foucault and kadan-limit order book as a market for liquidity

T hree equilib rium typesare ob tained - the type isd etermined b y three param eters: the d egree ofim patienceofthe patient trad ers,w hic h w e interpret asthe c ost ofexec utiond ela

Lim it O rd er B ook asa M arket f or

T hierry Fouc aul t

HE C Sc hoolofM anagem ent

1 rue d e la Lib eration

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P eteKyle,L eslieM arx,N arayanN aik,M aureenO 'H ara,ChristineP arlour,P atrikSandas, IlyaStrebulaev,A vi W ohl,andseminarparticipants atA msterdam,Emory,Illinois,Insead, Jerusalem,L B S,T elA viv,W hartonfortheirhelpfulcommentsandsuggestions.Comments

by participants atthe W FA 20 0 1 meeting and the G erzensee Symposium20 0 1 have been helpfulas well.T heauthors thankJ.N achmias Fund,and KrugerFoundation for¯nancial support.

Ab strac t

W e d evelop a d ynamic mod elofanord er-d rivenm arket populated b y d iscretionaryliquid ity trad ers T hese trad ersmust trad e, yet c anc hoose the type oford er andare fully strategic intheir d ec ision.T rad ersd i®er b y their im patience: lesspatienttrad ersd emand liquid ity, m ore patient trad ersprovid e it T hree equilib rium typesare ob tained - the type isd etermined b y three param eters: the d egree ofim patienceofthe patient trad ers,w hic h w e interpret asthe c ost ofexec utiond elay inprovid ingliquid ity; their proportioninthe population, w hic h d eterm inesthe d egree ofc om -petitionam ong the liquid ity provid ers; and the tic k siz e, w hic h isthe c ost ofthe

m inimalpric e im provem ent.Despite itssim plic ity, the m od elgeneratesa ric h set ofempiric alpred ic tionsonthe relationb etw eenm arket param eters,tim e to exec ution,and spread s.W e argue that the ec onom ic intuitionofthismod elisrob ust, thusits

m ainresultsw illremaininmore generalm od els

1 In trod uc tion

Lim it and m arket ord ersc onstitute the c ore ofany ord er-d rivenc ontinuoustrad ingsystem (suc h asthe NY SE , LondonStoc k E xc hange, E uronext, T okyo and T orontoStock E xc hanges, asw ellasallthe E CNs).1 A m arket ord er guaranteesanim m ed i-ate exec utionat the b est pric e availab le at the m om ent ofthe ord er arrivalat theexc hange.Ingeneral, a m arket ord er representsd em and for liquid ity (im med iac y ofexec ution).W ith a lim it ord er, a trad er c animprove hisexec utionpric e relative tothe market ord er pric e, b ut the exec utionisneither imm ed iate, nor c ertain.A lim itord er representssupply ofliquid ity to future trad ers.2

T he optim alord er c hoic e ultim ately involves a trad eo® b etw eenthe c ost ofa

d elayed exec utionand the c ost ofim m ed iate exec ution,w hic h (for sm alltransac tions)isd eterm ined b y the siz e ofthe insid e spread Intuitively w e expec t patient trad ers

to post lim it ord ersand supply liquid ity to im patient trad ers, w ho opt for m arketord ers.Inhissem inalpaper Dem setz (1968) stressesthe lim it ord ersasthe sourc e ofliquid ity,pointingout the trad e o® b etw eenlonger exec utiontim e and b etter pric es

O ur m od elfeaturesb uyersand sellersarriving sequentially E ac h trad er w ants

1 D omowitz (1 993)shows thatover30 important¯nancialmarkets in the world in the early90 's had some oforder-driven marketfeatures in theirdesign.T he importance oforder-driven markets around theworld has been steadilyincreasingsince.

2 W eignoreheremarketable limitorders.

to b uy or sellone unit ofa sec urity W e assum e that these are liquid ity trad ers,

i.e.they w illb uy/sellregard lessofpric e.How ever,they choose b etw eenm arket andlim it ord ersso asto m inim iz e their c ost oftrad ing.U ponarrival,the trad ersd ec id e

to plac e a market ord er or a lim it ord er, c onditionalonthe state ofthe b ook Ifsub m ittinga limit ord er the trad er c hoosesa pric e and b earsthe opportunity c ost ofpostponingthe trad e

U nder severalsim plifyingassum ptionsw e are ab le to d evelop a rec ursive m ethod

for c alculatingthe ord er plac em entsstrategiesand the expec ted tim e-to-exec utionforlim it ord ers.Ingeneral,inequilib rium ,patient trad ersprovid e liquid ity to im patienttrad ers.W e id entify3typesofequilib ria c harac terized b ym arked lyd i®erent d ynam ic s

for the lim it ord er b ook.T hese d ynam ic sturnout to b e very sensitive to the ratioofthe proportionofpatient trad ersto the proportionofim patient trad ers.Ac tuallythe larger isthisratio, the m ore intense isc om petitionam ong liquid ity suppliers

T hey are also in°uenced b y the d ispersionofw aiting c ostsac rosstrad ers.Som e ofour main¯ndingsc anb e summ arized asfollow s

²Lim it ord erstim e-to-exec utionare large w henthe proportionofpatient trad ersisrelatively large.T hise®ec t enhancesc om petitionam ong liquid ity provid ers

w ho sub m it m ore aggressive ord ersto shortentheir tim e-to-exec ution.Hence

m arkets w ith a relatively large proportionofpatient trad ers feature sm allerspread s

²Inord er to speed up exec ution, trad ersfrequently ¯nd optim alto underc ut

or outb id the b est quotesb y m ore thanone tic k T hishappensw hen(i) theproportionofpatient trad ersisrelatively large, (ii) w aiting c ostsare large or(iii) the tic k siz e issm all

²A d ec rease inthe tick size c anresult inlarger expec ted spread s Ac tually itgivesthe possib ility to trad ersto quote lessc om petitive pric esb y expanding

the set ofpric es.Ifc om petitionam ongliquid ity provid ersisw eak,they use thenew pric esand the average spread increases.

²A d ec rease inthe ord er arrivalrate c anresult insmaller expec ted spread s.Intuitively, suc h a d ec rease extends the expec ted tim e-to-exec utionfor limitord ers.T hise®ec t induc esliquid ity suppliersto plac e m ore aggressively pric edlim it ord ersw henthe insid e spread islarge

Insom e lim it ord er m arkets,d esignated m arket-m akersare required to enter b idand ask quotesinthe lim it ord er b ook T hisisthe c ase, for instance, inthe P aris

B ourse for m ed ium and sm allc apitaliz ationstoc ks.3 W e c onsid er the e®ec t

ofintro-d uc ingthistype oftraofintro-d er inour m oofintro-d el.W e show that the presence ofa traofintro-d er w ho

m onitorsthe market and oc c asionally sub m itslim it ord ers,c ansigni¯c antly alter theequilib rium Hisinterventionforc espatient trad ersto sub m it m ore aggressive o®ersinord er to speed up exec utionand hence narrow sthe spread s.T hisresult provid es

im portant guid ance for m arket d esign

O ur resultsc ontrib ute to the grow ing literature onlim it ord er m arkets.M ost ofthe m od elsinthe theoretic alliterature are foc used onthe optim alb id d ingstrategies

for lim it ord er trad ers (see e.g G losten(199 4 ), Chakravarty and Holden(19 95),

R ock (19 96), Seppi (19 97), B iais, M artim ort and R oc het (2 0 0 0 ), P arlour and Seppi(2 0 0 1)).T hese m od elsd o not analyz e the c hoic e b etw eenm arket and lim it ord ersandare static For thisreasonthey d o not d escrib e the interac tionsb etw eenim patience,tim e-to-exec utionand ord er plac em ent strategiesasw e d o inthispaper

P arlour (1998) and Fouc ault (199 9) stud y d ynam ic m od els.P arlour (19 98) show show the ord er plac em ent d ec isionisin°uenced b y the d epth availab le at the insid equotes.Fouc ault (199 9) analyz esthe im pac t ofthe risk ofb eing pic ked o® and therisk ofnonexec utionontrad ers' ord er plac em ent strategies Inb oth m od els, limitord er trad ersd o not b ear w aiting c ost.Hence tim e-to-exec utiond oesnot in°uence

3 In the P aris B ourse,the designated market-makers arerequired topostbid-and askquotes for aminimum numberofshares and theirspread cannotexceed 5% ofthestockprice.

trad ers'b id d ingstrategiesinthese m od elsw hereasit playsa c entralrole inthe presentartic le.4

W e are not aw are ofother theoretic alpapersinw hic h pric esand tim e-to-exec ution

for limit ord ersare jointly d eterm ined inequilib rium T im e-to-exec ution, how ever,isanim portant d imensionofm arket quality inlim it ord er m arkets(see SE C 19 97).Lo,M c K inlayand Zhang(2 0 0 1) estim ate variousec onom etric m od elsfor the time-to-exec utionoflim it ord ers.Som e oftheir ¯nd ingsare c onsistent w ith our results, e.g.the expec ted tim e-to-exec utionincreasesw ith the d istance b etw eenthe lim it pric eand the m id -quote.O ur m od elalso generatesnew pred ic tionsthat c ould b e tested

w ith d ata onac tualtime-to-exec utionfor lim it ord ers.For instance w e show that theaverage tim e-to-exec ution(ac rossalllim it ord ers) d ependson(i) the tick size, (ii)the ord er arrivalrate and (iii) the proportionofpatient trad ers.5 B iais, HillionandSpatt (1995) d escrib e the interac tionsb etw eenthe size ofthe insid e spread and theord er °ow 6 T hey ob serve that limit ord er trad ersquickly im prove the insid e spread

w henit islarge.Inour m od elthe am ount b y w hic h a limit ord er trad er underc utsoroutb id sthe b est o®ersd epend son(i) the insid e spread ,(ii) the proportionofpatienttrad ersand (iii) the ord er arrivalrate.T hese ¯ndingsprovid e guid ance for empiric alstud iesoflimit ord er m arkets.7

T he paper is organized as follow s Sec tion2 d escrib es the m od el Sec tion3

d erivesthe equilib rium ofthe lim it ord er market and provid esexam ples.InSec tion

(1 994), D omowitz and W ang(1 994)and H arris (1 995)whoconsidermodels with exogenous order

° ow U singqueuingtheory, D omowitz and W ang (1 994)analyze the stochastic properties ofthe book A ngel(1 994)and H arris (1 995)study how the optimal choice between market and limit orders varies accordingtodi®erentmarketconditions (e g.the state ofthe book,the rate oforder arrival ).W e use more restrictive assumptions than these authors.B utthese assumptions enable

us toendogenizetheorder° owand the time-to-execution forlimitorders.

5 L oetal.(20 0 1 )reportthatthere is alarge variation in mean time-to-execution across stocks.

A ccordingtoourmodel,thesevariations canbeexplainedbythefactthatstocks di®erwithrespect totradingactivityorticksize.

6 See alsoB enston,Irvineand Kandel(20 0 1 ).

7 Empiricalanalyses oflimitordermarkets include G oldstein and Kavajecz (20 0 0 ), H anda and Schwartz (1 996), H arris and H asbrouck (1 996), H olli¯eld, M illerand Sandas (20 0 1 a,b), Kavajecz (1 999)and Sandas (20 0 0 ).

4 w e explore the e®ec t ofa c hange intic ksize and a c hange intrad ers'arrivalrate on

m easuresofm arket quality.Sec tion5 presentssome extensions.Sec tion6c onclud es.Allproofs(exc ept for P roposition1) are inthe Appendix

2 1 T im ingand M arket Struc ture

Consid er a c ontinuousm arket for a single sec urity, organiz ed asa lim it ord er b ook

w ithout interm ed iaries.W e assum e that latent inform ationab out the sec urity value

d eterm inesthe range ofad m issib le pric es, how ever the transac tionpric e itselfisd term ined b y trad ersw ho sub mit m arket and lim it ord ers.8 Spec i¯c ally, at pric e Aoutsid e investorsstand read y to sellanunlim ited am ount ofsec urity, thusthe sup-ply at A isin¯nitely elastic W e also assum e that there existsanin¯nite d em and

e-for sharesat pric e B (B < A) M oreover, A and B are c onstant over tim e T heseassum ptionsassure that allthe pric esinthe lim it ord er b ookare inthe range [B ;A].9

T he goalofthism od elisto investigate the b ehavior ofthe lim it ord er b ook andtransac tionpric esw ithinthisinterval T hisb ehavior isd eterm ined b y the supplyand d em and ofliquid ity,or inother w ord sb yoptim alsub m issionofm arket and limitord ers

T hisisanin¯nite horiz onm od elw ith d iscrete tim e period s At the b eginningofevery period a trad er arrivesat the m arket and ob servesthe lim it ord er b ook

E ac h trad er must b uy or sellone unit ofthe sec urity.T hese liquid ity trad ershave a

d iscretiononw hic h type oford er to sub m it.E ac h trad er c ansub m it a market ord er

to ensure anim med iate trad e at the b est quote availab le at the tim e.Alternatively,

he c ansub mit a limit ord er,w hic h improvesthe pric e,b ut d elaysthe exec ution.W eassum e that trad ers'w aitingc ostsare proportionalto the tim e theyhave to w ait until

8 W ediscuss this modellingstrategybelow.

9 A similarassumption is used in Seppi (1 997)andP arlourand Seppi (20 0 1 ).

c om pletionoftheir transac tion.Hence trad ersfac e a trad e-o® b etw eenthe exec utionpric e and the tim e-to-exec utionw henthey c hoose b etw eenm arket and lim it ord ers.Inc ontrast w ith Ad m ati and P °eid erer (19 88) or P arlour (19 98), trad ers are notrequired to c arry their d esired transac tionb y a d ead line.

Allpric es(b ut not w aitingc ostsand trad ers'valuations) are plac ed ona d iscretegrid T he tic k size, w hic h isc hosenb y the exc hange d esigner, isd enoted b y ¢ > 0 Allthe pric esinthe m od elare expressed interm sofinteger m ultiplesof¢ W e

d enote b y a and b the b est ask and b id quotesw hena trad er c omesto the m arket

T he insid e spread at that tim e iss := a¡b.G iventhe setup w e know that a ·A,

b¸B ,and s ·K := A ¡B 10

B oth b uyers and sellers c anb e oftw o types w hic h d i®er b y the siz e oftheir

w aiting c osts.T ype 1 trad ers(the patient type) incur anopportunity c ost ofd1 foranexec utiond elay ofone period T ype 2 trad ers(the im patient type) incur a c ostofd2 (0 ·d1 < d2).T he proportionofpatient trad ersinthe populationisd enoted

b y µ (0 < µ < 1).P atient typesc anb e thought asinstitutionsb uildingup positions,

or other long-term investors Arb itragersor b rokersc ond uc ting agency trad esareexam plesofim patient trad ers

Lim it ord ers are stored inthe lim it ord er b ook and are exec uted insequence

ac c ord ing to price priority (e.g.sellord ersw ith the low est o®er are exec uted ¯rst).For trac tab ility, w e m ake the follow ing sim plifying assumptions ab out the m arketstruc ture

A.1: E ach trad er arrivesonly once,sub m itsa m arket or a lim it ord er and exits.Sub m itted ord ersc annot b e c ancelled or m od i¯ed

A.2 : T rad ersw ho sub mit lim it ord ersmust narrow the spread b y at least onetic k

1 0 N oticethata;b;s;A ;B ;K andallotherspreadsandprices thatfollowarepositiveintegers.T his

is sosince we use integermultiples ofthe tick size, ¢ ;instead ofdollarprices and dollarspreads Furthermore the modeldoes not require time subscripts on variables, thus they are omitted for brevity.

A.3: B uyersand sellersalternate w ith c ertainty,e.g.¯rst a b uyer arrives,thenaseller,thena b uyer,and so on.T he ¯rst trad er isa b uyer w ith prob ab ility 0 5.Assum ptionA.1 impliesthat trad ersinthe m od eld o not ad opt ac tive trad ingstrategiesw hic h m ay involve repeated sub m issionsand c ancellations T hese ac tivestrategiesrequire m arket m onitoring, w hic h isc ostly (e.g.b ec ause liquid ity trad ers'tim e isvaluab le) T he sec ond assum ptionim pliesthat limit ord er trad ersc annotqueue at the sam e pric e (note how ever that they queue at d i®erent pric essince limitord ers d o not d rop out ofthe b ook) W ith this assum ption, the insid e spread isthe only state variab le w hic h in°uences trad ers' ord er plac em ent strategies T hisgreatly simpli¯esthe d escriptionand the c harac teriz ationoftrad ers'ord er plac em entstrategies.T hisassum ptionislessrestric tive thanit m ay appear.InSec tion6, w eshow that w e c and ispense w ith assum ptionA2 ifpatient trad ers' w aiting c ost islarge enough.T he third assum ptionfac ilitatesthe c om putationoftrad ers' expec ted

w aiting tim e and isimperative to keep the m od eltrac tab le (see Sec tion3.1 for a

d iscussion)

Let pband psb e the pric espaid b y b uyersand sellers,respec tively.Inour m od el,asinAd m ati and P °eid erer (19 88) for instance, trad ersd o not have the optionnot

to trad e T hustheir only d ec isionisa c hoic e ofstrategy resulting ina trad e A

b uyer c aneither pay the low est ask a or sub m it a limit ord er w hich c reatesa newinsid e spread w ith siz e j.Ina sim ilar w ay, a seller c aneither rec eive the largest b id

b or sub m it a lim it ord er w hic h c reatesa new insid e spread w ith siz e j.T hisc hoic e

d eterm inesthe exec utionpric e:

pb= a¡j; ps= b+ jw ith j2 f0 ;:::;s ¡1g;

w here j= 0 representsa market ord er It isc onvenient to c onsid er j(rather than

pb or ps) asthe trad er'sd ec isionvariab le For b revity, w e say that a trad er usesa

\j-lim it ord er"w henhe postsa lim it ord er w hic h c reatesa spread w ith siz e j.T heexpec ted tim e-to-exec utionofa j-lim it ord er isd enoted b y T (j).Since the w aiting

c ostsare assum ed to b e linear inw aitingtim e,the expec ted w aitingc ost ofa j-lim itord er isdiT (j), i2 f1;2 g:Asa market ord er entailsim m ed iate exec ution, w e setT(0 ) = 0

W e assume that trad ersare riskneutral.T he expec ted pro¯t oftrad er i(i2 f1;2 g)

w ho sub m itsa j-lim it ord er is:

Vb¡pb¢ ¡diT (j) = (Vb¡a¢ ) + j¢ ¡diT (j) iftrad er iisa b uyer

ps¢ ¡Vs¡diT (j) = (b¢ ¡Vs) + j¢ ¡diT (j) iftrad er iisa seller

w here Vb,Vsare b uyers'and sellers'valuations,respec tively.T o justifythisc lassi¯c tionto b uyersand sellers,w e assum e that Vb> > A¢ ,and Vs< < B ¢ 11 E xpressionsinparenthesisrepresent pro¯tsassoc iated w ith m arket ord er sub m ission.T hese prof-itsare d etermined b y the trad er'svaluationand the b est quotesw henhe sub mitshism arket ord er It isimm ed iate that the optim alord er plac ement strategy w henthe insid e spread hassize s solvesthe follow ingoptim izationprob lem ,for b uyersandsellersalike:

a-max

W e w illshow that T(j) isnon-d ec reasing inj, inequilib rium Hence a b etterexec utionpric e (larger value ofj) isob tained at the c ost ofa larger expec ted w aitingtim e

A strategy for a trad er isa m appingthat assignsa j-lim it ord er,j2 f0 ;:::;s¡1g;

to every possib le spread s2 f1;:::;K g.T hus,a strategy d eterm inesw hic h ord er tosub m it giventhe siz e ofthe insid e spread At the b eginning ofthe gam e w e set:

a = A and b = B hence s = K :Let oi(:) b e the ord er plac ement strategy ofatrad er w ith type i A trad er'soptimalstrategy d epend sonfuture trad ers' ac tionssince they d eterm ine hisexpec ted w aitingtim e,T (¢):Consequently a subgame perfec tequilibrium ofthe trad ing game isa pair ofstrategies, o¤

w henthe expec ted w aiting time T(¢) isc om puted using the fac t that trad ersfollowstrategieso¤

Inm ost m arket mic rostruc ture m od els, quotes are d eterm ined b y agents w hohave no reasonto trad e, and either trad e for spec ulative reasons, or m ake m oneyprovid ing liquid ity For these value-m otivated trad ers, the risk oftrad ing w ith a

b etter inform ed agent isa c oncernand a®ec tsthe optim alord er plac em ent strategies.Inc ontrast, inour mod el, trad ershave a non-inform ationm otive for trad ing andare prec om m ited to trad e T he risk ofad verse selec tionis not anissue for theseliquid ity trad ers.R ather,they d etermine their ord er plac em ent strategy w ith a view

at m inim iz ingtheir transac tionc ost and b alance the c ost ofw aitingagainst the c ostofob taining im m ed iac y inexec ution.13 Inord er to foc us onthistrad e-o® inthesim plest w ay, w e propose a fram ew ork that allow sfor a simple d ic hotom y b etw een

\m ac ro" inform ation-b ased asset pric ing and m arket \m ic ro"struc ture W e assum ethat inform ation-related c onsid erationsd eterm ine the pric e range, rather thanthepric e itself.T he equilib rium inthe m arket for liquid ity provisiond eterm inesquotesinsid e thisrange.At thisstage w e d o not m od elthe d eterm inationofthisrange,b utrather assum e that it exists.For ¯xed incom e sec uritiesthese b oundariesare quitenatural, giventhe existence ofc lose sub stitutes Inc ase ofequitiesw e c onjec turethat thispric e range representsthe c onsensusam ongallanalysts/investors,yet isnot

1 2 T he rules ofthe game, as wellas allthe parameters are assumed to be common knowledge amongallthetraders.

1 3 H arris(1 998)andG losten(20 0 0 )alsoarguethatoptimalorderplacementstrategiesaredi®erent forliquiditytraders and value-motivated traders.

sub jec t to arb itrage (see Shleifer and Vishny 199 7).

T he trad e-o® b etw eenthe c ost ofim m ed iate exec utionand the c ost ofd elayedexec utionm ay b e relevant for value-m otivated trad ersasw ell How ever, it isvery

d i± c ult to solve d ynamic m od elsw ith asym m etric informationam ong trad ersw ho

c anstrategic ally c hoose b etw eenm arket and lim it ord ers.Infac t w e are not aw areofsuch d ynam ic m od els.14

3 E quil ib rium P attern s

Inthissec tionw e c harac terize the equilib rium strategiesfor eac h type oftrad er.Forgivenvalues ofthe param eters, the equilib rium is unique W e also c alculate thestationary prob ab ility d istrib utionofthe insid e spread inequilib rium T he d ynam ic softhe ord er °ow and the d istrib utionofthe insid e spread d epend on(i) the proportionofpatient trad ersrelative to the proportionofim patient trad ersand (ii) the d i®erenceinw aitingc ostsb etw eenpatient and im patient trad ers.T hislead susto d istinguish

b etw eenthree d i®erent typesofequilib ria.W e provid e examplesw hic h illustrate theattrib utesofeac h one ofthe three equilib rium types

3 1 E xpec ted W aitingT im e

Inord er to c harac terize the equilib rium ,w e ¯rst analyze the b ehavior ofthe expec ted

w aitingtime functionT (j).Suppose the trad er arrivingthisperiod c hoosesa j-limitord er.W e d enote b y ®k(j) the prob ab ility that the trad er arriving next period and

ob servinganinsid e spread w ith size jc hoosesa k-lim it ord er, k2 f0 ;1;:::;j¡1g.15

Clearly ®k(j) d ependsontrad ers'strategiesand

j ¡1 X k= 0

®k(j) = 1;8j= 1;:::;K ¡1:

1 4 Chakravarty and H olden (1 995)considerasingle period modelin which informed traders can choosebetweenamarketandalimitorders.G losten(1 994)orB iais etal (20 0 0 )considerlimitorder markets with asymmetricinformationbutdonotallowtraders tochoosebetween marketand limit orders.

1 5 R ecallthatk= 0 stands foramarketorder.

Assum ptionA.2 impliesthat a trad er w ho fac esa one tick spread sub m itsa m arketord er Consequently, the tim e-to-exec utionfor a 1-lim it ord er is one period , i.e.T(1) = 1 Next, w e estab lish a generalrec ursive formula for the expec ted w aitingtim e function.T hisformula linksthe expec ted w aitingtim e functionto trad ers'ord erplac em ent strategies(d escrib ed b y the ® s0).

Lem m a 1 If®0(j) > 0 ; the expec ted w aitingtime for the exec utionofa j-lim it ord erisgivenby the follow ingrec ursive form ula:

®k(j)T (k)

3

5 8j= 2 ;:::;K ¡1 and T(1) = 1 (2 )

T w o extrem e c asesare w orth em phasiz ing.T he ¯rst isw henno trad er sub mits

a market ord er w henhe fac esa spread w ith siz e j¤.Inthisc ase ®0(j¤) = 0 and theexpec ted w aiting tim e ofa j-lim it ord er, w ith j¸ j¤, isin¯nite.Suc h limit ord ers

w illnever b e sub m itted inequilib rium ,since they are d om inated b y a m arket ord er.Hence,inequilib rium ,the expec ted w aitingtim e oflimit ord ersisalw ays¯nite.T his

im pliesthat lim it ord ersexec ute w ith c ertainty.16 T he sec ond c ase isw henalltrad erssub m it a m arket ord er w henthey fac e a spread w ith siz e j¤¤.Inthisc ase T(j¤¤) = 1

It w illb ec om e apparent that no spread ssm aller thanj¤¤and larger thanj¤c anb e

ob served inequilib rium Inb etw een,there isa variety ofc asesinw hich som e trad ers

¯nd it optim alto sub m it lim it ord ers,w hile otherssub mit m arket ord ers

Assum ptionA.3 isused to ob tainthe expec ted w aiting function(E q.(2 )) T healternationofb uyersand sellersyieldsa sim ple ord ering of the queue ofun¯lledlim it ord ers(the b ook): a j-lim it ord er c annot b e exec uted b efore j0-lim it ord ers

w here j0< j:T hisisofc ourse true w henw e c onsid er tw o b uy or tw o selllimit ord ers

b ec ause ofpric e priority.W ithout A.3,thisw ould not b e true how ever ifthe j-limitord er and the j0-lim it ord er are inopposite d irec tion(a b uy ord er and a sellord er

for instance).T he ord ering im plied b y A.3 explainsw hy the expec ted w aiting tim e

1 6 H owever, execution may take place aftera very longtime.In fact,in any ¯nite time interval, theexecution probabilityofaj-limitorderis strictlysmallerthan1 ;ifT (j)> 1

hasa sim ple rec ursive struc ture W ithout thisrec ursive struc ture, it b ec omesvery

d i± c ult to c om pute the expec ted w aitingtim e functionand the m od elis(ingeneral)intrac tab le

3 2 E quil ib rium strategies

Although the trad ing gam e hasanin¯nite horizon, the nod esw ith one-tic k spreadserve asend-nod esinthe usual¯nite gam e trees, since everyb od y sub m it a m arketord er.T husw e c ansolve the game b y bac kw ard induc tion.T o see thispoint,c onsid er

a trad er w ho arrivesinthe market w henthe size ofthe insid e spread iss = 2 :T hetrad er hastw o choic es: either he sub m itsa m arket ord er or a 1-lim it ord er.T he latter

im proveshisexec utionpric e b y one tic kc om pared to a m arket ord er b ut resultsinaone period d elay inexec ution.Choosing the b est ac tionfor eac h type oftrad er, w e

d eterm ine ®k(2 ) (for k= 0 and k= 1).If®0(2 ) = 0 ,the expec ted w aitingtim e for a

2 -limit ord er isin¯nite.It follow sthat no spread larger thanone tickc anb e ob servedinequilib rium If®0(2 ) > 0 ;w e c ompute T (2 ) (using E q.(2 )) T henw e proc eed to

s = 3 and so forth.T hisind uc tive approac h isthe key to m ost resultsinthe paper

T hree resultsfollow im m ed iately.F irst,asthisisa gam e ofperfec t informationanequilib rium inpure strategiesalw aysexists.Sec ond,since thisisa one-play game foreac h trad er,there are no Nash equilib ria (inpure strategies) other thanthe sub -gameperfec t equilib ria that w e trac e b y b ac kw ard induc tion.And third ,the equilib rium isunique for anytie-b reakingrule.W e c hoose the follow ingrule.Ifa trad er isindi®erent

b etw eena j1-limit ord er and a j2-lim it ord er,w ith j1 < j2,he sub m itsthe lim it ord er

w ith the sm allest spread (inthisc ase the j1-lim it ord er)

W e proc eed b y provingresultsthat c harac teriz e the equilib rium T rad erssub mitlim it ord ersonly ifthey c anc over their w aitingc ost.Since lim it ord ersw ait at leastone period ,there isa spread b elow w hic h a trad er stric tlyprefersto use m arket ord ers

W e refer to thisspread asb eingthe trad er's\reservationspread "and w e d enote it jR

i

for trad er i(i2 f1;2 g).T histhe sm allest spread trad er iisw illingto estab lish w ith

a lim it ord er,and stillthe assoc iated expec ted pro¯t isgreater thanzero (d om inates

a m arket ord er) Inord er to give a form ald e¯nitionofthe reservationspread , letint(x) b e the largest integer sm aller thanor equalto x T he reservationspread oftrad er iis:17

1 = jR

2 Intuitively, trad ersare indistinguishab le ifthe tw o w aiting c osts fallinto the same c ellonthe grid :[0 ;¢ );[¢ ;2 ¢ );[2 ¢ ;3¢ );:::

P roposition1 Suppose trad ers' typesare indistinguishable (jR

1 = jR

2 = jR) then, inequilibrium alltrad erssubmit a market ord er ifs· jR and submit a jR-lim it ord erifs > jR

T he proofofP roposition1 issim ple and intuitive hence w e present it here insteadofrelegating it to the Appendix.Consid er a trad er w ho arrivesinthe market w henthe insid e spread iss > jR.Ifhe sub m itsa j-lim it ord er w ith jR < jthenthe nexttrad er sub m itsa jR-lim it ord er giventhe spec i¯c ationof trad ers' strategies T his

im pliesthat ®0(j) = 0 (i.e the w aiting tim e is in¯nite) for jR < j:T herefore aj-lim it ord er w ith jR < jc annot b e optim alsince it isnever exec uted Ifthe trad ersub m itsa jR-limit ord er; hisord er isc leared b y the next trad er.B y d e¯nitionofthereservationspread ,thisc hoic e d om inatesa m arket ord er.T hisestab lishesthat w henthe insid e spread islarger thantrad ers' reservationpric e, the optim alstrategy istosub m it a jR-limit ord er.F inallyc onsid er a trad er w ho arrivesinthe m arket w henthespread iss·jR.B y d e¯nitionofthe reservationspread ,the sub m issionofa m arket

waitingcostforatraderwith type i is d i :Itfollows thatthe smallestspread traderi can establish

ord er isa d om inant strategy for thistrad er.T hisc ompletesthe proofofP roposition1.

T he equilib rium w ith indistinguishab le trad ersisc harac teriz ed b y anoscillatingpattern.T he ¯rst,asw ellasevery od d -num b ered trad er afterw ard s,sub m itsa limitord er w hic h c reatesa spread w ith size jR T he sec ond, and every even-numb eredtrad er afterw ard s, sub mitsa m arket ord er.T he insid e spread oscillatesb etw eenKand jR and transac tionstake plac e only w henthe spread issmall.T rad e pric esareeither A¡jR ifthe ¯rst trad er isa b uyer, or B + jR, ifthe ¯rst b uyer isa seller

T he outc om e isc om petitive inthe sense that limit ord er trad ersalw aysquote theirreservationspread ,that isthe spread suc h that they just c over their w aitingc ost.18

After c harac teriz ing the ¯rst type ofequilib rium , w e proc eed b y assum ing thattrad ersare heterogeneous: jR

1 < jR

2 G iventw o spread sj1 < j2 w e d enote b yhj1;j2ithe set:fj1;j1+ 1;j1+ 2 ;:::;j2g,i.e.the set ofallpossib le spread sb etw eenj1 and j2(inclusive).Inpartic ular,the range ofallpossib le spread sish1;K i

P roposition2 Suppose trad ersare heterogeneous(jR

1 < jR

2 ) Inequilibrium thereexistsa c uto® spread sc2 hjR

2 ;Ki suc h that:

1.G ivena spread s2 h1;jR

1 i; patient and impatient trad erssubmit a market ord er

2 G ivena spread s2 hjR

1 + 1;sci; a patient trad er submitsa limit ord er and an

im patient trad er submitsa m arket ord er

3.G ivena spread s 2 hsc+ 1;Ki; patient and impatient trad erssubmit a limitord er

T he propositionshow sthat w henjR

reservationspread ofthe patient trad er, jR

1, representsthe sm allest spread ob servedinthe m arket.At the other end scisthe largest quoted spread inthe market.Limitord ersw hic h c reate a larger spread have anin¯nite w aiting tim e since no trad erssub m it a m arket ord er w henthe insid e spread islarger thansc Hence these limitord ersare never sub m itted T hisob servationperm itsusto restric t our attentionto

c asesw here sc= K ; for b revity.T hisequality holdstrue w henthe c ost ofw aitingforanim patient trad er issu± c iently large.19 U nder thisc onditionim patient trad ersal-

w aysd em and liquid ity (sub mit m arket ord ers),w hile patient trad erssupply liquid ity(sub m it lim it ord ers) w henthe insid e spread islarger thantheir reservationspread

P roposition3 Suppose sc= K :Any equilibrium exhibits the follow ingstruc ture:there exist q spread s, n1 < n2 < :::< nq, w ith n1 = jR

1 , nq= K and 2 ·q ·K ; suc hthat the optimalord er submissionstrategy isasfollow s:

²Animpatient trad er submitsa market ord er, for any spread inh1;K i

²A patient trad er submitsa market ord er w henhe facesa spread inh1;n1i andsubmits a nh-limit ord er w henhe faces a spread inhnh + 1;nh + 1i for h =1;:::;q¡1

Hence w hena patient trad er fac es aninsid e spread w ith size nh + 1 > jR

1; heresponds b y sub m itting a limit ord er w hic h im proves uponthe insid e spread b y(nh + 1 ¡nh) tic ks T hisord er estab lishesa new insid e spread equalto nh W henthe insid e spread isK ; it takesa streak ofq¡1 patient trad ersto b ring the insid espread to the c ompetitive leveljR

1 :Hence q d eterm inesthe m axim alnumb er oflimitord ersw hic h c anb e ob served inthe b ook.W e refer to q asthe length ofthe book:A

sm alllength ofthe b ookmeansthat patient trad ersquic kly m ake good o®erssince ittakesa few patient trad ersto b ringthe spread to the c om petitive level

1 9 Forinstance, s c = K ifj R

¸ K: Itis worth stressingthatthis condition is su±cientbutnot

Next w e analyz e the expec ted w aiting time inequilib rium Let r := 1¡µµ b e theratio ofthe proportionofpatient trad ers to the proportionofim patient trad ers.Intuitively, w henthis ratio is sm aller (larger) than1, liquid ity is c onsum ed m ore(less) quic kly thanit issupplied Asw e show b elow , thisratio d eterm inestrad ers'

b id d ingstrategiesand time-to-exec utionfor lim it ord ers

P roposition4 T he expec ted w aitingtime functioninequilibrium isgivenby:

T (n1) = 1 an d T (nh) = 1 + 2

h X k= 2

Another d eterm inant ofthe expec ted w aiting tim e isthe proportionofpatienttrad ersrelative to the proportionofim patient trad ers,r.T he intuitionisasfollow s.Notic e that h d eterm inesthe priority statusofa lim it ord er inthe queue ofun¯lledlim it ord ers.Ac tually annh-lim it ord er c annot b e exec uted b efore nh0-lim it ord ershave b eenexec uted ifh0< h (w henthese ord ersare present inthe b ook,ofc ourse)

W henr increases, the likelihood ofa m arket ord er d ec reases It follow s that theexpec ted w aitingtim e for the hth lim it ord er inthe queue enlarges.It turnsout thatthe rate ofincrease inthe w aitingtim e from one lim it ord er to the next inthe queue oflim it ord ersd ependsonr asw ell.Ac tually w henr > 1(r < 1) the m arginalexpec ted

w aiting tim e T (nh)¡T(nh ¡1) isnon-d ec reasing (non-increasing) inh Inthisc ase,

w e say that T(¢) is\c onvex"(\c oncave") inh T he next c orollary sum mariz estheseremarks

Corollary 1 T he expec ted w aitingtim e ofthe hth limit ord er inthe queue oflim itord ersincreasesw ith r, the ratio ofthe proportionofpatient trad ersto the proportionofim patient trad ers.T he expec ted w aitingtim e functionis\convex"w henr > 1; and

\concave" w henr < 1

W e show b elow that these propertiesofthe expec ted w aitingtim e fence trad ers' b id d ing strategies.Inthe next propositionw e expressthe spread sonthe equilib rium path, i.e n1;n2;::;nq, interm softhe exogenousparam eters De-

unctionin°u-¯ne ªh := nh ¡nh ¡1 for h ¸ 2 asthe spread improvement, w henthe insid e spreadhasa siz e equalto nh.T he spread im provem ent isthe num b er oftic ksb y w hic h atrad er narrow sthe spread w henhe sub m itsa lim it ord er.T he larger isthe spread

im provem ent,the m ore aggressive isthe lim it ord er

P roposition5 T he set ofequilibrium spread sisgivenby:

n1 = j1R; nq= K ;

nh = n1 +

h X k= 2

ªk h = 2 ;:::;q¡1;

w here

ªh = int(2 rh ¡1d1

¢ ) + 1and the length ofthe book, q isthe smallest integer suc h that:

j1R +

q X k= 2

T he previouspropositionshow sthat w henever,2 d1rh ¡1 ¸¢ ,a lim it ord er trad er

¯ndsoptim alto underc ut or to outb id the b est pric esb ym ore thanone tic k(ªh > 1)

B iais, Hillionand Spatt (19 95) ob serve that liquid ity suppliersfrequently im proveuponthe b est quotesb y severaltic ks O ur result id enti¯es four d eterm inantsforthe spread im provement w hic h c ould b e c onsid ered infuture em piric alinvestigation

T hese d eterminantsare: (i) the proportionofpatient trad ers, r, (ii) the per period

w aiting c ost,d1 (iii) the tic k siz e,¢ ,and (iv) the insid e spread W e analyz e eac h ofthese d eterm inantsinturn

W henr increases,the tim e-to-exec utionfor a givenpositioninthe queue oflim itord ersb ec om eslarger.Hence,other thingsequal,liquid ity suppliersb ear larger w ait-ingc osts(d1T):T rad ersreac t b y sub m itting m ore agressive ord ersto preem pt goodpositionsinthe queue oflim it ord ersand thereb yred uc e their time-to-exec ution.T hesame e®ec t operatesw hend1 increases.Inthisc ase,trad ersb ear larger w aitingc osts

b ec ause the per-period w aitingc ost islarger.T he sm aller isthe tic ksiz e,the sm alleristhe c ost ofim provinguponthe b est b id and askpric es.T husa sm aller tic kresultsinlarger spread im provem entsinterm softic ks

T he spread im provement, ªh, increases(d ec reases) w ith h w henr > 1 (r < 1):

T hism eansthat w henr > 1 the spread im provem ent increasesw ith the siz e oftheinsid e spread , w hile the opposite istrue w henr < 1 T he intuitionisasfollow s.Consid er the (h¡1)th trad er inthe queue ofun¯lled lim it ord ers.T histrad er'stime

to exec utionisT (nh ¡1) instead ofT(nh) for the trad er b ehind him inthe queue.Hencethe d i®erence inexpec ted w aitingc ost b etw eenthe hth and the (h ¡1)th positionsinthe queue oflim it ord ersisequalto (T(nh)¡T (nh ¡1))d1.Intuitively,thisshould b ethe \pric e"ofac quiringthe (h¡1)th positioninstead ofthe hthpositioninthe queue

T he d ollar spread im provem ent playsthe role ofthispric e and, for thisreason, itisapproximately equalto (T (nh)¡T(nh ¡1))d1:2 0 T hisshow sthat the shape ofthe

w aiting tim e functiond eterm inesthe relationship b etw eenthe spread im provem entand the insid e spread W henr > 1,the w aitingtim e functionisc onvexinh Henceliquid ity supplierso®er larger spread im provementsw henthe spread islarge.W hen

r < 1,the w aitingtim e functionisc oncave and liquid ity supplierso®er larger spread

im provem entsw henthe spread issm all

20 Infactobservethatª h ¢ ' 2r h ¡1 d 1 = (T (n h ) ¡T (n h ¡1 ))d 1 :T hedollarspreadimprovementis onlyapproximatelyequaltothedi®erenceinwaitingcostbecausethe setofprices is discrete.

Notic e that w henspread im provementsare larger than1 tic k,the trad ersd o not

m ake use ofallthe possib le pric esinequilib rium T hisim pliesthat the limit ord er

b ook features\holes", i.e.c asesinw hic h the d istance b etw eentw o c onsec utive ask

or b id pric esislarger thanone tic k.2 1

T he last part ofthe previousproposition(E q.(4 )) im pliesthat that the lengthofthe b ook d ec reasesw henspread im provem entsget larger Ac tually, lim it ord ertrad ersim prove onthe b est quotes b y a larger numb er ofticksso that a sm allernumb er ofpric esonthe grid are used T hism eansthat m ore c om petitive outc om esare expec ted w henthe length ofthe b ookissm all.T hisisthe c ase inpartic ular w hen

r¸1 b ec ause (a)spread im provem entsare large and (b ) liquid ityisnot c onsumed tooquic kly(w hic h leavestime for the insid e spread to narrow ).For thisreasonw e c alltheequilib rium w henr ¸ 1 a High Competition(HC) E quilibrium and the equilib rium

w henr < 1; a Low Com petition(LC) E quilibrium.U singthisterm inology,w e c lassifyallequilib ria inthree c ategoriesd escrib ed inT ab le 1

T ab le 1 - T hree equilib rium patterns

E quilib rium pattern Description Spec i¯c ation

O scillating Indistinguishab le T rad ers jR

1 = jR

2 ;8rSpread soscillate b etw eenK and jR:

1 < jR

2 ; r¸1High levelofc om petition

am ongliquid ity provid ers

21 H oles in the limitorderbook is a phenomenon documented by severalempiricalstudies: B ais, H illion and Spatt (1 995)- P aris B ourse;G oldstein and Kavajecz (20 0 0 )- N Y SE;H olli¯eld,

i-M iller,andSandas (20 0 1 a)-Stockholm;B enston,Irvine,andKandel(20 0 1 )-T oronto;andKandel,

L auterbach,and T kach(20 0 0 )-T elA viv.

3 3 E xam pl es

W e illustrate the three equilib rium patternsb y num eric alexam ples.T he tic k siz e is

¢ = $ 0 :12 5.T he low er pric e b ound ofthe b ook isset to B ¢ = $ 2 0 , and the upper

b ound isset to A¢ = $ 2 2 :5.T hus,the m axim alspread isK = 2 0 (K ¢ = $ 2 :5).T heparametersthat d i®er ac rossthe exam plesare presented inT ab le 2

T ab le 3presentsthe equilib rium strategy for patient (type 1) and im patient (type

2 ) trad ersineac h exam ple.E ac h entry inthe tab le presentsthe optimallim it ord er(interm softic ks) giventhe c urrent spread (0 standsfor a market ord er).2 2

T ab le 3 - E quilib rium strategies

22 T he equilibrium strategies in Examples 2 and 3 followfrom the formulae given in P roposition 5.

Current E xam ple 1 E xam ple 2 E xam ple 3

Spread T ype 1 T ype 2 T ype 1 T ype 2 T ype 1 T ype 2

2 = 2 , thuspatient and impatient trad ersare indistinguishab le

T he insid e spread oscillatesb etw eenthe m aximalspread of2 0 tic ksand the vationspread of2 tic ks InE xam ple 2 and 3, the trad ersare heterogeneoussince

im provem entsare larger thanone tic k for allspread sonthe equilib rium path inthis

23 T able3 speci¯es actions forspreads onando®theequilibrium path.T his is necessaryforafull speci¯cation oftheequilibrium strategy.

c ase.Inc ontrast, inE xam ple 3, spread im provem entsare equalto one tick inm ost

c ases.Hence the m arket w illappear m ore c om petitive inE xam ple 2 (r > 1) thanin

E xam ple 3(r < 1)

E xpec ted W aitingT ime

T he expec ted w aiting tim e functioninE xam ples2 and 3 isillustrated inF igure1.T his¯gure presentsthe expec ted w aitingtim e ofa lim it ord er asa functionofthespread it c reates Inb oth examplesthe expec ted w aiting tim e increasesw henw e

m ove from one reac hed spread to the next one, w hile it isc onstant over the spread s

w hich are not posted inequilib rium T he expec ted w aiting tim e issm aller at anyspread inE xam ple 3.T hisexplainsthe d i®erencesinb id d ingstrategiesinE xamples

2 and 3.W henr < 1, lim it ord er trad ersare lessaggressive b ec ause they expec t a

faster exec ution

B ookDynamic s

F igure 2 illustratesthe b ook resultingfrom 4 0 roundsofsim ulationm pendent d raw sfrom the d istrib utionoftrad ers' type.W e use the sam e realizations

akinginde-for E xam ples2 and 3and look at the d ynam ic softhe lim it ord er b ook

Asisapparent from F igure 2 , the insid e spread c onvergesm ore quic kly tow ard s

sm alllevelsinE xam ple 2 thaninE xam ple 3 Since the type realiz ationsinb oth

b ooksare id entic al, thisob servationisonly d ue to the fac t that patient trad ersuse

m ore aggressive limit ord ers,inord er to speed up exec ution,inE xam ple 2 Ifthe typerealiz ationsw ere not held c onstant,there w ould b e a sec ond forc e ac tinginthe same

d irec tion.W henr islarger than1,the liquid ity o®ered b y the b ookisconsumed lessrapid ly thanw henr issmaller than1.T hismeansthat the likelihood ofa m arketord er arriving w hile the spread islarge issm aller w henr > 1 T hise®ec t w ouldreinforc e the fac t that spread stend to b e sm aller inE xam ple 2 W e prove thispoint

m ore formallyinthe next sec tionb yd erivingthe prob ab ilityd istrib utionofthe insid espread

Figure 1 - Expected waiting time

Figure 2 - Book Simulation (same realizations of type arrivals for two examples)

Example 2 - Intense competition among liquidity suppliers ( r = 1.222)

Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Trader B2 S1 B1 S2 B2 S2 B1 S1 B2 S1 B2 S1 B1 S1 B2 S1 B1 S1 B1 S2 B1 S1 B1 S1 B2 S2 B1 S2 B1 S1 B1 S2 B2 S1 B2 S2 B2 S1 B1 S2

3 4 Distrib utionofSpread s

W e have so far estab lished the struc ture ofequilib rium strategies.O ur next step is

to d erive the prob ab ility d istrib utionofspread sind uc ed b y these strategies.Inthis

w ay, w e show that sm allspread sare m ore frequent inmarketsw here r > 1:T his

form aliz esthe intuitionthat c om petitioninthese m arketsism ore intense.W e alsouse the d istrib utionofspread sinord er to c alculate m easuresofm arket qualityinthenext sec tion

m arket ord er w henhe fac esa spread ofsiz e n1.Anim patient trad er alw ayssub mits

a m arket ord er (w e maintainthe assum ptionthat sc= K ) T hus, ifthe insid espread hassize nh (h = 2 ;:::;q¡1) the prob ab ility that itssize b ec om esnh ¡1 inthenext period isµ; and the prob ab ility that itssiz e b ec om esnh + 1 inthe next periodis1¡µ.Ifthe size ofthe insid e spread isn1 allthe trad erssub m it m arket ord ersand itssiz e b ec om esn2 w ith c ertainty.Ifthe siz e ofthe insid e spread isK thenitremainsunchanged w ith prob ab ility 1¡µ (a m arket ord er) or it d ec reasesto nq¡1

w ith prob ab ility µ (a lim it ord er).Hence the insid e spread isa ¯nite M arkov chain

w ith q¸2 states.T he q £q transitionm atrixofthisM arkov chain,d enoted b y W ;is:

W =

0 B B B B B

T he jth entry inthe hth row ofthism atrixgivesthe prob ab ility that the siz e of

the insid e spread b ec om esnjc ond itionalonthe insid e spread havingsiz e nh (h ;j=1;:::;q) A stationary d istrib utionofthis M arkov c hain, m ay b e regard ed as thelong term prob ab ility d istrib utionofthe insid e spread s.2 4 W e d enote the stationaryprob ab ilitiesb y u1;:::uq; w here uh isthe prob ab ility ofaninsid e spread w ith siz e nh:Lem m a 2 T he M arkov c haingivenby W hasa unique stationary d istribution.T hestationary probabilitiesare givenby:

r < 1) Inc ontrast, it skew ed tow ard low er spread sinE xam ple 2 (w here r > 1)

T hisob servationiseasily explained b y c onsid eringthe expressionsfor the stationaryprob ab ilities.For h ;h02 f2 ;3;:::;qg w ith h > h0,the previouslem m a impliesthat

w hich yieldsthe follow ingproposition

P roposition6 For a giventic ksize and givenvaluesofthe w aitingcosts

1.Ifr < 1 (LC equilibrium) , uh > uh 0for 1 ·h0< h · q:T hismeansthat the

d istributionofspread sisskew ed tow ard shigher spread sw henr < 1

2 Ifr > 1 (HC equilibrium) , uh < uh0for 2 ·h0< h ·q:2 5T hismeansthat the

d istributionofspread s isskew ed tow ard slow er spread sw henr > 1

24 See Feller(1 968).

25 T he inequality, u h < u h 0 ; does notnecessarily hold forh 0 = 1 ; even ifr > 1 A ctually the smallestinsidespread can onlybe reached from higherspreads whileotherspreads can be reached from bothdirections (n q = K can bereached eitherfrom n q ¡1 orfrom n q itself).T his implies that theprobabilityofobservingthe smallestpossiblespread is relativelysmallforallvalues ofr.

Figure 3 - Equilibrium spread distribution