Inform ationSharing,Liquid ity and T ransac tionCosts

inF loor-B ased T rad ingSystem s.

1

T hierry Fouc ault

HE C and CE P R

1,rue d e la Lib ¶eration

78351 J ouy enJ osas,France.

E m ail:f

ouc ault@hec .f

r

Laurence Lesc ourret

CR E ST and Doc torat HE C

15,B oulevard G ab rielP ¶eri

9 2 2 4 5 M alako®,France.

E mail:lescourr@ensae.f

r

Novemb er,2 0 0 1

1

W e thank G iovanni Cespa, A sani Sarkarand seminarparticipants atCR EST , L avalU ni-

versity, the A FFI20 0 0 conference, the EEA 20 0 0 conference, the FM A 20 0 1 M eetings and the
InternationalFinanceConference T unisie 20 0 1 .A llerrors areours.

Ab strac t

In

f

orm ationSharin

g, Liquid ity an

d T ran

sac tionCostsinF l

oor-B ased T rad ing

System s.

W e c onsid er inf

orm ationsharingb etw eentrad ers(\°oor b rokers") w ho possessd i®erent

typesofinf

ormation,namelyinf

orm ationonthe payo® ofa riskysec urityor inf

ormationon

the volume ofliquid itytrad inginthissec urity.W e interpret these trad ersasd ual-c apac ity

b rokersonthe °oor ofanexc hange.W e id entif

y c onditionsunder w hic h the trad ersare

b etter o® sharing inf

ormation. W e also show that inf

orm ationsharing im proves pric e

d iscovery, red uc esvolatility and low ersexpec ted trad ing c osts. Inf

orm ationsharing c an

im prove or im pair the d epth ofthe m arket, d epend ing onthe valuesofthe param eters.

O verallour analysissuggeststhat inf

orm ationsharing am ong °oor b rokersim provesthe

perf

orm ance of°oor-b ased trad ingsystem s.

K eyw ord s: M arket M ic rostruc ture,F loor-B ased T rad ingSystem s,O penO utc ry,Inf

or-

m ationSharing,Inf

orm ationSales.

J E L Classi¯c ationNumb ers: G 10 ,D82 .

1

In

trod uc tion

T he organiz ationoftrad ing onthe NY SE hasb eenrem arkab ly stab le since its¯rst c on-

stitutionin1817.T rad ingisc onduc ted through openoutc ry ofb id sand o®ersofb rokers

ac tingonb ehalfoftheir c lientsor f

or their ow nac c ount.

1

T histrad ingm echanism isnot

unique to the NY SE .E quity m arketslike the Frankf

urt Stoc k E xc hange and the AM E X

or d erivativesm arketslike the CB O T and the CB O E are °oor m arkets.

2

How ever °oor-

b ased trad ing m ec hanism sare endangered spec iesasthey are progressively replac ed b y

f

ully autom ated trad ing system s

3

. G iventhistrend tow ard autom ation, it isnaturalto

ask w hether °oor-b ased trad ing system sc anprovid e greater liquid ity and low er exec u-

tionc oststhanautomated trad ingsystem s.T hisquestionisofparam ount importance f

or

m arket organizersand trad ers. Inf

ac t, it hasb eenhotly d eb ated b etw eenmemb ersof

E xc hangesw ho c onsid ered sw itc hingf

rom °oor to elec tronic trad ing

4

.Inord er to survive

°oor-b ased trad ing m ec hanism smust outperf

orm autom ated trad ing system salong som e

d imensions.

Autom ated trad ing system sd ominate °oor-b ased trad ing systems inm any respec ts.

F irst °oor m arketsare more expensive to operate (see Dom ow itz and Steil(19 99 )).Sec -

ond physic alspac e lim itsthe numb er ofpartic ipantsin°oor marketsb ut not inautomated

trad ingsystem s.F inally trad ersw ithout anac c essto the °oor are at aninf

orm ationald is-

ad vantage c om pared w ith the trad ersonthe °oor.T hisd isad vantage islikelyto exac erb ate

agency prob lem sb etw eeninvestorsand their b rokers(Sarkar and W u (199 9)).

B y d esign, °oor-b ased marketsf

oster person-to-personc ontac ts. Hence the ab ility of

m arket partic ipantsto share inf

ormationisgreater inthese markets.T hisf

eature isof

ten

view ed asb eingone ad vantage,ifnot the unique one,of°oor-b ased trad ingsystem s.

5

For

instance Harris(2 0 0 0 ),p.8,pointsout that

1

O fcourse,many tradingrules have been changed since the creation ofthe N Y SE.B utithas always

beena° oormarket.SeeH asbrouck,So¯anos andSosebee(1 993)foradetaileddescriptionofthetrading
rules on the N Y SE.

2

In Frankfurt,the° ooroperates in parallelwith an electronictradingsystem.

3

T heM arch¶

eµ

aT ermeInternationaldeFrance(M A T IF),theT orontoStockExchangeandT heL ondon

InternationalFinancialFutures and O ptions Exchange (L IFFE)shutdown their° oorin 1 997,1 998 and
20 0 0 ,respectively.

4

See the Economist(July 31 st1 999):\A home grown revolutionary"and the Economist(A ugust26th

20 0 0 ):\ O utofthepits".

5

Covaland Shumway (1 998)show thatthe levelofnoise on the ° oorofCB O T '

s 30 yearT reasury

B ond futures a®ects price volatility.T his alsosuggests thatperson toperson contacts on the ° oorhave
an impacton priceformation.

1

`F loor-based trad ingsystem sd om inate elec tronic trad ingsystemsw henbrokers

need to exc hange inf

ormationabout their c lientsto arrange their trad es.'

Inf

orm ationsharing isa f

unctionofthe °oor w hic h isd i± c ult to replic ate inelec tronic

trad ing system s. T hese system susually restric t the set ofm essagesthat c anb e sent b y

users(generally trad ersc anonly post pric esand quantities).Furtherm ore trad inginthese

system sisinm ost c asesanonym ous. T hisf

eature preventstrad ersf

rom d eveloping the

reputationofhonestly sharinginf

orm ationthrough enduringrelationships.

Inf

orm ationsharingonthe °oor c antake plac e b etw eentw o typesofpartic ipants.F irst

°oor-b rokersc anexc hange inf

orm ationontheir trad ing motivationsw ith m arket-makers.

B enveniste,M arc usand W hilelm (199 2 ) m od elthistype ofinf

orm ationsharingand show

that it m itigatesad verse selec tion.Sec ond °oor-b rokersc anc ommunic ate w ith other °oor-

b rokers.For instance,So¯anosand W erner (1997),p.

6 notic e that

`Inad d ition, by standinginthe c row d , °oor brokersmay learnabout ad d itional

broker-represented liquid ity that is not re°ec ted inthe spec ialist quotes: °oor

brokers w illof

tenexc hange inf

orm ationontheir intentions and capabilities,

espec ially w ith competitorsw ith w hom they have good w orkingrelationships.'

O ur purpose inthispaper isto analyz e thistype ofinf

orm ationsharing.At ¯rst glance,

inf

orm ationsharing am ong °oor b rokersispuzz ling.Inf

ac t standard m od elsw ith asym -

m etric inf

orm ation(e.

g.K yle (19 85)) show that inf

orm ed trad ersw ant to hid e their inf

or-

m ationrather thand isclose it to potentialc om petitors.Furtherm ore,inf

orm ationsharing

reinf

orc esinf

orm ationalasymetriesb etw eenthose w ho share inf

orm ationand those w ho

d o not.It istheref

ore not ob viousthat it should im prove m arket quality.Hence w e ad -

d resstw o questions.F irst, isit optimalf

or °oor b rokersto share inf

orm ationw ith their

c om petitors? Sec ond, w hat isthe e®ec t ofinf

orm ationsharing am ong °oor b rokerson

the overallperf

orm ance ofthe m arket? Inpartic ular w e stud y the impac t ofinter-°oor

b rokersc om munic ationonstandard m easuresofm arket quality, nam ely pric e volatility,

pric e d iscovery,m arket liquid ity and trad ingc osts.

W e m od el°oor trad ingand inf

orm ationsharingusingK yle (1985)'sm od elasa w orkhorse.

AsinR oÄell(199 0 ),w e assum e that trad ers(°oor b rokers) have ac c essto tw o typesofin-

f

orm ation: (i) f

undamentalinf

orm ationw hic h isinf

orm ationonthe payo® ofthe sec urity

and (ii) non-f

undam entalinf

orm ationw hic h isinf

orm ationonthe volume ofliquid ity(non-

inf

orm ed ) trad ing.W e c onsid er the possib ility f

or tw o °oor b rokersendow ed w ith d i®erent

2

typesofinf

orm ation(one hasf

undam entalinf

orm ationand the other hasnon-f

undamental

inf

orm ation) to share inf

orm ation.M ore spec i¯c ally w e assum e that °oor b rokershave in-

f

orm ationsharingagreem ents(they f

orm a \c lique").Anagreem ent spec i¯esthe prec ision

w ith w hic h eac h b roker reportshisor her inf

orm ationto the other b roker.Af

ter rec eiving

f

undam entalor non-f

undam entalinf

orm ation,the b rokersina c lique pooltheir inf

orm ation

ac c ord ingto the term softheir agreem ent just b ef

ore sub m ittingtheir ord ersf

or exec ution.

W e estab lish the f

ollow ingresults.

²T here isa w id e range ofparam etersfor w hich it isoptim alfor °oor b rokersto share

their inf

orm ation(i.e.their expec ted pro¯tsare larger w ith inf

orm ationsharing).

²Inform ationsharing c anim prove or impair the d epth ofthe market, d epending on

the valuesofthe parameters.

²Inform ationsharingalw aysred uc esthe aggregate trad ingc ostsfor liquid ity trad ers.

How ever w heninf

orm ationsharingim pairsm arket d epth,som e liquid ity trad ersare

hurt.

²Inform ationsharingoc c ursat the expense ofthe °oor b rokersw ho are not part to

the inf

orm ationsharingagreem ent.

²Inform ationsharingimprovespric e d iscovery and red uc esm arket volatility.

Intuitively inf

orm ationsharing intensi¯esc om petitionb etw een°oor b rokersand inthis

w ay it low ersthe totalexpec ted pro¯tsofall°oor b rokers(red uc esthe aggregate trad ing

c osts). Inf

orm ationsharing also c hanges the alloc ationoftrad ing pro¯ts am ong °oor

b rokers.M ore spec i¯c ally the °oor b rokersw ho share inf

orm ationc apture a larger part of

the totalexpec ted pro¯ts, at the expense of°oor b rokersw ho d o not share inf

orm ation.

T hese tw o e®ec ts explainw hy inf

orm ationsharing c ansimultaneously b ene¯t liquid ity

trad ersand the °oor b rokersw ho share their inf

orm ation. O verallinf

ormationsharing

b etw een°oor b rokersisanad vantage f

or °oor-b ased trad ingsystem ssince it resultsin(a)

low er trad ingc osts, (b ) f

aster pric e d iscovery and (c ) low er pric e volatility.Interestingly,

inline w ith our result, Venkataram an(2 0 0 0 ) ¯ndsthat trad ing c ostsonthe NY SE are

low er thanonthe P arisB ourse (anautom ated trad ingsystem ),c ontrollingf

or d i®erences

instocksc harac teristic s.

6

6

T heissen (1 999)compares e®ective bid-askspreads in an automated tradingsystem (X etra)and the

° oorofthe FrankfurtStock Exchange forstocks thattrade in both systems.H e ¯nds thatthe average

3

O ur analysisisrelated to the literature oninf

ormationsales(e.g.Ad m atiand P °eid erer

(19 86), (19 88) and F ishm anand Hagerty (199 5)). Inc ontrast w ith this literature, w e

assum e that the m ed ium f

or inf

orm ationexc hange isinf

orm ation, not m oney. Ac tually

inour m od el, the trad er w ho rec eivesinf

orm ationrew ard sthe inf

orm ationprovid er b y

d isclosinganother type ofinf

orm ation.Hence w e c onsid er °oor-b ased system sasm arkets

f

or trad ingsharesand f

orum to b arter inf

orm ation.Another im portant d i®erence isthat w e

c onsid er c om munic ationofinf

orm ationonthe volum e ofliquid ity trad ing.W e show that

it m ay b e optim alto `

sell'(b arter) suc h aninf

orm ationand that salesofnon-f

undam ental

inf

orm ationhave anim pac t onmarket quality.

T he m od elisd escrib ed inthe next sec tion.Sec tion3 show sthat it c anb e optim alf

or

°oor b rokersto share inf

orm ation.Sec tion4 analyz esthe im pac t ofinf

orm ationsharing

onvariousm easuresofm arket perf

orm ance.Sec tion5 c onclud es.T he proof

sw hic h d o not

appear inthe text are inthe Appendix.

2

T he M od el

2 .

1

Inf

orm ationSharin

gAgreem en

ts

T he T rad ingCrow d .

W e c onsid er a m od eloftrad inginthe market f

or a riskysec urityw hic h isb ased onK yle

(19 85). T he ¯nalvalue ofthe sec urity, w hic h isd enoted ~

v, isnorm ally d istrib uted w ith

m ean¹ and a variance ¾

2

v

that w e norm alize to 1.T his¯nalvalue ispub lic ly revealed at

d ate 2 .T rad inginthissec uritytakesplac e at d ate 1.At thisd ate,investorssub mit m arket

ord ersto b uy or to sellsharesofthe sec urity.T he exc essd em and (supply) isc leared at

the pric e posted b y a c ompetitive and risk-neutralm arket m aker.

T he trad ing \c row d " f

or the sec urity isc om posed ofN + 1 °oor b rokers.

7

At tim e

1, there are tw o typesof°oor b rokers: (i) N f

undam entalspec ulatorsand (ii) one non

f

undam entalspec ulator, B .Fundam entalspec ulatorshave inf

orm ationonthe ¯nalvalue

ofthe sec urity.For simplic ity,asinK yle (19 85),w e assum e that theyperf

ec tlyob serve this

¯nalvalue,just b ef

ore sub m ittingtheir ord ersat d ate 1.B roker B , the non-f

undamental

quoted spreads on the° oorcan belargerorsmallerthan in theautomated tradingsystem,dependingon
thestockcharacteristics.O naveragethequotedspreads areequal.T his is consistentwithourresultthat
theimpactofinformation sharingon marketdepth is ambiguous.

7

T hemarket-makercan alsobeconsidered as beinga° oorbrokerwhohas noinformation.

4

spec ulator, rec eivesord ersf

rom liquid ity trad ers. W e d enote ~

x

B

the totalquantity that

b roker B m ust exec ute onb ehalfofliquid ity trad ers.Asa w hole, liquid ity trad ershave

a net d em and equalto ~

x = ~

x

0

+ ~

x

B

shares. W e assume that ~

x

0

and ~

x

B

are norm ally

and independently d istrib uted w ith m eans0 and variances¾

2

0

and ¾

2

B

respec tively. W e

norm alize the variance ofthe ord er °ow d ue to liquid ity trad ing,¾

2

x

,to 1,i.e.

:

¾

2

x

= ¾

2
B

+ ¾

2

0

= 1:

Inthisw ay,¾

2

B

c anb e interpreted asb roker B 'sm arket share ofthe totalord er °ow f

rom

liquid ity trad ers.T he rem ainingpart ofthe ord er °ow c anb e seenasb einginterm ed iated

b y °oor b rokersw ho d o not trad e f

or their ow nac c ount or asb eing routed elec tronic ally

to the °oor.

8

,

9

B oth typesofspec ulatorsc anengage inproprietary trad ing. Inpartic ular b roker B

c anac t b oth as anagent (she c hannels a f

rac tionofliquid ity trad ers' ord ers) and as

a principal(she sub mitsord ersf

or her ow nac c ount). T hisprac tic e isknow nas`

d ual-

trad ing' and isauthoriz ed insec uritiesm arkets(see Chakravarty and Sarkar (2 0 0 0 ) f

or a

d iscussion).

10

M od elsw ith d ual-trad inginclud e R Äoell(19 90 ),Sarkar (199 5) or F ishm anand

Longsta® (199 2 ).Inthese m od els,asinthe present artic le,b rokersengaged ind ual-trad ing

exploit their ab ility to ob serve ord erssub m itted b y uninf

orm ed (liquid ity) trad ers.

11

None

ofthese m od elshasc onsid ered inf

orm ationsharing off

und am entaland nonf

undam ental

inf

orm ationam ong b rokers, how ever.O ur purpose isto stud y the e®ec tsofthisac tivity.

Asargued inthe introd uc tion,thistype ofinf

orm ationexchange isa d istinctive f

eature of

°oor m arkets.T he spec ulatorsw ith f

undam entalinf

ormationc anb e seenasb rokersw ho

exc lusively trad e f

or their ow nac c ount (like scalpersand loc alsind erivativesm arkets).

T hey c ould also b e seenasb rokersw ho have no c ustom ers' ord ersto exec ute at d ate 1.

It isreasonab le to assume that the ord er °ow f

rom liquid ity trad ersisindependent

ac rossb rokers (f

or instance b rokers have d i®erent c lients). Inc ontrast, signals onthe

f

undam entalvalue ofthe sec urity are c orrelated . For these reasons, w e assum ed that

only one °oor b roker ob servesthe non-f

undam entalinf

ormation, ~

x

B

,w hereasseveral°oor

8

In the U .

S, fullline brokerage houses engage in proprietary trading activities.D iscountbrokers do

not,however.

9

Forinstance, on the N Y SE,orders can reach amarket-makerthrough ° oorbrokers orelectronically

through asystem called SuperD ot.

1 0

Forinstance, Chakravarty and Sarkar(20 0 0 )observe thatin the N Y SE potentialdualtraders are

nationalfulllinebrokeragehouses andtheinvestmentbanks.

1 1

See also M adrigal(1 996). W e borrowthe distinction between `fundamental'vs. `

non-fundamental'

speculators from this author.

5

b rokersob serve the f

undamentalinf

orm ation,~

v.W e have analyz ed the m od elw henthere is

m ore thanone non-f

und am entalb roker (w ith independent ord er °ow ) and b rokersperf

ec tly

share inf

orm ation(inf

orm ationsharingisd escrib ed b elow ).T he presentationofthe m od el

ism ore c om plex b ut the c onclusionsare qualitatively sim ilar to those w e ob taininthe

c ase w ith only one non-f

undamentalb roker. O ne reasonf

or w hic h the m od elis more

c om plexisthat the numb er ofc liques(groupsofpaired b rokersw ith d istinct inf

ormation)

isendogenous. Inequilib rium , thisnum b er c anb e sm aller thanthe m aximum possib le

numb er ofc liques.For instance ifthere isanequalnumb er, N , off

und am entaland non-

f

undam entalb rokers, the numb er ofc liquesc anb e sm aller thanN . Inpartic ular, w ith

perf

ec t inf

ormationsharing, thisisnec essarily the c ase w hen¾

2

0

= 0 . Inthisc ase, the

aggregate ord er °ow c hanneled b y the non-f

undam entalb rokersw ho are not a± liated to

a c lique playsthe role of~

x

0

inthe present artic le.

In

f

orm ationSharin

g.

W e mod elinf

orm ationsharing asf

ollow s.W e assume that the nonf

undam entalspec -

ulator, B , hasanagreem ent to share inf

orm ationw ith one f

und am entalspec ulator, S.

Ac c ord ing to thisagreem ent, b ef

ore trad ing at d ate 1, the non-f

und am entalspec ulator

sendsa signal

^

x = ~

x

B

+ ~

´;

to the f

undam entalspec ulator.Inexc hange,the f

undam entalspec ulator sendsa signal

^

v = ~

v + ~

";

to the nonf

undam entalspec ulator.T he random variab les~

´ and ~

" are independently and

norm allyd istrib uted w ith meanzero and variances¾

2

´

and ¾

2

"

,respec tively.W e ref

er to the

inverse of¾

2

´

(resp.¾

2

"

) asthe prec isionofthe signalsent b y b roker B (S).T he larger is¾

2

´

(¾

2

"

),the lessprec ise isthe signalsent b y spec ulator B (spec ulator S) and hence the low er

isitsinf

orm ative value.T w o polar c asesare ofpartic ular interest.F irst there isperf

ec t

inf

ormationsharingif¾

2

´

= ¾

2

"

= 0 .Sec ond there isno informationsharingif¾

2

´

= ¾

2

"

=

1 .

In-b etw eenthese tw o c ases, there isinf

orm ationsharing b ut it isim perf

ec t (at least one

spec ulator d oesnot perf

ec tly d isclose hisor her inf

orm ation). T he inf

orm ationsetsof

spec ulatorsB and S at d ate 1 are d enoted y

B

= (~

x

B

; ^

x;^

v) and y

S

= (~

v; ^

x; ^

v),respec tively.

Inreality °oor b rokersare likely to exc hange inf

orm ationw ith the b rokersw ith w hom

they have enduring relationships.Inthisc ase their d ec isionto share inf

orm ationw ith a

6

givenb roker must b e b ased onthe long-term (average) b ene¯tsofinf

ormationsharing.For

thisreason,w e assum e that the spec ulatorsd ec id e to share inf

ormationb yc om paringtheir

ex-ante (i.e.prior to rec eivinginf

orm ation) expec ted pro¯tsw ith and w ithout inf

orm ation

sharing.W e say that inf

orm ationsharingispossib le ifthere existsa pair (¾

2

´

,¾

2

"

) suc h that

the expec ted pro¯tsofspec ulator S and B are larger w henthere isinf

orm ationsharing.

Insec tion3,w e id entif

y param eters' valuesf

or w hic h inf

orm ationsharingispossib le.

R em arks.

It isw orth stressingthat w e f

oc usonthe possib ilityofaninf

orm ationsharingagreem ent

b ut not onitsimplementation. Inpartic ular, w e d o not ad d ressenf

orc em ent issues. In

that, w e f

ollow the literature oninf

orm ationsalesw here the quality ofthe inf

orm ation

w hich issold isassum ed to b e c ontrac tib le.

12

W e also assum e that the inf

orm ationsharing

agreem ent and its c harac teristic s (¾

2

´

;¾

2

"

) are know nb y allpartic ipants (includ ing the

m arket-m aker).T hisc om monknow led ge assum ptionisalso standard inthe literature on

inf

orm ationsales.

2 .

2

T he equil

ib rium ofthe F l

oor M arket

Inthissec tion,w e d erive the equilib rium ofthe trad ingstage at d ate 1,giventhe c harac -

teristic softhe inf

ormationsharingagreement b etw eenspec ulatorsB and S.T hen,inthe

next sec tion,w e analyze w hether or not it isoptim alf

or B and Sto exc hange inf

orm ation.

W e d enote b y Q

S

(y

S

) and Q

B

(y

B

), the ord erssub m itted b y spec ulatorsS and B , re-

spec tively.Inthe set off

undamentalspec ulators, w e assignindex1 to spec ulator S.An

ord er sub m itted b y the other f

undamentalspec ulatorsi= 2 ;:::;N isd enoted Q

i

(~

v).T he

totalexc essd emand that must b e c leared b y the c om petitive m arket m aker istheref

ore

O =

i= N

X

i= 2

Q

i

(~

v) + Q

S

(y

S

) + Q

B

(y

B

) + ~

x:

Asthe market maker isassum ed to b e c om petitive,he setsa pric e p(O ) equalto the asset

1 2

See A dmati and P ° eiderer(1 986),(1 988).Some papers have shown howincentives contracts can be

used toinduce an information providertotruthfully revealthe quality ofhis signal(see A llen (1 990 )or
B hattacharya and P ° eiderer(1 985)). R eputation e®ects may also help to sustain information sharing
agreements (seeB enabou andL aroque (1 992)).

7

expec ted value c onditionalonthe net ord er °ow ,i.e.

p(O ) = E(~

v

jO ):

(1)

Anequilib rium c onsistsoftrad ing strategiesQ

S

(:), Q

B

(:), Q

i

(:);i= 2 ;:::;N and a c om -

petitive pric e f

unctionp(:) suc h that (i) eac h trad er'strad ingstrategy isa b est response to

other trad ers' strategiesand (ii) the d ealer'sb id d ing strategy isgivenb y E quation(1).

13

For givenc harac teristic s, (¾

2

´

;¾

2

"

), ofaninf

ormationsharing agreement, the next lemm a

d escrib esthe unique linear equilib rium ofthe trad inggam e.

Lem m a 1 : T he trad ingstage hasa unique linear equilibrium w hic h isgivenby

p(O ) =

¹ + ¸O ;

(2 )

Q

S

(y

S

) =

a

1

(~

v

¡¹)+ a

2

(^

v

¡¹)+ a

3

^

x;

(3)

Q

i

(~

v) =

a

0

(~

v

¡¹);i= 2 ;:::;N

(4 )

Q

B

(y

B

) =

b

1

~

x

B

+ b

2

^

x + b

3

(^

v

¡¹),

(5)

w here coe± c ientsa

1

;a

2

;a

3

;a

0

;b

1

;b

2

;b

3

and ¸ are

a

1

=

3(¾

2

v

+ ¾

2

"

)

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

;

a

2

=

¡

¾

2

v

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

;

a

3

=

¡

¾

2

B

3

¡

¾

2

B

+ ¾

2

´

¢;

a

0

=

2 ¾

2

v

+ 3¾

2

"

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

;

b

1

=

¡

1
2

;

b

2

=

¾

2

B

6

¡

¾

2

B

+ ¾

2

´

¢;

b

3

=

2 ¾

2

v

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

;

1 3

M oreprecisely,weconsidertheP erfectB ayesian Equilibriaofthe tradinggame.

8

and

¸ (¾

2

"

;¾

2

´

) =

6

q

¾

2

v

¡

¾

2

B

+ ¾

2

´

¢

(4 (N + 1)¾

4

v

+ (12 N + 5)¾

2

v

¾

2

"

+ 9 N ¾

4

"

)

(2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

q

¾

2

B

¡

4 ¾

2

B

+ 9¾

2

´

¢

+ 36¾

2

0

¡

¾

2

B

+ ¾

2

´

¢:

T rad erspurc hase (sell) the sec urity w hentheir estimationofthe asset value isab ove

(b elow ) the unconditionalexpec ted value.Hence,the c oe± c ientsa

1

,a

0

and b

3

are positive.

Nonf

undam entalinf

ormationisalso a sourc e ofpro¯t.Intuitively liquid itytrad ers'ord ers

c reate tem porary pric e pressures. B rokersw ith non-f

undam entalinf

orm ationare aw are

ofthese pric e pressures. T hey c anpro¯t f

rom thisknow led ge b y selling (b uying) high

(low ) w henliquid ity trad ers b uy (sell). M ore f

orm ally suppose that the f

undamental

spec ulators(b ut not the market m aker) d o not expec t c hangesinthe sec urity value (i.

e.

~

v = ¹).Suppose also that B and S perf

ec tly share inf

orm ationand that liquid ity trad ers

sub m it b uyord ers.T hese ord erspush the pric e upw ard b ec ause the m arket m aker c annot

d istinguish liquid ity ord ersf

rom inf

orm ed ord ers.Spec ulatorsB and Show ever know that

the c orrec t value ofthe sec urityis¹ .Inantic ipationofthe upw ard pressure onthe c learing

pric e,theysub m it sellord ers.B ysym metry,theysub m it b uyord ersw henliquid itytrad ers

sub m it sellord ers.T hisexplainsw hy c oe± c ientsb

1

and a

3

are negative.T hism eansthat

°oor b rokersB and S partly ac c om m od ate liquid ity trad ers' ord ersand red uc e the ord er

°ow imb alance that must b e exec uted b y the m arket-maker.A similar e®ec t isob tained

inR Äoell(1990 ) and Sarkar (19 95).

T he previousd iscussionshow show spec ulatorsc anpro¯t b oth f

rom f

und am entaland

nonf

undam entalinf

ormation. Hence there isa b ene¯t to exc hange f

undam ental(non-

f

undam ental) inf

orm ationf

or non-f

undam ental(f

und am ental) inf

ormation. Inf

orm ation

sharing isc ostly, how ever. Ac tually spec ulatorsS and B d eprec iate the value oftheir

private inf

orm ationw henthey share it.Consid er spec ulator B f

or instance.Ifshe d oes

not share inf

orm ation(¾

2

´

= +

1 ), she ac c om mod ateshalfofthe ord er °ow she rec eives

(since b

1

=

¡1=2 ).Ifshe sharesinformationthenb rokersB and S (instead ofb roker B

alone) provid e liquid ityto the ord ersc hanneled b yb roker B .For instance ifthere isperf

ec t

inf

orm ationsharing theneac h b roker ac c om mod atesone third ofthe ord ersrec eived b y

b roker B (since b

1

+ b

2

=

¡1=3 and a

3

=

¡1=3 w hen¾

2

´

= 0 ).T hisc ompetitionf

or the

provisionofliquid ity hastw o e®ec ts. F irst, b roker B trad essm aller quantities.Sec ond,

the ord er imb alance that must b e exec uted b y the m arket-m aker issm aller.Hence, f

or a

9

givenpric e sched ule (a ¯xed ¸ ), pric esreac t lessto the ord er °ow .

14

Inf

ac t spec ulator

B red uc esher trad e size w henshe sharesinf

orm ation(b

2

hasa signopposite the signof

b

1

) prec isely to m itigate thise®ec t.T hese tw o e®ec ts(sm aller trad e siz e/sm aller ab solute

pric e m ovem ents) red uc e spec ulator'sB pro¯tsonnon-f

undamentalinf

orm ation.T hisis

the c ost ofsharingnon-f

undam entalinf

orm ation.

A sim ilar argum ent holdsf

or spec ulator S. He d eprec iatesthe value off

undam ental

inf

orm ationw henhe sharesit w ith spec ulator B .Inord er to m itigate thise®ec t,he ad justs

histrad ingstrategy to the message he send sto spec ulator B .T hisexplainsw hy a

2

hasa

signopposite a

1

.

T o sum up,inf

orm ationsharinghasb ene¯tsand c osts.Inf

ormationsharingisa sourc e

ofpro¯tssince it allow seac h b roker to trad e ona new type ofprivate inf

orm ation.B ut the

b rokersob tainnew inf

orm ationonly ifthey d isclose allor part oftheir inf

orm ation.T his

isc ostlysince it red uc esthe trad ingpro¯tsthat c anb e mad e onthe inf

orm ationoriginally

possessed b y a b roker.Inthe next sec tionw e show that the b ene¯t ofinf

orm ationsharing

c anoutw eight itsc ost.

3 IsInf

orm ationSharingP ossib l

e?

Inthissec tion, w e id entif

y c asesinw hic h spec ulatorsB and S are b etter o® w henthey

share inf

orm ation. W e start b y c onsid ering the e®ec t ofthe prec isionsw ith w hic h the

spec ulatorsB and S share their inf

orm ationonthe m arket d epth (m easured b y ¸

¡1

).

15

It

turnsout that thise®ec t isim portant to interpret the results.

Lem m a 2 : T he d epth ofthe m arket (i.e. ¸

¡1

) isa® ec ted by the prec isionsw ith w hic h

the f

undamentaland the nonf

undam entalspec ulatorsshare their inf

ormation.

1.T he m arket d epth increasesw ith the prec isionofthe signalsent bybroker S(

@¸

@¾

2

"

> 0 ),

2 .T he market d epth d ec reasesw ith the prec isionofthe signalsent by broker B (

@¸

@¾

2

´

<

0 ).

1 4

Inordertoconveytheintuitionwetake¸ as given.H owevertheslopeofthepricescheduleis a®ected

by information sharing. A s shown below (L emma 2)sharing non-fundamentalinformation enlarges ¸.
T his mitigates theloss in pro¯tduetotheseconde®ect(smallerpricechanges).

1 5

T hemarketdepth is theorder° ownecessarytochangetheprice by1 unit.T helargeris the market

depth,the greateris the liquidityofthe market.A ctually,when ¸ is small,the market-makeraccommo-
dates largeorderimbalances withoutsubstantialchanges in prices.

10

Notic e that anincrease inthe quality ofthe inf

orm ationprovid ed b y B to S enlarges¸,

that isit d ec reasesthe d epth ofthe m arket. T he intuitionf

or thisresult isasf

ollow s.

E xc hange ofnon-f

und am entalinf

orm ationincreasesthe role of°oor b rokers(B and S) in

the provisionofliquid ity.T o see thispoint, let Q

T

= Q

B

+ Q

S

b e the totaltrad e siz e of

spec ulatorsB and S and c onsid er their expec ted totaltrad e siz e c ontingent on~

x

B

= x

B

.

W e ob tain

E(Q

T

j~x

B

= x

B

) = (b

1

+ b

2

+ a

3

)(x

B

) =

¡(

1
2

+

¾

2

B

6(¾

2

B

+ ¾

2

´

)

)(x

B

):

(6)

T he sm aller is¾

2

´

,the larger isthe f

rac tion(

jb

1

+ b

2

+ a

3

j) ofthe ord ersrec eived byb roker B

w hich isac c omm od ated b y spec ulatorsSand B .Asa c onsequence the d ealer partic ipates

lessto liquid itytrad es.Inthissense the exc hange ofnon-f

und am entalinf

orm ation`

siphons'

uninf

orm ed ord er °ow aw ay f

rom the m arket-m aker.T husthissiphone® ec t increaseshis

exposure to inf

orm ed trad ingand the pric e sched ule b ec om essteeper.

16

Interestingly anincrease inthe quality ofthe inf

orm ationprovid ed b y S to B has

exac tly the opposite e®ec t: it im provesthe d epth ofthe m arket.Inthisc ase, the e®ec t

ofinf

orm ationsharingisto increase c ompetitionam ongf

undam entaltrad ers.Hence they

scale b ack their ord er siz e (a

1

and a

0

d ec rease w hen¾

2

"

d ec reases).T hise®ec t red uc esthe

m arket-m aker'sexposure to inf

orm ed trad ing and thereb y m akesthe pric e sched ule less

steep.

W e d enote spec ulator j'sex-ante expec ted pro¯t (i.

e.b ef

ore ob servinginf

orm ation) b y

¦

j

(¾

2

´

;¾

2

"

;N ) .U singLem m a 1,w e ob tainthe f

ollow ingresult.

Lem m a 3 : F or givenvaluesof¾

2

"

and ¾

2

´

, the expec ted trad ingpro¯ tsfor spec ulatorsB

1 6

In equilibrium informed traders scale backtheirordersize when ¸ increases.B utthis is insu±cient

tocompensatethereduction in uninformed tradingdue tothe siphone®ect.

11

and S are

¦

S

(¾

2

´

;¾

2

²

;N ) =

Ã

¾

2

v

(¾

2

v

+ ¾

2

"

)(4 ¾

2

v

+ 9 ¾

2

"

)

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

2

+

¸¾

4

B

9

¡

¾

2

B

+ ¾

2

´

¢

!

d ef

=

¦

S
f

(¾

2

´

;¾

2

"

;N ) + ¦

S
n f

(¾

2

´

;¾

2

"

;N );

an d ;

¦

B

(¾

2
´

;¾

2
"

;N ) =

Ã

4 ¾

4

v

(¾

2

v

+ ¾

2

"

)

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

2

+

¸¾

2

B

¡

4 ¾

2

B

+ 9 ¾

2

´

¢

36

¡

¾

2

B

+ ¾

2

´

¢

!

d ef

=

¦

B

f

(¾

2

´

;¾

2

"

;N ) + ¦

B

n f

(¾

2

´

;¾

2

"

;N ):

E ach spec ulator'sexpec ted pro¯tshastw o c om ponents: (i) the expec ted pro¯t she or he

ob tainsb y trad ing onf

undam entalinf

orm ation(¦

j
f

) and (ii) the expec ted pro¯t she or

he ob tainsb y trad ing onnon-f

undam entalinf

orm ation(¦

j
nf

). Aninf

orm ationsharing

agreem ent isviab le ifand only ifb oth spec ulatorsB and Sare b etter o® w henthey share

inf

orm ation.Hence aninf

orm ationsharingagreem ent ispossib le ifand only ifthere exists

a pair (¾

2

´

;¾

2

"

) such that

¡

B

¡

¾

2

´

;¾

2

"

;N

¢

d ef

= ¦

B

(¾

2

´

;¾

2

"

;N )

¡¦

B

(

1 ;1 ;N ) > 0 ;

(7)

and

¡

S

¡

¾

2

´

;¾

2

"

;N

¢

d e f

= ¦

S

(¾

2

´

;¾

2

"

;N )

¡¦

S

(

1 ;1 ;N ) > 0 :

(8)

T he ¡s' m easure the expec ted surplusassoc iated w ith the inf

orm ationsharingagreem ent

f

or spec ulatorsB and S.

P roposition1 : T he set ofparametersf

or w hic h spec ulatorsB and S share information

isnon-empty.

W e estab lish the result b y provid ing 3 num eric alexam ples. For eac h exam ple, w e

report inT ab les1, 2 and 3 b elow the b reak-d ow nofthe trad ing pro¯tsf

or the d i®erent

partic ipantsw ith and w ithout inf

orm ationsharing. W e also c ompare the m arket d epth

w ith and w ithout inf

orm ationsharing. T he exam ples have b eenc hosenb ec ause they

illustrate d i®erent phenom ena that w e w illd iscussinthe rest ofthe paper.T he trad ing

pro¯tsare scaled b y ¾

2

v

and ¾

2

x

that w e norm aliz e to 1 throughout the paper.

12

P roof

:

E xam pl

e 1: ¾

2

0

= 0 ,¾

2

"

= 0 ,¾

2

´

= 2 =3,N = 2 .

P ro¯tsand d epth

Inf

orm ationSharing No In

f

orm ationSharin

g

(N

¡1)£¦

i

f

i

6

= S;B

0 :0 589

0 :1178

¦

S

f

0 :0 589

0 :1178

¦

S

nf

0 :0 70 7

0

¦

B

f

0 .0589

0

¦

B

nf

0 :1767

0 :2 357

T otalE xpec ted P ro¯ts 0 :4 2 4 2

0 :4 714

M arket Depth (¸ )

1:0 60 7

0 :94 2 8

T ab l

e 1

Inthisc ase w e ob tainthat

¡

S

= ¦

S
f

+ ¦

S
nf

¡¦

S

(

1 ;1 ) = 0 :0 589 + 0 :0 70 7¡0 :1178 = 0 :0 118;

and

¡

B

= ¦

B
f

+ ¦

B
nf

¡¦

B

(

1 ;1 ) = 0 :0 589 + 0 :1767¡0 :2 357= 0 :

O b serve that the totalsurplusf

or spec ulatorsB and S ispositive and equalto

¡

S

+ ¡

B

= 0 :0 118;

b ut that the totalsurplusf

or allspec ulatorsisnegative and equalto

(N

¡1)¡

i

+ ¡

S

+ ¡

B

= (0 :0 589

¡0 :1178)+ 0 :0 118 = ¡0 :0 4 71:

(¡

i

d enotesthe d i®erence inthe expec ted pro¯t w ith and w ithout inf

orm ationf

or a spec -

ulator d i®erent f

rom S or B .

)

E xam pl

e 2 : ¾

2

0

= 0 :6,¾

2

"

= 0 ,¾

2

´

= 0 ,N = 10 .

13

P ro¯tsand Depth

Inf

orm ationSharing No In

f

orm ationSharin

g

(N

¡1)£¦

i

f

i

6

= S;B

0 :1815

0 :2 165

¦

S

f

0 :0 2 0 2

0 :0 2 4 1

¦

S

nf

0 :0 153

0

¦

B

f

0 .02 0 2

0

¦

B

nf

0 :0 153

0 :0 34 4

T otalE xpec ted P ro¯ts 0 :2 52 5

0 .274 9

M arket Depth (¸)

0 .34 4 3

0 .34 36

T ab l

e 2

Inthisc ase w e ob tainthat

¡

S

= ¦

S
f

+ ¦

S
n f

¡¦

S

(

1 ;1 ) = 0 :0 114 ;

and

¡

B

= ¦

B
f

+ ¦

B
nf

¡¦

B

(

1 ;1 ) = 0 :0 0 11

O b serve that the totalsurplusf

or spec ulatorsB and S ispositive (0 :0 12 5) b ut that the

totalsurplusf

or allspec ulatorsisnegative (

¡0 :0 2 2 4 :)

E xam pl

e 3: ¾

2

0

= 0 :6,¾

2

"

= 0 ,¾

2

´

= 1:3,N = 10 .

P ro¯tsand Depth

Inf

orm ationSharing No In

f

orm ationSharin

g

(N

¡1)£¦

i

f

i

6

= s;B

0 :1874

0 :2 165

¦

S

f

0 :0 2 0 8

0 :0 2 4 1

¦

S

nf

0 :0 0 35

0

¦

B

f

0 .02 9 8

0

¦

B

nf

0 :0 2 9 0

0 :0 34 4

T otalE xpec ted P ro¯ts 0 :2 615

0 :2 74 9

M arket Depth (¸)

0 .033

0 .034

T ab l

e 3

Inthisc ase w e ob tainthat

¡

S

= ¦

S
f

+ ¦

S
n f

¡¦

S

(

1 ;1 ) = 0 :0 0 0 3;

14

and

¡

B

= ¦

B
f

+ ¦

B
nf

¡¦

B

(

1 ;1 ) = 0 :0 155

O b serve that the totalsurplusf

or spec ulatorsB and Sispositive and equalto ¡

S

+ ¡

B

=

0 :0 158.T he totalsurplusf

or allspec ulatorsisnegative and equalto

¡0 :0 134 :

Inallthe exam ples, the joint expec ted pro¯tsofspec ulatorsB and S increase w hen

they share inf

orm ation.Notic e that thisisa nec essary c onditionf

or inf

orm ationsharing.

Ac tually E quations(7) and (8) im ply that

¦

B

(¾

2

´

;¾

2

"

;N ) + ¦

S

(¾

2

´

;¾

2

"

;N ) > ¦

S

(

1 ;1 ;N )+ ¦

B

(

1 ;1 ;N ):

At the sam e tim e, there isa d ec line inthe joint expec ted pro¯tsofthe spec ulatorsw ho

d o not share inf

ormation.E ventually the totalexpec ted pro¯tsf

or allthe spec ulatorsare

low er inallthe exam ples(thisisalw aysthe c ase; see P roposition5 inSec tion4 ).Insum

inf

orm ationsharingisa w ay f

or spec ulatorsB and S to sec ure a larger part ofa sm aller

`cake'.T he f

allintotalpro¯tsisnot surprising: inf

orm ationsharingincreasesc om petition

b etw een°oor b rokers.T he surprisingpart isthat the joint expec ted pro¯tsofspec ulators

B and S c anincrease d espite the d ec line inthe totaltrad ing pro¯tsf

or the spec ulators.

T hisiskey since thisisa nec essary c onditionf

or inf

ormationsharing.W e now provid e an

explanationf

or thisob servation.T he explanationisquite c om plexb ec ause severale®ec ts

interplay.

Consid er the f

ollow ingratio

r

1

(¾

2

"

;¾

2

´

)

d ef

=

E(Q

T

j~v = v)

E(Q

T

j~v = v;¾

2

"

=

1 ;¾

2

´

=

1 )

:

T hisratio c om paresthe expec ted totaltrad e size (Q

T

) ofthe c lique f

orm ed b y spec ulators

B and Sc ond itionalonf

undamentalinf

orm ationw ith and w ithout aninf

orm ationsharing

agreem ent.U singLem m a 1,w e c anw rite thisratio as

r

1

(¾

2
"

;¾

2
´

) =

a

1

(¾

2

"

;¾

2

´

) + a

2

(¾

2

"

;¾

2

´

) + b

3

(¾

2

"

;¾

2

´

)

a

1

(

1 ;1 )

:

Hence r

1

> 1 m eans that the c lique f

orm ed b y B and S trad es m ore aggressively on

f

undam entalinf

orm ationw henthere isinf

orm ationsharingthanw henthere isnot.U sing

15

the expressionsf

or a

1

,a

2

and b

3

giveninLem m a 1,w e eventually ob tain

r

1

(¾

2

"

;¾

2
´

) = (

¸(

1 ;1 )

¸ (¾

2

²

;¾

2

´

)

)(

(4 ¾

2

v

+ 3¾

2

²

)(N + 1)

2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

²

):

As¸(

1 ;1 ) > ¸(0 ;1 ) (Lemm a 2 ),it isim m ed iate that r

1

(0 ;

1 ) > 1.B y c ontinuity,this

inequality also holdstrue f

or other valuesof¾

2

²

and ¾

2

´

. Hence there exist inf

orm ation

sharingagreementsw hic h ind uc e the c lique f

orm ed b y B and Sto trad e m ore aggressively.

Inturnthisf

orc esspec ulatorsw ho are not part ofthe c lique to shad e their totaltrad e

size.T o see thispoint c onsid er the f

ollow ingratio

r

2

(¾

2

²

;¾

2

´

)

d ef

=

E((N

¡1)Q

i

j~v = v)

E((N

¡1)Q

i

j~v = v;¾

2

"

=

1 ;¾

2

´

=

1 )

= (

¸ (

1 ;1 )

¸(¾

2

²

;¾

2

´

)

)(

(2 ¾

2

v

+ 3¾

2

²

)(N + 1)

2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

²

);

w here (N

¡1)Q

i

isthe totaltrad e siz e ofspec ulatorsd i®erent f

rom B and S.U singLem m a

2 ,w e d ed uc e that r

2

increasesw ith ¾

2

´

.T hisim pliesthat

r

2

(¾

2

²

;¾

2

´

)

·r

2

(¾

2

²

;

1 ):

U singthe expressionsf

or ¸(

1 ;1 )and ¸(¾

2

²

;

1 ) giveninthe proofofLem m a 2 ,w e ob tain

17

r

2

(¾

2
²

;

1 ) < 1 8¾

2

²

<

1 :

W e c onclud e that r

2

(¾

2

²

;¾

2

´

) < 1.T hism eansthat inf

orm ationsharingagreem entsf

orc e the

spec ulatorsw ho are not part ofthe c lique to trad e lessaggressively ontheir inf

orm ation.

Hence the spec ulatorsw ho share inf

orm ationappropriate a larger share ofthe totalpro¯ts

w hich d erive f

rom trad ing onf

undam entalinf

orm ation.

18

For thisreason, inf

orm ation

sharing enlargestheir joint expec ted pro¯t onf

undam entalinf

orm ation.T hisisthe c ase

f

or instance inE xamples2 and 3.

Now c onsid er the e®ec t ofinf

orm ationsharing onthe pro¯tsw hich d erive f

rom non-

f

undam entalinf

ormation.O nthe one hand,there are m ore spec ulatorsw ho ac c om m od ate

the ord er °ow b rokered b y B . T hise®ec t d ec reasesthe levelofexpec ted pro¯t onnon-

f

undam entalinf

ormation.O nthe other hand the exc hange ofnonf

undam entalinf

orm ation

1 7

T heproofrequires straightforward manipulations and is availableupon request.

1 8

N oticethatspeculators inourmodelarelikeCournotcompetitors.InCournotcompetition,each¯rm

would like to committo trade a largersize than itdoes in equilibrium. T his commitmentwould force
other¯rms totradeinsmallersizes.Inthis waythecommitted¯rm cancapturealargershareofthetotal
pro¯ts. Intuitively sharing fundamentalinformation is a way to make this commitmentcredible. T his
e®ecthas been pointedoutbyFishman and H agerty(1 995)in amodelofinformation sale.

16

d ec reasesthe market d epth and thise®ec t increasespro¯tsf

rom non-f

undam entalspec -

ulationasc anb e seenf

rom Lem m a 3. It turnsout that there are c ases(f

or instance

E xam ple 1) inw hic h the sec ond e®ec t d ominatesand the joint expec ted trad ingpro¯tsof

spec ulatorsS and B onnon-f

und am entalinf

orm ationare larger w henthere isinf

orm ation

sharingor

¦

B
n f

(¾

2

´

;¾

2

"

;N ) + ¦

S
n f

(¾

2

´

;¾

2

"

;N )

¡¦

B
nf

(

1 ;1 ;N ) ¸0 ; for ¾

2

´

<

1

and

¾

2

"

<

1

O b serve that thisc anoc c ur onlyw heninf

orm ationsharingim pairsm arket d epth (increases

¸).InE xam ple 3,inf

orm ationsharingim provesm arket d epth and the joint expec ted pro¯t

onnon-f

undam entalinf

orm ationd ec reases.

T o sum up, there are tw o reasonsw hy inf

orm ationsharing c anincrease the joint ex-

pec ted pro¯tsofspec ulatorsB and S:

²Sharingfundamentalinformationallow sthe c oalitionform ed b y b rokersSand B to

trad e m ore aggressively onf

undam entalinf

ormationand to c apture thereb y a larger

share ofthe totalpro¯tsf

rom spec ulationonf

undam entalinf

ormation.

²Sharingnon-fundam entalinform ationc anred uc e the m arket d epth.T hisim pliesthat

pric esreac t m ore to ord er imb alances.Larger totalexpec ted pro¯tsf

rom spec ulation

onnon-f

undam entalinf

orm ationf

ollow s.

T he prec isionsw ith w hic h the spec ulatorsshare their inf

orm ationd etermine how the

surplus (¡

S

+ ¡

B

) c reated b y inf

orm ationsharing is split b etw eenb rokers B and S.

For instance, c onsid er E xam ples2 and 3. T he value of¾

2

´

islarger inE xam ple 3, b ut

otherw ise the valuesofthe param etersare id entic alinthe tw o exam ples.T he surplusf

or

spec ulator B (S) islarger (low er) inE xam ple 3 thaninE xam ple 2 .Inline w ith intuition,

f

or a ¯xed value of¾

2

"

,spec ulator B (S) pref

ersto provid e (rec eive) aninf

orm ationoflow

(high) quality. Hence spec ulatorsB and S have c on°ic ting view sover the inf

orm ation

sharingagreem entsw hic h should b e c hosen.It isalso w orth stressingthat the siz e ofthe

surplusc reated b y inf

orm ationsharingd epend sonthe prec isionsw ith w hic h trad ersshare

inf

orm ation. For instance the joint surplusissm aller inE xam ple 2 thaninE xam ple 3.

Inthispaper, w e d o not stud y how trad ersselec t the c harac teristic softheir inf

orm ation

sharing agreem ent (¾

2

²

and ¾

2

´

). T hisisnot nec essary b ec ause our statem entsregard ing

m arket perf

orm ance (next sec tion) only d epend sonthe existence ofinf

orm ationsharing

agreem ents,not onthe spec i¯c valuesc hosenf

or ¾

2

"

and ¾

2

´

.

17

W e now c onsid er inm ore d etailsinf

orm ationsharingagreementsinw hich spec ulatorsB

and Sperf

ec tly share inf

ormation(¾

2

"

= ¾

2

´

= 0 ).P erf

ec t inf

orm ationsharingisofinterest

b ec ause it isrelatively easy to im plement.Ac tually,ifthere isperf

ec t inf

orm ationsharing,

B know s w hich quantity S should trad e and vic e versa (inour m od elthey optimally

trad e the sam e quantity).Consequently, one spec ulator c and etec t cheating b y the other

spec ulator b y ob servinghisor her trad e size.

P roposition2 : For N

¸ 2 , there exist tw o c ut-o® values (i) ¾

2

0

(N ) and (ii) ¾

¤2

0

(N )

suc h that perf

ec t inf

ormationsharingispossible ifand only if¾

2

0

2 [¾

2

0

(N );¾

¤2

0

(N )].Fur-

therm ore the c uto® valuesincrease w ith N and are suc h that 0 < ¾

2

0

(N ) < ¾

¤2

0

(N ) < 1.

T he propositionshow s that perf

ec t inf

orm ationsharing ispossib le ifb roker B d oes

not c hannela too large or a too sm allf

rac tionofthe ord er °ow f

rom liquid ity trad ers.

O b serve that pro¯tsm ad e onnon-f

und am entalinf

orm ation(¦

j
n f

) are proportionalto the

am ount ofliquid itytrad ingb rokered b yB (¾

2

B

= 1

¡¾

2

0

).Hence ¾

2

0

d eterm inesthe value of

non-f

undam entalinf

orm ation.P erf

ec t inf

orm ationsharingc antake plac e w henthisvalue

isneither too large, nor too sm all. Ifthe value ofnon-f

undamentalinf

orm ationislarge

(¾

2

0

< ¾

2

0

(N )),the c ost ofd isclosingher inf

orm ationperf

ec tlyf

or B (sm aller pro¯tsonnon-

f

undam entalinf

orm ation) islarge c om pared to the b ene¯t (the possib ility to pro¯t f

rom

f

undam entalinf

orm ation).Inord er to attenuate thisc ost,B must theref

ore send a noisy

signalto S.W henthe value ofnon-f

und am entalinf

orm ationissm all(¾

2

0

> ¾

¤2

0

(N )), the

b ene¯t ofperf

ec t inf

ormationsharing issm allf

or the f

und am entalspec ulator.T heref

ore

he ref

usesto perf

ec tly d isclose hisinf

orm ation.

T he larger isthe numb er off

undam entalspec ulators, the sm aller must b e the f

rac tion

ofliquid ity trad ers' ord er °ow b rokered b y B to sustaina perf

ec t inf

orm ationsharing

agreem ent (¾

2

0

(N ) increasesw ith N ).Ac tually the pro¯tsf

rom f

und am entalinf

orm ation

d ec rease w ith the numb er off

undamentalspec ulators.T he value off

und am entalinf

orm a-

tionistheref

ore sm allw henN islarge.Hence b roker B ac c eptsto perf

ec tly d isclose her

inf

orm ationonly ifthe value ofnon-f

undam entalinf

orm ationisitselfsm all.T he last part

ofthe propositionim pliesthat f

or allvaluesofN ,there exist valuesof¾

2

0

< 1 such that a

perf

ec t inf

orm ationsharingagreem ent c anb e sustained .F igure 1 plots¾

2

0

(N ) and ¾

¤2

0

(N )

f

or d i®erent valuesofN

¸2 and show sw henperfec t informationsharingispossib le.

19

1 9

T hecuto®values ¾

2

0

(N )and ¾

¤2

0

(N )areimplicitlyde¯nedin theproofofP roposition 2.

18

R em ark.Inthe mod elw e assum e that b rokers' rolesare ¯xed : one hasf

undamental

inf

orm ationand the other hasnon-f

undam entalinf

orm ation.Another possib ility isthat

the rolesare random ly alloc ated b ef

ore trad ingand unkw now nat the tim e b rokersd ec id e

to share inf

orm ation.For sim plic ity, assum e that eac h b roker inthe c lique hasanequal

prob ab ility to b e the b roker endow ed w ith non-f

undamentalinf

orm ation. Inthisc ase,

b rokersagree to share inf

orm ationi®

¦

B

(¾

2

´

;¾

2

"

;N ) + ¦

S

(¾

2

´

;¾

2

"

;N ) > ¦

S

(

1 ;1 ;N )+ ¦

B

(

1 ;1 ;N ):

T hisc ond itionisalw ayssatis¯ed w hen(¾

2

"

;¾

2

´

) are suc h that Conditions(7) and (8) are

satis¯ed . Hence ifaninf

orm ationsharing agreem ent ispossib le w henb rokers' rolesare

¯xed ,it isstillpossib le w henb rokers' role are randomly c hosen.

4

Inf

orm ationSharin

g and M arket P erf

orm ance

Inthissec tion, w e analyze the e®ec tsofinf

orm ationsharing ontrad itionalm easuresof

m arket quality: (1) the inf

orm ationale± c iency ofpric es(m easured b y V ar(~

v

jp)), (2 )

pric e volatility (measured b y V ar(~

v

¡p)),(3) m arket d epth (m easured by ¸) and (4 ) the

expec ted trad ingc ostsb orne b y liquid ity trad ers(i.e.their expec ted losses,E(~

x(p

¡~v))).

T hese aspec tsofm arket perf

ormance play a prom inent role inthe d eb atesregard ing the

d esignoftrad ingsystem sand have attrac ted c onsid erab le attentioninthe literature (see

M ad havan(19 96) or Vives(19 95) f

or instance).

P rop osition3 : P rices are more inf

orm ative (V ar(~

v

j p) smaller) and less volatile

(V ar(~

v

¡p) smaller) w henthere isinformationsharing.

T he intuitionb ehind thisresult issim ple.W henspec ulatorsSand B share inf

orm ation,

the numb er ofspec ulatorstrad ing onf

undam entalinf

orm ationincreases.It f

ollow sthat

the aggregate ord er °ow ism ore inf

ormative. For thisreason, pric esare m ore ac c urate

pred ic torsofthe ¯nalvalue ofthe sec urity and pric e d iscovery isim proved .

W e now examine the im pac t ofinf

orm ationsharing onthe d epth ofthe m arket. As

show nb y Lem m a 2 , anincrease inthe prec isionw ith w hic h spec ulator S transm itshis

inf

orm ationim proves market d epth. How ever, anincrease inthe prec isionw ith w hich

spec ulator B transm itsher inf

orm ationim pairsm arket d epth (b ec ause ofthe siphone®ec t).

19

Hence the im pac t ofinf

ormationsharing onm arket d epth c anb e positive or negative.

O fc ourse, f

or the param eterssuc h that inf

orm ationsharing oc c urs, one e®ec t c ould b e

d om inant. How ever E xam ples2 and 3 inthe previoussec tionshow that thisisnot the

c ase.Inthese exam ples, ¾

2

²

and ¾

2

´

are suc h that (i) inf

orm ationsharing isoptim aland

(b ) inf

ormationsharing im pairs m arket d epth (E xam ple 2 ) or im proves market d epth

(E xam ple 3).T he next propositionc onsid ersthe e®ec t ofperf

ec t inf

orm ationsharing on

m arket d epth.T o thisend,w e d e¯ne

¹

¾

2

0

(N ) =

1

¡h

2

(N )

8h

2

(N )

¡3

< 1;

w here h (N ) =

2 (N + 2 )

p

N

3(N + 1)

p

N + 1)

< 1:

P roposition4 : P erf

ec t informationsharingim provesm arket d epth ifand only if¾

2

0

¸

¹

¾

2

0

(N ).

Hence perf

ec t inf

ormationsharing im provesm arket d epth w henb roker B rec eivesa suf

-

¯c iently sm allf

rac tionofthe totalord er °ow (¾

0

¸ ¹¾

2

(N )). R ec allthat w henthere is

perf

ec t inf

ormationsharing,¾

2

0

must b e larger thana threshold (¾

2

0

(N )).F igure 2 d epic ts

¹

¾

2

(N ) (d otted line) w henN increases. Asit c anb e seen, there are valuesof¾

2

0

and N

suc h that perf

ec t inf

ormationsharing oc c ursand im pairsmarket liquid ity (allthe values

b elow the d otted line and ab ove the plainline).

2 0

Notic e that the market d epth isrelated to the b id -ask spread .Ac tually a b uy ord er of

size q pushesthe pric e upw ard b y ¸ q w hereasa sellord er ofthe sam e siz e pushesthe pric e

d ow nw ard b y ¸ q.Hence

s(q) = p(q)

¡p(¡q) = 2 ¸q;

c anb e interpreted asthe b id -askspread f

or anord er ofsiz e q inour m od el(see M ad havan

(19 96)). T he spread increases w ith ¸. Ac c ord ingly the im pac t ofinf

orm ationsharing

onb id -ask spread sisamb iguous. Interestingly em piric alstud iesw hic h c om pare b id -ask

spread s in°oor-b ased trad ing system s and autom ated trad ing system shave not f

ound

that spread sw ere system atic ally low er inone trad ingvenue.For instance,severalstud ies

(K of

m anand M oser (199 7), P irrong (199 6) and Shyy and Lee (19 95)) have c om pared

the b id -ask spread s onLIF F E (w henit w as a °oor market) and DT B (anautom ated

20

Forlargevalues ofN ,thedi®erence(¹

¾

2

0

(N )

¡¾

¤2

0

(N ))becomes smallerandsmallerbutis neverzero.

T hatis even forN large, there are values for¾

2

0

such thatperfectinformation sharingtakes place and

impairs marketdepth.

2 0

trad ingsystem ) f

or the sam e sec urity(nam elythe G erm anB und f

uturesc ontrac t).K of

m an

and M oser (19 97) ¯nd that spread sare equalinthe tw o m arkets; P irrong (199 6) reports

narrow er spread sonDT B w hereasShyy and Lee (19 95) ¯nd sm aller spread sonLIF F E .

InApril1997, the T oronto Stoc k E xc hange c losed its trad ing °oor and introd uc ed an

elec tronic trad ingsystem .G ri± thset al.(19 98) c om pare b id -askspread sf

or stockslisted

onthe T oronto Stoc k E xc hange b ef

ore and af

ter the sw itc h to the autom ated trad ing

system .T hey d o not ¯nd signi¯c ant c hangesinquoted spread s.

F inally w e c onsid er the e®ec tsofinf

orm ationsharingonthe aggregate expec ted trad ing

c ostsf

or the liquid ity trad ers.T hese expec ted trad ingc ostsare

E(TC ) = E(~

x(p

¡~v)) =

E(~

x

B

(p

¡~v))

|

{

z

}

O rd e rs c

h an ne le d by B

+

E(~

x

0

(p

¡~v))

| {

z

}

O rd ers notc

h anne led by B

:

Inthe last expression,w e d istinguish b etw eenthe expec ted trad ingc ostsf

or the liquid ity

trad ersw ho send their ord ersto b roker B and the expec ted trad ing c ostsf

or those w ho

d o not.U singLemm a 1,w e ob tainthat

E(~

x

B

(p

¡~v)) = E(~x

B

E(p

¡~v j~x

B

= x

B

)) = ¸E(~

x

2
B

(1+ b

1

+ b

2

+ a

3

)) = ¸

Ã

2 ¾

2

B

+ 3¾

2

´

6

¡

¾

2

B

+ ¾

2

´

¢

!

¾

2
B

;

and

E(~

x

0

(p

¡~v)) = ¸¾

2
0

:

Hence w e rew rite the expec ted trad ingc ostsas

E(TC ) =

¸g(¾

2

´

)¾

2

B

| {

z }

O rd ers c

h an nele d by B

+ ¸¾

2
0

;

w ith g(¾

2

´

) =

µ

2 ¾

2

B

+ 3¾

2

´

6

(

¾

2

B

+ ¾

2

´

)

¶

. T he ratio g(¾

2

´

) increasesw ith ¾

2

´

. Hence w heninf

orm ation

sharingim provesmarket d epth,it also d ec reasesthe expec ted trad ingc ostsf

or al

lliquid ity

trad ers: (1) the liquid itytrad ersw hose ord ersare c hanneled through b roker B and (2 ) the

other liquid ity trad ers. For instance, w ith perf

ec t inf

orm ationsharing thisoc c ursw hen

¾

2

0

2 [¹¾

2

0

(N );¾

¤2

0

(N )].

W heninf

orm ationsharing im pairsm arket d epth (increases¸), the expec ted trad ing

c ostsofthe liquid ity trad ersw ho d o not send their ord er to b roker B increase.How ever

the expec ted trad ing c ostsf

or the liquid ity trad ersw ho use B 'sservic esd ec line d espite

2 1

the d ec rease inm arket d epth.Ac tually inf

orm ationsharingincreasesc ompetitionam ong

trad ersprovid ingc ounter-partiesto B 'sc lients.T heref

ore a sm aller f

rac tionofthe ord ers

sub m itted b y B 'sc lientsmust b e exec uted against the m arket-m aker w henspec ulators

S and B share nonf

undam entalinf

orm ation(see E quation(6)). T he next proposition

show sthat the red uc tioninthe expec ted trad ing c ostsf

or B 'sc lientsalw aysd om inates

the increase inexpec ted trad ingc ostsf

or the other liquid ity trad ers.

P roposition5 : T he expec ted trad ing costs borne by the liquid ity trad ers are alw ays

sm aller w henthere isinf

ormationsharing.

T he trad inggam e isa zero-sum gam e inthism od el.T hisim pliesthat the expec ted trad ing

c ostsb orne b y liquid ity trad ersare equalto the spec ulatorsaggregate expec ted pro¯ts.

Let ¦

a

(¾

2

´

;¾

2

"

;N ) b e spec ulators' aggregate expec ted pro¯ts.W e have

E(T C ) = ¦

a

(¾

2

´

;¾

2
"

;N )

d ef

= ¦

S

+ ¦

B

+ (N

¡1)¦

i

;

w here ¦

i

(¾

2

´

;¾

2

"

;N ) isthe expec ted pro¯t ofa spec ulator w ho isnot part to the inf

orm ation

sharing agreem ent. R ec allthat a nec essary c onditionf

or inf

orm ationsharing isthat it

increasesthe joint expec ted pro¯tsofspec ulatorsB and S,i.e.¦

S

+ ¦

B

.Since inf

orm ation

sharingd ec reasesthe aggregate expec ted pro¯tsofallspec ulators,it f

ollow sthat the joint

expec ted pro¯t ofspec ulatorsi

2 f2 ;:::;N g d ec reases.T herefore,the c oncom itant d ec rease

intrad ingc ostsf

or liquid ity trad ersand increase intotalexpec ted pro¯tsf

or spec ulators

S and B oc c ur at the expense ofthe spec ulatorsw ho d o not share inf

orm ation.O b serve

that thisc annot happenw henthere isa single f

undamentalspec ulator (N = 1).Inf

ac t

inthisc ase, it ispossib le to show that there are no valuesf

or the param etersf

or w hich

inf

orm ationsharingisoptim alf

or B and S.

O verallthe resultsofthissec tionshow how inf

orm ationsharingonthe °oor c anim prove

the quality of°oor-b ased m arketsalong severald im ensions. Inf

orm ationsharing m akes

pric e more inf

orm ative,lessvolatile and f

ostersc om petitionb etw een°oor b rokers,so that

ultim ately the aggregate trad ingc ostsb orne b y the trad ersw ithout anac c essto the °oor

are low er.

2 2

5 Concl

usion

Inthispaper w e have analyz ed pre-trad e inf

orm ationsharing b etw eentw o tw o trad ers

endow ed w ith d i®erent types ofinf

orm ation, nam ely f

undam entalor non-f

undam ental

inf

orm ation.W e ¯nd that there are c asesinw hic h the tw o trad ersare b etter o® sharing

their inf

orm ation.Inf

orm ationsharing im provespric e d iscovery and d ec reasesvolatility.

W e also show that inf

ormationsharing d ec reases the aggregate expec ted trad ing c osts

b orne b y liquid ity trad ers.F inally the e®ec t ofinf

orm ationsharingonm arket d epth and

b id -ask spread sisamb iguous.

F loor-b ased trad ing systems are d esigned insuc h a w ay that they greatly f

ac ilitate

inf

orm ationsharing am ong °oor b rokers. O verallour resultsshow how thisf

eature c an

im prove their perf

ormance.Aninterestingquestionisw hether the b ene¯tsb rought up b y

inf

orm ationsharingare outw eighted b yinherent d isad vantagesof°oor-b ased system s(such

aslac k oftransparency or larger operatingc osts).T hisissue islef

t f

or f

uture researc h.

2 3

R ef

erences

[1] Ad m atiA.,and P °eid erer P .(1986),\A M onopolistic M arket f

or Inf

orm ation",J our-

nalofE conom ic T heory,39,4 0 0 {4 38.

[2 ] Ad m atiA.,and P °eid erer P .(1988),\Sellingand T rad ingonInf

orm ationinF inancial

M arkets",Am ericanE conomic R eview ,78(2 ),9 6{10 3.

[3] AllenF .(19 90 ),\T he M arket f

or Inf

orm ationand the O riginofF inancialInterm ed i-

ation",J ournalofF inancialInterm ed iation,1,3{30 .

[4 ] B enab ou, R .and Laroque, M .(1992 ), \U sing privileged inf

orm ationto m anipulate

m arkets: insid ers,gurusand c red ib ility",Quarterly J ournalofE conomic s,92 1{9 58.

[5] B enveniste L.M .,M arc usA.

J .,and W ilhelm W .

J .(19 92 ), \W hat'sspec ialab out the

spec ialist ?",J ournalofF inancialE conom ic s,32 ,61{86.

[6] B hattacharya and P °eid erer (19 85), \Delegated portf

olio m anagement", J ournalof

E conom ic T heory,36,1{2 5.

[7] Chakravarty,S.and Sarkar,A.(2 0 0 0 ),\A M od elofB roker'strad ingw ith Applic ations

T o O rd er F low Internaliz ation",f

orthc om inginR eview ofF inancialE conomic s.

[8] Coval, J .and Shumw ay, T .(199 8), \Issound just noise", W orking paper, U niversity

ofM ichigan.

[9] Dom ow itz , Iand Steil, B .(1999 ), \Autom ation, trad ing c ostsand the struc ture of

the Sec uritiesT rad ing Industry", B rookings-W hartonP apersonF inancialServices,

2 ,33-9 2 .

[10 ] F ishm anM .and Hagerty K .(19 95), \T he Incentive to SellF inancialM arket Inf

or-

m ation",J ournalofF inancialInterm ed iation,4 ,95{115.

[11] F ishm an,M .and F .Longsta® (19 92 ),\DualT rad inginFuturesm arkets",J ournalof

F inance,4 7,64 3{669.

[12 ] G ri± ths, M .

, Smith, B .

, T urnbull, A.and R .W hite (1998), \Inf

orm ationF low sand

O penO utc ry: E vid ence ofIm itationT rad ing", J ournalofInternationalF innac ial

M arkets, Institutionsand M oney,8,10 1{116.

2 4

[13] Harris(2 0 0 0 ), L., \F loor versusautom ated trad ing systems: a survey ofthe issues",

LHarris.U SC.

ed u/Ac rob at/°oor.

[14 ] Hasb rouc k, J ., So¯anos, G .and Soseb ee, D.(199 3), \NY SE System sand T rad ing

P roc ed ures",NY SE W orkingP aper,93-0 1.

[15] K of

m an,P .and M oser,J .(19 97),\Spread s,inf

orm ation°ow sand transparencyac ross

trad ingsystems",Applied F inancialE conom ic s,7,2 81{2 94 .

[16] K yle A.S.(1985), \Continuous Auc tions and Insid er T rad ing", E conom etrica, 6,

1315{1335.

[17] M ad havan,A.(1996),\Sec urity pric esand m arket transparency",J ournalofF inan-

c ialInterm ed iation,5,2 55{2 83.

[18] M ad rigalV.(199 6),\Non-Fundam entalSpec ulation",J ournalofF inance,51(2 ),553{

576.

[19] P irrong, C.(199 6), \M arket liquid ity and d epth onc om puteriz ed and openoutc ry

trad ing system s: a c omparisonofDT B and LIF F E B und c ontrac ts", J ournalofFu-

turesM arkets,16,519{54 3.

[2 0 ] R ÄoellA.(19 90 ),\Dual-Capac ity T rad ingand the Quality ofthe M arket",J ournalof

F inancialInterm ed iation,1,10 5{12 4 .

[2 1] Sarkar, A.(199 5) \Dualtrad ing: w inners, losers and m arket impac t", J ournalof

F inancialIntermed iation,4 ,77-93.

[2 2 ] Sarkar, A.and W u, L.(199 9) \Con°ic t ofinterestsand exec utionquality off

utures

°oor trad ers",W orkingP aper,New -Y ork F E D.

[2 3] Shyy, G .and Lee, J .H., (19 95) \P ric e transmissionand inf

orm ationasym m etry in

B und f

uturesm arkets: LIF F E vs.DT B ",J ournalofFuturesM arkets,15,87-99 .

[2 4 ] So¯anos G .

, and W erner I.(19 97), \T he trad es ofNY SE F loor B rokers", NY SE

W orkingP aper 9 7-0 4 .

[2 5] T heissen,E .(19 99 ),\F loor versusSc reenT rad ing: E vid ence f

rom the G erm anStoc k

M arket",W orkingP aper CR 69 0 ,G roupe HE C.

2 5

[2 6] VivesX.(199 5),\T he Speed ofInf

orm ationR evelationina F inancialM arket M ec ha-

nism ",J ournalofE conomic T heory,67,178{2 0 4 .

[2 7] Venkataraman,K .(2 0 0 0 ) \Autom ated versusF loor T rad ing: AnAnalysisofexec ution

Costsonthe P arisand New -Y ork E xc hanges," W orking P aper, SouthernM ethod ist

U niversity.

2 6

6 Append ix

P roofofLem m a 1

Step1: T he optim altrad in

g strategy f

or sp ec ul

ator S.Let y

S

= (~

v;^

v; ^

x) b e the

inf

orm ationset ofspec ulator S.T he latter c hooseshism arket ord er,Q

S

,so asto m axim iz e

hisexpec ted pro¯t

¼

S

(y

S

) = E(Q

S

(~

v

¡p(~

O ))

jy

S

):

T he ¯rst ord er c onditionyields

Q

S

(y

S

) =

(~

v

¡¹)¡¸ £E

h

Q

B

(y

B

) +

P

j= N
j= 2

Q

i

(~

v) + ~

x

0

+ ~

x

B

jy

S

i

2 ¸

:

(9 )

Notic e that

E

¡

Q

i

(~

v)

jy

S

¢

= a

0

(~

v

¡¹);

and

E

¡

Q

B

(y

B

)

jy

S

¢

= b

1

E (~

x

B

j^

x) + b

2

^

x + b

3

(^

v

¡¹ );

and

E (~

x

B

j^

x) =

¾

2

B

^

x

¾

2

B

+ ¾

2

´

:

Sub stitutingthese expressionsinE quation(9) yields

Q

S

(y

S

) =

(~

v

¡¹ )

2 ¸

¡

1
2

£

·

(N

¡1)a

0

(~

v

¡¹) + b

3

(^

v

¡¹ )+ (b

1

+ 1)

¾

2

B

¾

2

B

+ ¾

2

´

^

x + b

2

^

x

¸

=

µ

1

2 ¸

¡

(N

¡1)a

0

2

¶

(~

v

¡¹)¡

b

3

2

(^

v

¡¹ )¡

1
2

µ

(b

1

+ 1)

¾

2

B

¾

2

B

+ ¾

2

´

+ b

2

¶

^

x:

Hence,

a

1

=

µ

1

2 ¸

¡

(N

¡1)a

0

2

¶

a

2

=

¡

b

3

2

a

3

=

¡

1
2

µ

(b

1

+ 1)

¾

2

B

¾

2

B

+ ¾

2

´

+ b

2

¶

Step 2 : T he optim altrad ingstrategy f

or spec ul

ator i,i

6

= S.

2 7

Spec ulator ic hooseshismarket ord er,Q

i

,so asto m axim iz e hisexpec ted pro¯t

¼

i

(v) = E(Q

i

(~

v

¡p(~

O ))

j~v = v):

T he ¯rst ord er c onditionyields

Q

i

(~

v) =

(~

v

¡¹)¡¸ £E

h

Q

S

(y

S

) +

P

j= N
j= 2

Q

j

(~

v) + Q

B

(y

B

) + ~

x

j~v = v

i

¸

(10 )

W e f

oc usonsymm etric trad ing strategiesf

or allthe spec ulatorsi

6

= S. T hisim poses

Q

j

(~

v) = Q

i

(~

v);

8j6

= i.Sub stitutingQ

j

b y Q

i

inE quation(10 ) yields

Q

i

(~

v) =

(~

v

¡¹)

N ¸

¡

1

N

£

¡

E

£

Q

S

(y

S

)

j~v

¤

+ E

£

Q

B

(y

B

)

j~v = v

¤¢

:

(11)

Furthermore

E

¡

Q

S

(y

S

)

j~v = v

¢

= (a

1

+ a

2

)(~

v

¡¹);

and

E

¡

Q

B

(y

B

)

j~v = v

¢

= b

3

(~

v

¡¹ ):

Consequently

Q

i

(~

v) =

µ

1

N ¸

¡

(a

1

+ a

2

+ b

3

)

N

¶

(~

v

¡¹ ):

(12 )

W e d ed uc e that

a

0

=

1

N ¸

¡

(a

1

+ a

2

+ b

3

)

N

:

(13)

Step 3: T he optim altrad ingstrategy f

or sp ec ul

ator B .W e d enote y

B

= (~

x

B

;^

v; ^

x),

the inf

ormationset ofspec ulator B .She c hoosesher market ord er,Q

B

,so asto m axim iz e

¼

B

(y

B

) = E(Q

B

(~

v

¡p(~

O ))

jy

B

):

2 8

T he ¯rst ord er c onditionyields

Q

B

(y

B

) =

E (~

v

j^

v)

¡¹ ¡¸ £E

·

Q

S

(y

S

) +

N

P

i= 2

Q

i

(~

v) +

P

x

i

jy

B

¸

2 ¸

=

E (~

v

j^

v)

¡¹ ¡¸ £E

£

Q

S

(y

S

) + (N

¡1)Q

i

(~

v) + ~

x

jy

B

¤

2 ¸

:

W e notic e that

E

¡

Q

S

(y

S

)

jy

B

¢

= a

1

E (~

v

¡¹ j^

v) + a

2

(^

v

¡¹)+ a

3

^

x;

and

E

¡

Q

i

(~

v)

jy

B

¢

= a

0

E (~

v

¡¹ j^

v);

and that

E (~

v

¡¹ j^

v) =

¾

2

v

¾

2

v

+ ¾

2

"

(^

v

¡¹ ):

Sub stituting these expressionsinthe ¯rst ord er c onditionf

or Spec ulator B yields(af

ter

som e algeb ra)

Q

B

(y

B

) =

¡

~

x

B

2

¡

a

3

2

^

x +

1

2 ¸

µ

¾

2

v

¾

2

v

+ ¾

2

"

¡¸a

2

¡¸ (a

1

+ (N

¡1)a

0

)

¾

2

v

¾

2

v

+ ¾

2

"

¶

(^

v

¡¹ ):

(14 )

Hence,

b

1

=

¡

1
2

;

b

2

=

¡

a

3

2

;

b

3

=

1

2 ¸

µ

¾

2

v

¾

2

v

+ ¾

2

"

¡¸a

2

¡¸ (a

1

+ (N

¡1)a

0

)

¾

2

v

¾

2

v

+ ¾

2

"

¶

:

Steps1 to 3 give us9 equationsw ith 9 unknow ns(a

1

, a

2

etc ..

.). Solving thissystem of

2 9

equationsyield

a

1

=

3(¾

2

v

+ ¾

2

"

)

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

;

a

2

=

¡

¾

2

v

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

a

3

=

¡

¾

2

B

3

¡

¾

2

B

+ ¾

2

´

¢;

a

0

=

2 ¾

2

v

+ 3¾

2

"

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

;

b

1

=

¡

1
2

;

b

2

=

¾

2

B

6

¡

¾

2

B

+ ¾

2

´

¢;

b

3

=

2 ¾

2

v

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

;

Step 4 .Com putationof¸ .R ec allthat

p(O ) = E(~

v

j~

O = O ):

G ivenspec ulators' trad ingrules,

O = Q

S

(y

S

) + (N

¡1)Q

i

(~

v) + Q

B

(y

B

) + ~

x

= (a

1

+ (N

¡1)a

0

)(~

v

¡¹ )+ (a

2

+ b

3

)(^

v

¡¹ )+ (a

3

+ b

2

) ^

x + (b

1

+ 1) ~

x

B

+ ~

x

0

:

Hence ~

O isnorm ally d istrib uted ,w ith m eanzero.Consequently

p(O ) = ¹ + ¸ O ;

w ith

¸ =

C ov(~

v; ~

O )

V ar( ~

O )

:

(15)

Now

c

ov

³

~

v; ~

O

´

= (a

1

+ (N

¡1)a

0

+ a

2

+ b

3

)¾

2

v

=

(2 ¾

2

v

(N + 1) + 3N ¾

2

"

)¾

2

v

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

;

(16)

30

and

V ar (O ) =

(a

1

+ (N

¡1)a

0

)

2

¾

2

v

+ (a

2

+ b

3

)

2

¡

¾

2

v

+ ¾

2

"

¢

+ 2 (a

1

+ (N

¡1)a

0

)(a

2

+ b

3

)¾

2

v

+ (a

3

+ b

2

)

2

¡

¾

2
B

+ ¾

2

´

¢

+ (b

1

+ 1)

2

¾

2

B

+ 2 (a

3

+ b

2

)(b

1

+ 1)¾

2

B

+ ¾

2

0

=

(a

1

+ (N

¡1)a

0

+ a

2

+ b

3

)

2

¾

2

v

+

µ

b

3

2

¶

2

¾

2

"

+

µ

a

3

2

+

1
2

¶

2

¾

2
B

+

³a

3

2

´

2

¾

2

´

+ ¾

2

0

=

¾

2

v

(2 ¾

2

v

(N + 1) + 3N ¾

2

"

)

2

+ ¾

2

v

¾

2

"

(¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

))

2

¡

5¾

4

B

36

¡

¾

2

B

+ ¾

2

´

¢+

¾

2

B

4

+ ¾

2

0

:

W e d ed uc e that

¸ =

6

q

¾

2

v

¡

¾

2

B

+ ¾

2

´

¢

(4 (N + 1)¾

4

v

+ (12 N + 5)¾

2

v

¾

2

"

+ 9N ¾

4

"

)

(2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

q¡

¾

2

B

¡

4 ¾

2

B

+ 9¾

2

´

¢

+ 36¾

2

0

¡

¾

2

B

+ ¾

2

´

¢¢:

(17)

P roofofLem m a 2

W e w rite the equilib rium value of¸ inthe f

ollow ingw ay:

¸

¡

¾

2

"

;¾

2

´

¢

=

6

p

¾

2

v

(4 (N + 1)¾

4

v

+ (12 N + 5)¾

2

v

¾

2

"

+ 9N ¾

4

"

)

(2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

£

q

¾

2

B

+ ¾

2

´

q

¾

2

B

¡

4 ¾

2

B

+ 9¾

2

´

¢

+ 36¾

2

0

¡

¾

2

B

+ ¾

2

´

=

6

£¸

1

¡

¾

2

"

¢

£¸

2

¡

¾

2

´

¢

:

It f

ollow sthat

@¸

¡

¾

2

"

;¾

2

´

¢

@¾

2

´

=

6

£¸

1

¡

¾

2

"

¢

£

@¸

2

¡

¾

2

´

¢

@¾

2

´

=

¡

15

£¸

1

(¾

2

"

)

£(¾

2

B

)

2

q

¾

2

B

+ ¾

2

´

¡

¾

2

B

¡

4 ¾

2

B

+ 9¾

2

´

¢

+ 36¾

2

0

¡

¾

2

B

+ ¾

2

´

¢¢

3
2

< 0 ;

@¸

¡

¾

2

"

;¾

2

´

¢

@¾

2

"

=

6

£¸

2

¡

¾

2

´

¢

£

@¸

1

(¾

2

"

)

@¾

2

"

=

6

£¸

2

¡

¾

2

´

¢

£¾

4

v

(3(7N

¡5)¾

2

"

+ 2 (5N

¡2 )¾

2

v

)

2 (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

2

p

¾

2

v

(4 (N + 1)¾

4

v

+ (12 N + 5)¾

2

v

¾

2

"

+ 9N ¾

4

"

)

> 0 :

31

W e also ob serve that

lim

¾

2

"

! 1

¸

¡

¾

2

"

;¾

2

´

¢

=

6

£¸

2

¡

¾

2

´

¢

£

p

N ¾

2

v

(N + 1)

;

lim

¾

2

´

! 1

¸

¡

¾

2

"

;¾

2

´

¢

=

2

£¸

1

¡

¾

2

"

¢

£

1

p

¾

2

B

+ 4 ¾

2

0

:

Consequently,

¸(

1 ;1 ) = lim

¾

2

´

! 1

¾

2

"

! 1

¸

¡

¾

2

"

;¾

2

´

¢

= 6

£

1

p

9¾

2

B

+ 36¾

2

0

£ lim

¾

2

"

! 1

¸

1

¡

¾

2
"

¢

=

2

p

N ¾

2

v

(N + 1)

p

¾

2

B

+ 4 ¾

2

0

:

(18)

P roofofLem m a 3

W e d enote b y ¼

j

(y

j

), spec ulator j'sexpec ted pro¯t givenhisinf

orm ationset y

j

prior to

trad ingat d ate 1 and b y¦

j

(¾

2

´

;¾

2

"

;N ),hisex-ante expec ted pro¯t,that isb ef

ore ob serving

inf

orm ation.Notic e that

¼

j

(y

j

) = Q

j

£E(~v ¡¹ ¡¸ ~x ¡¸Q

¡j

¡¸Q

j

jy

j

):

(19 )

T he ¯rst ord er c onditionf

or spec ulator jim poses(see the proofofLem m a 1) that

2 ¸ Q

j

= E(~

v

¡¹ ¡¸ ~x ¡¸Q

¡j

jy

j

):

(2 0 )

Hence,w e d ed uc e f

rom E quations(19 ) and (2 0 ) that ¼

j

(y

i

) = ¸(Q

j

)

2

.It f

ollow sthat

¦

j

= E(¼

j

(y

j

)) = ¸

£V ar(Q

j

):

T hisim pliesthat

¦

S

(¾

2

´

;¾

2

"

;N ) = ¸

¡

a

2

1

V ar~

v + a

2

3

V ar^

x

1

+ a

2

2

V ar^

v + 2 a

1

a

2

c

ov (~

v;^

v)

¢

;

w hich yield (usingthe expressionsf

or a

1

,a

2

and a

3

)

¦

S

(¾

2

´

;¾

2

"

;N ) =

Ã

¾

2

v

(¾

2

v

+ ¾

2

"

)(4 ¾

2

v

+ 9¾

2

"

)

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

2

+

¸¾

4

B

9

¡

¾

2

B

+ ¾

2

´

¢

!

:

32

W e d e¯ne

¦

S
nf

d ef

=

¸ ¾

4

B

9

¡

¾

2

B

+ ¾

2

´

¢;

and

¦

S
f

d ef

=

µ

¾

2

v

(¾

2

v

+ ¾

2

"

)(4 ¾

2

v

+ 9 ¾

2

"

)

¸ (2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

2

¶

:

W e proc eed exac tly inthe same w ay f

or spec ulator B .

P roofofP roposition2

T he f

ollow inglem m a isusef

ulf

or the proof

.

Lem m a 4 : Inabsence ofinformationsharing, spec ulator S hasa larger expec ted pro¯ t

thanspec ulator B (¦

B

(

1 ;1 ;N ) ·¦

S

(

1 ;1 ;N )) i®

¾

2

0

¸

(N

¡1)

4 + (N

¡1)

:

P roof

: W e have

¦

B

(

1 ;1 ;N ) =

¸(

1 ;1 )¾

2

B

4

;

(2 1)

and

¦

S

(

1 ;1 ;N ) =

1

¸ (

1 ;1 )(N + 1)

2

:

(2 2 )

U singE quation(18) (proofofLem ma 2 ) w e ob tainthat ¦

B

(

1 ;1 ;N ) ·¦

S

(

1 ;1 ;N ) i®

¾

2

0

>

(N

¡1)¾

2

B

4

:

T henthe result f

ollow sf

rom the f

ac t that ¾

2

B

= 1

¡¾

2

0

.

W henthere isperf

ec t inf

orm ationsharing,spec ulatorsB and Shave the sam e expec ted

33

pro¯tsgivenb y

¦

S

(0 ;0 ;N ) = ¦

B

(0 ;0 ;N ) =

1

¸ (0 ;0 )(N + 2 )

2

+

¸(0 ;0 )¾

2

B

9

:

(2 3)

It f

ollow sthat perf

ec t inf

orm ationsharingispossib le i®

1

¸(0 ;0 )(N + 2 )

2

+

¸ (0 ;0 )¾

2

B

9

¸M axf¦

B

(

1 ;1 ;N );¦

S

(

1 ;1 ;N )g:

(2 4 )

Case 1.¾

2

0

¸

(N

¡1)

4 + (N

¡1)

:Inthisc ase,usingLemm a 4 ,w e c anrew rite Condition(2 4 ) as

1

¸(0 ;0 )(N + 2 )

2

+

¸(0 ;0 )¾

2

B

9

¸¦

S

(

1 ;1 ;N );

w hich yields(usingE quation(2 2 )),

1

¸ (0 ;0 )(N + 2 )

2

+

¸(0 ;0 )¾

2

B

9

¸

1

¸(

1 ;1 )(N + 1)

2

(2 5)

It f

ollow sf

rom the expressionof¸ (inthe proofofLem m a 2 ) that

¸(0 ;0 ) =

3

p

N + 1

(N + 2 )

p

¾

2

B

+ 9¾

2

0

;

and ¸(

1 ;1 ) isgivenb y equation(18).U sing these expressionsand the fac t that ¾

2

B

=

1

¡¾

2

0

,w e rew rite (af

ter som e algeb ra) E quation(2 5) as

µ

(N + 1)
(N + 2 )

¶

G (N ;¾

2

0

)

¸0 ;

w ith

G (N ;¾

2

0

) =

"

N + 1
N + 2

+

(N + 1)

2

(1

¡¾

2

0

)

(N + 2 )(1 + 8¾

2

0

)

¡

3

p

(N + 1)(1 + 3¾

2

0

)

2

p

N (1 + 8¾

2

0

)

#

:

Notic e that G (N ;:) d ec reasesw ith ¾

2

0

.Furtherm ore G isstric tly positive f

or ¾

2

0

=

(N

¡1)

4 + (N

¡1)

and negative f

or ¾

2

0

= 1.W e c onclud e that there existsa c uto® ¾

¤2

0

(N )

2 (

(N

¡1)

4 + (N

¡1)

;1) suc h

that Condition(2 5) issatis¯ed i® ¾

2

0

· ¾

¤2

0

(N ). T hisc uto® isim plic itly d e¯ned asthe

solutionof

G (N ;¾

2

0

) = 0 :

(2 6)

34

AsG (:;:) increasesw ith N and d ec reasesw ith ¾

2

0

, w e d ed uc e that ¾

2

¤

0

(N ) increasesw ith

N .

Case 2 .¾

2

0

<

(N

¡1)

4 + (N

¡1)

:Inthisc ase,usingLemm a 4 ,w e c anrew rite Condition(2 4 ) as

1

¸ (0 ;0 )(N + 2 )

2

+

¸(0 ;0 )¾

2

B

9

¸¦

B

(

1 ;1 ;N );

w hich yields(usingE quation(2 1)),

1

¸ (0 ;0 )(N + 2 )

2

+

¸(0 ;0 )¾

2

B

9

¸

¸(

1 ;1 )¾

2

B

4

(2 7)

U sing the expressionsf

or ¸ (0 ;0 ) and ¸(

1 ;1 ), after som e m anipulations, w e rew rite the

previousc onditionas

F (N ;¾

2

0

)

d ef

=

Ã

3(N + 2 )

p

N (1 + 8¾

2

0

)

2

p

(N + 1)(1 + 3¾

2

0

)

¡(N + 1)

!

(1

¡¾

2

0

)

1 + 8¾

2

0

¡1 ·0 :

W e ob serve that F (N ;:) d ec reasesw ith ¾

2

0

. Furtherm ore F > 0 f

or ¾

2

0

= 0 and F < 0

f

or ¾

2

0

=

(N

¡1)

4 + (N

¡1)

.It f

ollow sthat there existsa c uto® ¾

2

0

(N )

2 (0 ;

(N

¡1)

4 + (N

¡1)

) suc h that f

or

¾

2

0

¸¾

2

0

(N ),Condition(2 4 ) issatis¯ed .T hisc uto® isim plic itly d e¯ned asthe solutionof

F (N ;¾

2

0

) = 0 :

AsF (:;:) increasesw ith N and d ec reasesw ith ¾

2

0

, w e d ed uc e that ¾

2

0

(N ) increasesw ith

N .Furtherm ore w e have

0 < ¾

2
0

(N ) <

(N

¡1)

4 + (N

¡1)

< ¾

2

¤

0

(N ) < 1:

P roofofP roposition3

Step 1: P ric esare m ore in

f

orm ative w henthere isinf

orm ationsharin

g.R ec all

that ~

v and ~

p are norm ally d istrib uted and that ~

p(O ) = ¹ + ¸O .T heref

ore

V ar(~

v

j~p(O ) = p) = ¾

2

v

¡

C ov

2

(~

v; ~

O )

V ar( ~

O )

:

U singE quations(15) and (16) w hic h appear inthe proofofLem m a 1,w e ob tainthat

35

V ar(~

v

j~p(O ) = p) = ¾

2

v

¡¸C ov(~v; ~

O ) = ¾

2

v

¡

(2 ¾

2

v

(N + 1) + 3N ¾

2

"

)¾

2

v

(2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

:

It isim m ed iate that V ar(~

v

j~

p(O ) = p) increasesw ith ¾

2

"

and d oesnot d epend on¾

2

´

.T his

m eansthat inf

orm ationsharing (a d ec rease in¾

2

"

and ¾

2

´

) m akesequilib rium pric esm ore

inf

orm ative.

Step 2 : P ric esare l

essvol

atil

e w henthere isin

f

orm ationsharin

g.

O b serve that

V ar(~

v

¡p) = E(E((~v ¡p)

2

j~p = p)):

As~

p = E(~

v

j~p),the previousequality im pliesthat

V ar(~

v

¡p) = E(V ar(~v j~p = p)):

F inally since ~

v and ~

p are norm ally d istrib uted ,V ar(~

v

j~p = p) isc onstant so that

V ar(~

v

¡p) = V ar(~v j~

p = p):

Hence pric esare lessvolatile w henthere isinf

orm ationsharing since pric esare more in-

f

orm ative inthisc ase.

P roofofP roposition4

Consid er the f

ollow ingratio

H (N ;¾

2

0

) =

¸(0 ;0 )

¸ (

1 ;1 )

:

P erf

ec t inf

orm ationsharingimprovesm arket liquid ity ifand only if

H (N ;¾

2

0

) < 1:

U singthe expressionf

or ¸ giveninthe proofofLem m a 2 ,w e ob tain

H (N ;¾

2

0

) =

3(N + 1)

p

(N + 1)(1 + 3¾

2

0

)

2 (N + 2 )

p

N (1 + 8¾

2

0

)

:

36

It isim m ed iate that H (N ;:) d ec reasesw ith ¾

2

0

.Furtherm ore H (N ;1) < 1 and H (N ;0 ) > 1.

T heref

ore there existsa threshold ¹

¾

2

0

(N ) suc h that H < 1 i® ¾

2

0

> ¹

¾

2

0

(N ).T histhreshold

solves

H (N ;¾

2

0

) = 1:

Solvingthisequation,w e d ed uc e that

¹

¾

2

0

(N ) =

1

¡h

2

(N )

8h

2

(N )

¡3

;

w here h (N ) =

2 (N + 2 )

p

N

3(N + 1)

p

N + 1)

< 1.Ash (N )

¸2 =3,w e have ¹¾

2

0

< 1.

P roofofP roposition5

T he expec ted trad ingc ostsf

or the liquid ity trad ersw henthere isinf

ormationsharingare

E (C T

e

) = ¸

Ã

6¾

2

0

¡

¾

2

B

+ ¾

2

´

¢

+

¡

2 ¾

2

B

+ 3¾

2

´

¢

¾

2

B

6

¡

¾

2

B

+ ¾

2

´

¢

!

:

U singthe expressionf

or ¸,w e rew rite thisequationas

E (C T

e

) =

p

¾

2

v

(4 (N + 1)¾

4

v

+ (12 N + 5)¾

2

v

¾

2

"

+ 9N ¾

4

"

)

¡

6¾

2

0

¡

¾

2

B

+ ¾

2

´

¢

+

¡

2 ¾

2

B

+ 3¾

2

´

¢

¾

2

B

¢

(2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

q¡

¾

2

B

+ ¾

2

´

¢£

¾

2

B

¡

4 ¾

2

B

+ 9 ¾

2

´

¢

+ 36¾

2

0

¡

¾

2

B

+ ¾

2

´

¢¤ :

W henthe b rokersd o not share their inf

orm ation,then

E (C T

ne

) =

E [(P(O )

¡~v)£~x]= ¸

n e

µ

¾

2
0

+

1
2

¾

2

B

¶

=

p

¾

2

v

N

(N + 1)

£

(2 ¾

2

0

+ ¾

2

B

)

p

¾

2

B

+ 4 ¾

2

0

:

W e d enote © the d i®erence b etw eenthe expec ted trad ingc ostsw henthere isinf

orm ation

sharingand w henthere isno inf

orm ationsharing.Hence

©

¡

N ;¾

2

"

;¾

2

´

¢

= E (C T

e

)

¡E (C T

ne

)

37

Straightf

orw ard m anipulationsshow that

p

¾

2

v

(4 (N + 1)¾

4

v

+ (12 N + 5)¾

2

v

¾

2

"

+ 9N ¾

4

"

)

(2 (N + 2 )¾

2

v

+ 3(N + 1)¾

2

"

)

<

p

¾

2

v

N

(N + 1)

(2 8)

Now c onsid er the f

ollow ingf

unction

Ã

¡

¾

2

´

¢

=

¡

6¾

2

0

¡

¾

2

B

+ ¾

2

´

¢

+

¡

2 ¾

2

B

+ 3¾

2

´

¢

¾

2

B

¢

2

¡

¾

2

B

+ ¾

2

´

¢£

¾

2

B

¡

4 ¾

2

B

+ 9¾

2

´

¢

+ 36¾

2

0

¡

¾

2

B

+ ¾

2

´

¢¤¡

(2 ¾

2

0

+ ¾

2

B

)

2

¾

2

B

+ 4 ¾

2

0

:

As¾

2

0

= 1

¡¾

2

B

,w e rew rite the previousequationas

Ã

¡

¾

2

´

¢

=

¡

6

¡

¾

2

B

+ ¾

2

´

¢

¡¾

2

B

¡

4 ¾

2

B

+ 3¾

2

´

¢¢

2

¡

¾

2

B

+ ¾

2

´

¢£

36

¡

¾

2

B

+ ¾

2

´

¢

¡¾

2

B

¡

32 ¾

2

B

+ 2 7¾

2

´

¢¤¡

(2

¡¾

2

B

)

2

4

¡3¾

2

B

:

O b serve that

à (0 ) =

¾

2

B

(

¡7+ 11¾

2

B

¡4 ¾

4

B

)

(9

¡8¾

2

B

)(4

¡3¾

2

B

)

< 0 ,since ¾

2

B

2 [0 ;1]

and

lim

¾

2

´

! 1

Ã

¡

¾

2

´

¢

= 0 :

and

Ã

0

¡

¾

2

´

¢

=

¾

4

B

¡

176¾

8

B

+ 14 4 ¾

6

B

¡

2 ¾

2

´

¡3

¢

¡72 ¾

2

B

¾

2

´

¡

5¾

2

´

¡7

¢

+ 9 ¾

4

B

¡

13¾

4

´

¡88¾

2

´

+ 2 8

¢

+ 2 52 ¾

4

´

¢

¡

¾

2

B

+ ¾

2

´

¢

2

£

36

¡

¾

2

B

+ ¾

2

´

¢

¡¾

2

B

¡

32 ¾

2

B

+ 2 7¾

2

´

¢¤

2

(2 9 )

Now w e rem ark that if¾

2

B

2

£

0 ;

2 1
2 2

¤

, thenÃ

0

¡

¾

2

´

¢

> 0 and theref

ore Ã

¡

¾

2

´

¢

< 0 . If

¾

2

B

2

£

2 1
2 2

;1

¤

,thenthere isa unique value of¾

2

´

suc h that Ã

0

= 0 .T hisvalue is

¹

¾

2

´

=

2 ¾

2

B

(2 2 ¾

2

B

¡2 1)

3(14

¡13¾

2

B

)

:

Hence à hasonly one extremum and thisextremum isa m inimum since

38

Ã

00

¡

¹

¾

2

´

¢

=

2 7(14

¡13¾

2

B

)

4

62 5¾

8

B

(2 ¾

2

B

¡1)

> 0 ;

W e d ed uc e that

8¾

2

´

and

8¾

2

B

,Ã

¡

¾

2

´

¢

< 0 .W e c onclud e that

¡

6¾

2

0

¡

¾

2

B

+ ¾

2

´

¢

+

¡

2 ¾

2

B

+ 3¾

2

´

¢

¾

2

B

¢

q¡

¾

2

B

+ ¾

2

´

¢£

¾

2

B

¡

4 ¾

2

B

+ 9¾

2

´

¢

+ 36¾

2

0

¡

¾

2

B

+ ¾

2

´

¢¤<

(2 ¾

2

0

+ ¾

2

B

)

p

¾

2

B

+ 4 ¾

2

0

:

(30 )

U singInequality(2 8) and Inequality(30 ),w e d ed uc e that ©

¡

N ;¾

2

"

;¾

2

´

¢

< 0 w hich m eans

that the expec ted trad ingc ostsare alw ayslow er w henthere isinf

orm ationsharing.

39

FIGURE 1: Is Perfect Information Sharing Possible?

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

2

5

10

15

20

25

30

35

40

45

50

Number of Fundamental Speculators

s

2 0

NO

YES

NO

σ

2

0

(N)

σ

2

0

*

(N)

FIGURE 2: Does Perfect Information Sharing Improve Liquidity?

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

2

5

10

15

20

25

30

35

40

45

50

Number of Fundamental Speculators

s

2

0

YES

NO

σ

2

0

(N)

σ

2

0

*

(N)