomponens;heoldephinhemrke ndhedisribuionofhdephcrosshemulipleprices. Webeginwihsomenoion.Lehemidquoeimebedenoedbym.Nex,wedenoe gridfornpricesonheskndbidsides.thei h skpriceonhegridisdenoedby p i nd hei h b bidpriceisdenoedby p i. p1 ishefirspriceorbovehemidquoewhichdeph cn be lised nd similrly, p1 is he firs price below he midquoe which deph cn be lised.wewillrehegridsbeingequllyspcedsohechconsecuivepriceonhesk sideisfixedunibovehepreviousprice.thegridccounsforhefchvilbleprices inmosmrkesreresricedofllonvluesfixedicksizes.hencehesmllesincremen considered would be h of he ick size.lrger incremens could be considered s well. Finlly,wedefineheolnumberofshresvilbleinechpricebin.Onheskside, i denoesheoldephvilbleinhei h bin. 1 isheshresvilbleinhelimiorderbook pricespwhere p1 p p2ndfori>1 i denoesheshresvilblepricespwhere pi p pi 1.Asimilrnoionisusedforhebidsideofhemrkewhereb i denoeshe shresvilbleinhelimiorderbookonhebidside.thegridisnchoredhemidquoeso hegridhsimesubscrip.inbohcses,lrgervluesofiressociedwihpricesfurher wyfromhemidquoe.

4 Ourgolisospecifymodelforheexpecedshresvilbleinechbingivenheseof hemrkendperhpschrcerisicsofhesse.forsmllnumberofbins(smlln)he dephcouldbemodeledbysndrdimeseriesechniquessuchsvar.thesepproches quicklybecomeinrcblewhennismorehn1or2sincelrgenumberoflgsrelikely needed inroducing very lrge number of prmeers o be esimed.addiionlly, i is difficuloinerpreheindividulprmeersofvarinherelevnconexofliquidiynd onewouldhveoresoroimpulseresponsefuncions.wekedifferenpprochh decomposesheprobleminowocomponens.defineheolshresinhelimiorderbook overhefirsnbinss D N i1 i.wedecomposehemodelforhelimiorderbookino shpendlevelcomponens.givenheolshresinhelimiorderbook,define 1) i E D i D s he expeced frcion of he deph D in bin i, ime.given he ol shres, he expeceddephinbiniimeisgivenby 2) E D D i i Differences in deph cross bins re driven by he erms. Hence, his decomposiion sepreshemodelforhelimiorderbookinoshpecomponendescribedbyhe snd levelgivenbyheoverlldeph, D.Ingenerl,bohheshpeofhelimiorderbooknd he ol shres vilble, D, will depend on chrcerisics of he sse nd mrke g D F denoe condiions.lef 1 denoeninformionsevilbleime1,ndle 1 modelforheimevryingolshres.wenowcngenerlize1)nd2)ollowforime imevringprobbiliies,imevryingolshres, D,ndimevryinglimiorderbook: i 3) i E D, F 1 D Theonesephed,prediceddephishengivenby 4) i 1 i 1 E F g D F dd D Hence,helimiorderbookcnbemodeledusingmulinomilmodelfor3)ndunivrie g D F.Thelerisunivrieserieshcouldbemodeledwih imeseriesmodelfor 1

5 sndrd ime series ime series models such s n ARMA model.the new pr here is hereforeofindgoodmodelforhemulinomilprobbiliies. Thegolinspecifyinghemulinomilmodelisofindmodelhfishedwell,isesily inerpreed,ndllowsfornobelrgewihourequiringlrgenumberofprmeers.the limiorderbookclerlyexhibisdependenceespecillywhenviewedovershorimeperiods. The model mus herefore be specified in flexible wy so h he shpe depends on he hisoryofhelimiorderbook. Our model is formuled using mulinomil probi model. For he probi model, he mulinomilprobbiliiesredeerminedbyresunderhenormldensiyfuncion.these probbiliies re ime vrying when he men nd vrince of he Norml densiy re ime 2 vrying.specificlly,givenmen ndvrince heprobbiliyisgivenby: p m p m i i i1 Where ishecumulivedisribuionfuncionfornorml(, ).Ifhegridisseon icks,henhiswouldcorrespondohefrcionofhedephhliesonhei h ickbovehe midquoes. This prmeerizion is convenien o inerpre.clerly s increses, he cener of he disribuionmoveswyfromhemidquoe.therefore,lrgervluesof ressociedwih deph over he modeled region lying, on verge, furher from he midquoe.this would 2 correspond o less liquid mrke.as increses, he Norml densiy becomes fler hereforespredingouheprobbiliymoreevenlycrosshenbins.as goesoinfiniy heprobbiliiesbecomeequl.anincreseordecreseineiherhemenorhevrinceis hereforeesilyinerpreedinermsofvergedisncehhedephliesfromhemidquoe ndhowspredouhedephiscrosshenbins. Wenowurnodynmicsofhedisribuionwhichredrivenbyhedynmicsofhemennd vrince.sinceheshpeofhelimiorderbookwillbehighlydependen,especillyovershor ime inervls, we begin wih he simples version of he model using n uoregressive srucureforhemenndvrince.aechimeperiod,wecnclculehecenerofhe empiricl disribuion of he deph.this is given by 2 1 n i i D i 1 x p m 2.The difference n beween he cul men nd he prediced men is given by x i pi m i1.

6 1 n i i D i 1 2 Similrly,wecncompuehevrinceofhedephcrosshebinss 2 s p x n 2 ndhessociederrorisgivenby ln ln 2 s i pi x.ifhemodeliscorrecly i1 specifiedhenboherrorermswillbeuncorreled,lhoughhelerwillnobemenzero. Theseerrorsreusedobuildnuoregressivemodelforheimevryingmenndvrince h in urn dice he ime vrying probbiliies in he mulinomil.specificlly, simple modelforhedynmicofhemenisgivenby: Similrly,simplemodelforhedynmicsofhevrinceisgivenby: ln( ) ln( ) Clerlyhigherordermodelscouldbeconsidered.Addiionlly,ohervribleshcpurehe seofhemrkecouldbeincludedswell.theexplicidependenceofhecurrenmennd vrinceonhepsmenndvrincellowsforpoenilpersisenceinheseries.theerror ermsllowheupdingodependonhedifferencesbeweenheexpecedndculmen ndvrince.inhenexsecion,weurnomodelesimion. III. ModelEsimion Thedononesideofhemrkeconsisofhenumberofshresvilbleinechbin.We proceedoesimeprmeersforhemenndvrincedynmicsvimodelbymximum likelihood.ifechshresubmiedechimeperiodcouldbeviewedniiddrwsfrom mulinomildisribuionwihprobbiliiesgivenbyhe i shenhelikelihoodssociedwih hehperiodisgivenby: l n n This ssumes h he shres re iid drws which is surely flse.orders re submied in pckesofmulipleshres,ypicllyinincremensof100shres.ifllordersweresubmiedin pckesof100shreshenhelikelihoodforhe h observionwouldbegivenby: l n n i where i. 100

7 Henceweconsrucheloglikelihoods: L ln T n 1 i1 2 Givenniniilvlueof 0 nd 0 hesequencemulinomilprobbiliiescnbesequenilly upded nd he likelihood evlued for ny se of prmeers nd mximized.under he usulregulriycondiionsheesimeswillbeconsisenndsympoicllynorml. IV. D ThedconsisoflimiordershweresubmiedhroughheArchipelgoExchnge.This exchnge hs since been bough by NYSE nd is now clled ARCA. As of Mrch, 2007, ArchipelgoishesecondlrgesECNinermsofshresrded(bou20%mrkeshrefor NASDAQsocks).OurdconsissofonemonhoflllimiorderssubmiedinJnury2005. Thedconinsheypeofordercion;dd,modifynddelee. Add correspondso newordersubmission. Modify occurswhennorderismodifiedeiherinisprice,number ofshres,orifnorderisprillyfilled. Delee signifieshnorderwscncelled,filled, orexpires.thedlsoconinsimesmpdownohemillisecond,hepricendorder size,ndbuyorsellindicor,socksymbol,ndexchnge. We exrc orders for single sock Google (GOOG).Only orders submied during regulr hours(9:30o4:00)reconsidered.fromheorderbyorderdweconsruchecomplee limiorderbookeveryminue.thisresulsin390observionsperdy.forreference,he vergerdepriceforgoogleoverhemonhiscloseo$200.figure1presensploofhe dephechcenmovingwyfromhemidquoefromonecenoforycens.theplo revelspekeddisribuion,wihispekround1520censwyfromhemidquoe.of coursehisisnuncondiionldisribuion. The limi order book d is merged wih Trdes nd Quoes (TAQ) d for he sme ime period.fromhisdwecreeseverlvriblesreledordingndvoliliy.psorder flowshouldbereledofuureorderflowndhereforefuurelimiorderplcemen.for everyminue,weconsruchelogrihmofhevergerdesizeoverhemosrecen15 minueperiod.addiionlly,weconsrucheolnumberofrdesexecuedoverhemos recen15minueperiod.bohreindicionsofhedegreeofmrkeciviy.welsocree relizedvoliliymesureconsrucedbysummingsqured,oneminueinervlreurnsover he15mosrecenminues.finlly,hebidskspredrnscionimesisvergedover he15mosrecenminues. i i

8 Inprinciple,wecouldmodeldephouhroughnydisncefromhemidquoe.Wefocusour enioninhisnlysisohedephouhrough30cens.weggregeheshreswihin lrger5cenbinsndhereforehve6binsonhebidsidend6binsonheskside.our modelingsregyhssepremodelsforhebidndsksideofhemrke.inournlysis, wefocusonhesksideonly. V. Resuls Webeginwihsomesummrysisicsforheminuebyminued.Aechminue,we hve observed deph in he firs 6, 5cen bins, 1, 2,, 6.I is ineresing o ssess he dependencesrucureinhisvecorimeseries.specificlly,ifwesckhedephimeino vecor x where he firs elemen of x is 1 nd he ls elemen is 6, we consruc he uocorrelionsofhevecorx forlgs0hrough3minues.thesmpleuocorrelionsre presenedinfigure2.theuocorrelionsresisicllydifferenfromzeroiflrgerhn T inbsoluevlue. All uocorrelions re posiive indicing he deph he prices ends o move ogeher. Dephnerhedigonlendsobemorehighlycorreledhdephwyfromhedigonl indicinghhecorrelionbeweenclosebinsislrgerhnhecorrelionbeweenbins hrefrpr.thedigonloruocorrelionsofhesmeelemenofhevecorx end o hve he highes of ll correlions.alhough no presened, he generl posiive nd significncorrelionssrucureconinuesouhroughlg10(or10minues). Wenowesimehemodelforhedisribuionofhedephcrosshebins,hemulinomil probi. We begin by esiming simple, firs order model presened in Secion II, 2 2 specificlly nd ln( ) 0 1ln( 1) 2 1. The prmeer 2 2 esimes re given by nd ln( ).06.96ln( 1).06 1.All prmeersresignificnhe1%level.bohhemenndhevrinceexhibiverysrong persisence indicing h he verge disnce of he deph from he midquoe is highly persisensishedegreespredofhedephcrossbins.theauoregressiveermisner1 forbohmodels.allcoefficiensresignificnhe1%level. Anurlesofhemodelisocheckifheonesephedforecserrorsforhemennd vrinceequions( nd )reuncorreled.thenullofwhienoiseseriescnbeesed byexminingheuocorrelionsofheseinsmpleerrors.weperformljungboxeson hefirs15uocorrelionsssociedwihheerrorsforhemenequionndhevrince equion.the pvlues re.53 nd.06 respecively.hence his simple firs order model

9 ppers o do resonbly good job of cpuring he dependence in he shpe of he limi orderbook. Iisineresingoseehsimplefirsorderversionofhemodelcncpurehesubsnil dependence in he shpe of he limi order book.we now urn our enion o ddiionl mrke fcors h migh influence he dynmics of he limi order book.glosen (2000) predics h higher rding res should resul in deph clusering round he midquoe. Compeiionmongrdersinncivemrkeledsomorelimiordersbeingplcedner hemidquoe.similrly,rosu(2008)proposesheoreiclmodelforhedynmicsofhelimi orderbookhsuggesshlsopredicshmoredephshouldcluserroundmidquoe when mrke civiy is high.following Glosen nd Rosu, we should expec he men o decrese,ndhevergedisnceofhedephmovecloserohemidquoeinperiodsofhigh rdingres. Periodsofhighvoliliyressociedwihgreerunceriny.Inperiodsofhighunceriny here migh be higher probbiliy of rding gins beer informed gens. Clssic microsrucureheorypredicswideningofbidskspredswhenheprobbiliyofrding ginsbeerinformedgensishigher.wemighhereforeexpechdephshouldmove wyfromhemidquoeinperiodsofhighvoliliy.ahesmeime,highvoliliyinhe ssepriceincresesheprobbiliyhlimiorderfrfromhecurrenpricegesexecued. This migh lso serve s n incenive for rders o seek superior execuion by plcing limi orders furher from he curren price.boh of he ides imply h in periods of higher voliliy, he men verge disnce of he deph from he midquoe should increse.we migh lso expec h he disribuion of deph should flen.hence we migh expec he menndvrinceoincreseinperiodsofhighssepricevoliliy. In ligh of hese economic rgumens, we nex esime models h condiion on recen rnscionhisoryndvoliliy.specificlly,weusehernscionvolumeoverheps15 minues, he number of rdes over he ls 15 minues nd he relized minuebyminue voliliyoverhels15minues.addiionlly,weincludesomeohereconomicvriblesh reofineresincludinghevergespredoverhels15minuesndhepricechngeover hels15minues.weincludellheseeconomicvribleswihinhefirsorderimeseries modelesimedbove.thecoefficiensofheeconomicvriblesrepresenedinble1. Allvriblesresignificnhe1%level. Webeginwihdiscussionofherelizedvoliliy.Relizedvrincehsposiivecoefficien inhemenequionindicinghwhenhevoliliyofhessepriceincresesheverge disnceofhedephendsomovewyfromhemidquoe.thisisconsisenwihheboh ides,nmelyincresedlikelihoodofrdingginsbeerinformedgensmovesdepho moreconservivepriceshccounforhisrisk.iislsoconsisenwihheidehhigh

10 voliliyincreseshelikelihoodofdephfurherfromhemidquoegeingexecuedsome poin in he fuure. Similrly, he coefficien on he voliliy is posiive in he vrince equion.thisindicesfleningofhedisribuionsohhedephismoreevenlyspred overhebins. Nex,considerherdesizendrdingrevribles.Weseehlrgervergerdesize endsomovehedephcloserohemidquoe.higherrdingresendomovehedeph furher from he midquoe, on verge.the effec of rde size nd rding res re boh posiiveonhevrince.lrgerrdesizemybeindiciveoflrgerdephposedhebes bidndskprices.sincehedephiscorreled,hismighsimplybeindiciveoflrgedeph heskfollowinglrgerdephhesk.therdingresrelileesieroinerpre becusehereislessofdireclinkbeweenrdingresndquniieshebessk.the posiivesignhereindiceshdephendsomovewyfromhemidquoeduringperiods of high rnscion res.addiionlly, he posiive sign on boh vribles in he vrince equion indices h he deph is more evenly disribued during high rding res nd lrgervergesize.overll,heevidencedoesnosupporhepredicionsofglosenorhe modelofrosu. Widerspredsressociedwihmoreunceriny.Aswihvoliliy,wemighexpech dephshouldmovewyfromhemidquoeinperiodsofgreerunceriny.indeed,hesign on he spred is posiive boh for he men equion nd for he vrince equion.rising pricesendobessociedwihdephmovingwyfromhemidquoendhedisribuion becomingmoreevenlydisribued. Nex,weesimemodelforhesecondcomponenofhemodel,nmelyhelevelofhe deph D onhesksideofhemrke.specificlly,wespecifynarma(2,2)modelforhe logrihmofhedeph: D c D D rv ln ln ln where iswhienoisendrvisherelizedvoliliyoverhels15minues.theohereconomic vribles re no significn so hey re no included in he finl mode for he level.the esimed modelis: ln D ln D.28ln D rv TheinsmpleresidulspssLjungBoxeswih15lgs.Theprocessislsohighlypersisen. Alhough he ohereconomic vribles re insignificn he relized voliliy is significn he1%levelndimplieshhelevelofdephendsoincresefollowingperiodsofhigher voliliy.combiningheresulsforhedisribuionndhelevel,weseehheolnumber

11 ofshresinhefirs30censendsoincresefollowinghighvoliliyperiods,buhhe disribuionofhedephshifswyfromhemidquoendflensou.figure3presens ploofheprediceddephundervergecondiionsforllvriblesexcephevoliliywhich isvriedfromvergeohe5 h percenile(low)o95 h percenile(high). VI. Conclusions. We propose model for limi order book dynmics.themodel is formuled in wy h sepreshemodelingprobleminomodelforhelevelofhedephndmodelforhe disribuionofhedephcrossspecifiedbins.thedecomposiioncombinedwihheuseof convenien probi model llows he dynmics o be inerpreed in priculrly simple wy. Specificlly we model he level, verge disnce of he deph from he midquoe, nd he flnessorspredofhedephcrosshebins.themodelforhelevelofhedephcnbe ken from off he shelf processes.the new pr here is he model for he ime vrying mulinomildisribuion. Weshowhsimplelowordermodelsforheprobirebleocpurehesrongemporl dependenceinheshpeofhedisribuionofhedeph.moreineresingly,welsoconsider severleconomicvribles.wefindhhighervoliliypredicshheoverlllevelofhe dephwillincrese,buhdephmoveswyfromhemidquoendhedisribuionends oflenou,becomingmoredisperse. Conrry o he predicions of Glosen (2000) nd Rosu (2008) we find evidence h higher mrkeciviy,smesuredbyrdingres,endsomovedephwyfromhemidquoe ndflenhedisribuion.

12 References Biis,B.,P.Hillioin,ndC.Sp(1995), AnEmpiriclAnlysisofheLimiOrderBookndheOrder FlowinhePrisBourse,JournlofFinnce,50, Coppejnds,M.,ndI.Domowiz(2002), AnEmpiriclAnlysisofTrdes,Orders,ndCncellionsin LimiOrderMrke, Discussionpper,DukeUniversiy. Glosen,L.(1994) Isheelecronicopenlimiorderbookinevible?,JournlofFinnce49, Hll,A.ndNHusch(2004) OrderAggressivenessndOrderBookDynmics Workingpper UniversiyofCopenhgen Rnldo,A.(2004), OrderAggressivenessinLimiOrderBookMrkes, JournlofFinncilMrkes,7, 5374 Rosu,I.(2008) ADynmicModelofheLimiOrderBook,ReviewofFinncilSudies forhcoming.

13 Figure1.Disribuionofdephmesuredincenswyfrommidquoe.

14 ˆ ˆ ˆ ˆ Figure2.Auocorrelionsofdephindifferenbinsonheskside.

15 Figure3.Predicedlimiorderbookundervergecondiionssvoliiyvriesfromlowo high.

16 ModelforMen ModelforVrince RelizedVrince TrdeSize Spred TrdingRe PriceChnge Tble1.Esimedcoefficiensforeconomicvribles.