Information Sharing, Liquidity and Transaction Costsin Floor-Based Trading Systems

I f ationSharin ,Liqu i an T ranac tionCos norm g id ty d s ts i l nF oor-B as T rad i gSys s1 ed n tem T hi erry Fou au t c l Lau c e Les ou ren c rret HE C an CE P R d CR E ST an Doc torat HE C d 1,ru d e l Li ¶rati e a b e on 15,B ou evard G ab ri ¶ri l elP e 51 J ou enJ os ,Frane 83 y as c 2 M al o®,Frane ak c E m :ou au t@hec r lf c l f E mai :es ou ll c rr@enae.r s f Novemb er,2 0 1W e t a k G io a n Ce , A sa i Sa k se a hn v n i spa n r a nd rpa t ipa t a r ic n s tCR EST , L a a niv lU v r y t e A FFI20 0 c n e e e t e EEA 20 0 c n e e c , t e FM A 20 M e t g a d t e e sit , h o f r nc , h o frn e h e in s n h In e n t n lFin n eCo e e c T u isie 20 A l r o s a eo r t r a io a ac nf r n e n le r r r u s Ab s trac t I f ati norm onShari g L qu d i an T ranac ti n , i i ty d s onCos nF l tsi oor-B as T rad i g ed n Sys s tem W e c oni er i f ati hari gb etw eentrad ers(\°oor b rok " w ho pos es i sd norm ons n ers ) s sd ®eren t typesofnormati ,n if on amel normati yi f ononthe pay of ri k ec u tyor i f ati on o® a s ys ri norm on the vol me ofi i i u lqu d tytrad i gi n nthi ec u ty e i terpret thes trad ersasd u -c apac i ss ri W n e al ty b rok ersonthe °oor ofanex han e.W e i en f c on i on n er w hi h the trad ersare c g d ti y d ti su d c b etter o® s n i f hari g normati W e alo s on s how that i f ati norm ons n i proves pri e hari g m c d i c overy red u esvol lty an l ersex ted trad i g c os I f ati s , c ati i d ow pec n ts norm ons n c an hari g i mprove or i mpai the d epth ofthe m ark d epen i g onthe val esofthe param eters r et, d n u O veral r an ysss g es lou al i u g tsthat i f ati norm ons n am on °oor b rok hari g g ersi provesthe m perf ane of°oor-b as trad i gs tem s orm c ed n ys K eyw ord s: M ark M i ros c tu et c tru re,F l oor-B as T rad i gSys s penO u ry nored n tem ,O tc ,I f m ati onShari g normati n ,I f onSal es J E L Cl s¯c ati as i onNu ers: G 10 ,D82 mb I trod u ti n c on T he org i ati anz onoftrad i g onthe NY SE hasb eenrem ark l s l sne i n ab y tab e i c ts¯rs c on t s tu oni ti ti n1817 rad i gi on u ted throu h openou ry ofb i san o®ersofb rok T n sc d c g tc d d ers ac ti gonb ehal n fofthei c len r i tsor f thei ow nac c ou t T hi or r n1 strad i gm echans i ot n i m sn u i e to the NY SE E qu ty m ark nqu i etslk the Fran frt Stoc k E x han e an the AM E X ie ku c g d or d eri vesm ark vati etslk the CB O T an the CB O E are °oor m ark How ever °oorie d ets2 b as trad i g m ec hans sare en an ered s i ed n im d g pec esasthey are prog svel repl ed b y res i y ac fly au ated trad i g s tem s G i u l tom n ys venthi stren tow ard au ati , i i atu d tom on t sn ralto as w hether °oor-b as trad i g s tem sc anprovi e g k ed n ys d reater lqu d i an l er ex u i i ty d ow ec ti onc os tsthanau tomated trad i gs tem s hi n ys T squ ti sof es oni param ou t i portane f n m c or m ark org i et anzersan trad ers I ac t, i hasb eenhotl d eb ated b etw eenm emb ersof d nf t y E x han esw ho c oni ered s i hi gf °oor to el tronc trad i g I c g sd w tc n rom ec i n nord er to s rvi u ve °oor-b as trad i g m ec hans smu t ou ed n im s tperf orm au ated trad i g s tem sal g s e tom n ys on om di meni s son Au ated trad i g s tem sd omi ate °oor-b as trad i g s tem s i tom n ys n ed n ys nm an res ts y pec F i t °oor mark rs etsare more ex sve to operate (s Dom ow i an Stei peni ee tz d l(19 99 )).Sec on physc als e lm i d i pac i tsthe n mb er of u parti i tsi c pan n°oor m ark etsb u n i t ot nau ated tom trad i gs tem s i aly trad ersw i n ys F n l thou anac c es the °oor are at ani f ati ald i t sto norm on s ad van e c om pared w i the trad ersonthe °oor.T hi i ad van e i i el ex erb ate tag th sd s tag slk yto ac ag c y prob l sb etw eeni ves en em n torsan thei b rok d r ers(Sark an W u (199 9)) ar d B y d esg , °oor-b as mark in ed etsf ter pers -to-pers os on onc on ts Hene the ab i i of tac c lty m ark parti i tsto s et c pan hare i f normati sg oni reater i nthes m ark T hi eatu i ten e ets sf re sof vi ed asb ei gon ad van e,i n the u i e on ew n e tag f ot nqu e,of °oor-b as trad i gs tem s5 For ed n ys i s c e Harri 0 ),p ntan s(2 8,poi tsou that n t O fc u se o r ,ma y t a in u e h v be n c a g d sin e t e c e t n o h N Y SE u n r d gr l s a e e h n e c h r a io ft e B tith s aw y a l as be na° o rma k t eH a o c ,So a o a dSo be 993)f rad t il dd sc ipt no h r d g e o r e Se sbr u k ¯ n s n se e( o e a e e r io ft et a in r l s o t e N Y SE ue n h In F a k u t h o ro r t s in pa al lw h a e e t o ict a in r n f r,t e° o pe a e r le it n l c r n r d gsy e st m T h a c ảàT e meIn e n t na eF a c M A T IF) h o o t o kEx h n ea dT h o d n eM r h a r e t r a io ld r n e( ,t eT r n oSt c cag n eL n o In e n t n lFin n ia u u e a d O pt n Ex h n e ( IFFE)sh td w t e o rin 997 998 a d t r a io a a c lF t r s n io s c a g L u o n h ir° o ,1 n 20 0 ,r spe t e y e c iv l Se t e Ec no e h o mist( l 31 st1 999)\ A h me g o n r v l t n r "a d t e Ec n mist( u u Ju y : o r w e ou io a y n h oo A g st26t h 20 0 ) O u ft epit :\ to h s" Co a n Sh mw y ( 998)sh w t a h l v lo o o t e ° o ro v la d u a o h tt e e e fn ise n h o fCB O T ' 30 y a r a r s e rT e su y B o d f t r s a e t pr e v l t it T h asosu g st t a n u u e ® c s ic oa il y is l g e s h tpe so t r n ope so c n a t o t e ° o rh v r n o t cs n h o ae a impa to pr ef r t n n c n ic o ma io `l F oor-bas trad i gs s ed n y temsd om in el tron trad in y tem sw henbrok ate ec ic gs s ers n to ex hane i f eed c g normati onabou their c l tsto arrane their trad es' t ien g I f ati norm ons n i fnti hari g sa u c onofthe °oor w hi i i c u t to replc ate i ec tronc ch sd ± l i nel i trad i g s tem s T hes s temsu u l res c t the s ofm es ag n ys e ys s aly tri et s esthat c anb e s t b y en u ers(g eraly trad ersc anony pos pri esan qu ti es s en l l t c d an ti ).Fu rtherm ore trad i gi n nthes e s tem si nm os c as ys si t esan ym ou T hi eatu preven on s sf re tstrad ersf rom d evel n the opi g repu onofhon tl s n norm ati tati es y hari gi f onthrou h en u n ati s ps g d ri grel onhi I f ati hari gonthe °oor c antak pl e b etw eentw o typesof norm ons n e ac parti i ts i t c pan F rs °oor-b rok ersc anex han e i f ati c g norm ononthei trad i g m oti on i m ark r n vati sw th et-m ak ers B en i te,M arc u vens san W hi el (199 ) m od elthi d lm stype ofi f ati norm ons n d s hari gan how that i m i g t ti atesad vers s ec ti Sec on °oor-b rok e el on d ersc anc om mu i ate w i other °oornc th b rok For i s c e,So¯an ers ntan osan W ern (1997 n c e that d er ),p oti `nad d i on by s d i gi I ti , tan n nthe c row d , °oor brok ersm ayl earnabou ad d ition t al brok er-repres ted lqu d i that i ot re°ec ted inthe s ials qu : °oor en i i ty sn pec i t otes brok w i lof ers l tenex hane i f ationontheir in tion an capabil , c g norm ten s d ities es i l w i competi pec aly th torsw i w hom they have g w ork grel th ood in ationhips' s O u pu r rpos i e nthi spaper i sto an yz e thi al stype ofi f ati norm ons n At ¯rs gane, hari g t l c i f ati norm ons n am on °oor b rok hari g g ersi spu ln I ac t s d ard m od el i as zz i g nf tan sw th ym m etri i f ati c norm on(e K yl (19 85)) s g e how that i f ed trad ersw an to hi e thei i f norm t d r norm ati onrather thand i c l e i to poten alc om peti Fu s os t ti tors rtherm ore,i f ati norm ons n hari g rei f esi f ati alas norc norm on ymetri esb etw eenthos w ho s e hare i f ati norm onan thos w ho d e n ot.I i t stheref n ob vi sthat i s l i prove m ark qu i Hene w e ad ore ot ou t hou d m et alty c d res o qu ti s i t, i t opti alf °oor b rok stw es on.F rs si m or ersto s hare i f ati norm onw i thei th r c om peti ? Sec on , w hat i tors d sthe e®ec t ofi f ati norm ons n am on °oor b rok hari g g erson the overal lperf ane ofthe mark orm c et? I nparti u ar w e s d y the i pac t ofi ter-°oor c l tu m n b rok ersc om mu i ati nc onons d ard m eas resofm ark qu i , n el pri e vol lty tan u et alty am y c ati i , pri e d i c overy ark lqu d i an trad i gc os c s ,m et i i ty d n ts W e mod el °oor trad i gan i f ati s n sn yl (1985)' od el n d norm on hari gu i gK e sm asa w ork hors e Asi oÄl nR e l(199 ),w e as u that trad ers(°oor b rok ) have ac c es s me ers sto tw o typesofi n f ati : (i fn norm ati orm on ) u d tali f onw hi i norm ati ch si f ononthe pay ofthe s u ty o® ec ri an (i ) n -fn am en i f ati d i on u d tal norm onw hi h i norm ati c si f ononthe vol m e ofi i i u lqu d ty(n on i f ed ) trad i g e c oni er the posi i i f tw o °oor b rok norm n W sd sb lty or ersen ow ed w i d i d th ®eren t typesofnorm ati if on(on hasfn am en i f ati e ud tal norm onan the other hasn -fn am en d on u d tal i f ati ) to s norm on hare i f ati M ore s i aly w e asu e that °oor b rok norm on pec ¯c l sm ershave i n f ati hari gag orm ons n reem en ts(they f a \c lqu ).Anag orm i e" reem en s i t pec ¯esthe prec i i son w i w hi h eac h b rok reportshi th c er sor her i f ati norm onto the other b rok er.Af rec ei n ter vi g fn or n -fn am en i f ati ,the b rok n c lqu pool r i f ati ud tal on u d talnorm on ersi a i e thei norm on ac c ord i gto the termsof r ag n thei reem en j s b ef s b m i n t ut ore u tti gthei ord ersf ex u on r or ec ti W e es ls the f l i gres l tab i h olow n u ts ²T here i w i e ran e of sa d g param etersf w hi i i or ch t sopti alf °oor b rok m or ersto s hare thei i f ati r norm on(i e.thei ex ted pro¯tsare l er w i i f ati r pec arg th norm ons n ) hari g ²I f ati norm ons n c ani prove or i pai the d epth ofthe m ark d epen i g on hari g m m r et, d n the val esofthe parameters u ²I f ati norm ons n w aysred u esthe ag reg trad i gc os or lqu d i trad ers hari gal c g ate n tsf i i ty How ever w heni f ati norm ons n m pai hari gi rsm ark d epth,s e lqu d i trad ersare et om i i ty hu rt ²I f ati norm ons n oc c u hari g rsat the ex s ofthe °oor b rok pene ersw ho are n part to ot the i f normati ons n reem en hari gag t ²I f ati norm ons n mprovespri e d i c overy an red u esm ark vol lty hari gi c s d c et ati i I tu ti y i f ati n i vel norm ons n i teni hari g n s¯esc om peti onb etw een°oor b rok ti ersan i d nthi s w ay i l ersthe totalex ted pro¯tsofal t ow pec l°oor b rok ers(red u esthe ag reg trad i g c g ate n c os ) I f ts normati ons n alo c han es the aloc ati hari g s g l onoftrad i g pro¯ts am on °oor n g b rok M ore s i aly the °oor b rok ers pec ¯c l ersw ho s hare i f ati norm onc aptu a l er part of re arg the totalex ted pro¯ts at the ex s of°oor b rok pec , pene ersw ho d o n s ot hare i f ati norm on T hes tw o e®ec ts ex hy i f ati e pl nw norm ons n c ansmu tan sy b en lqu d i hari g i l eou l e¯t i i ty trad ersan the °oor b rok d ersw ho s hare thei i f ati O veral norm ati r norm on li f ons n hari g b etw een°oor b rok ersi sanad van e f °oor-b as trad i gs tem ssne i res l n(a) tag or ed n ys ic t u tsi l er trad i gc os , (b ) f ter pri e d i c overy an (c ) l er pri e vol lty n ow n ts as c s d ow c ati i I teres n l, ti gy i i e w i ou res l Venataram an(2 0 ) ¯n sthat trad i g c os nln th r u t, k d n tsonthe NY SE are l er thanonthe P ari ou e (anau ated trad i gs tem ),c on ln or d i ow sB rs tom n ys troli gf ®erenes c i toc k harac teri ti s6 ns sc sc T h isse ( 999)c mpa e e e t e bid a e n o r s ® c iv - skspr a s in a a t ma e t a in ed n u o t d r d gsy e ( e r )a d t e st m X t a n h ° o ro h F a k u tSt c Ex h n e f rst c s t a r d in bo h sy e o ft e r n f r o k c a g o o k h tt a e t st ms.H e ¯ d t a h a e a e n s h tt e v r g O u an yssi r al i srel ated to the lteratu oni f ati al i re norm ons es(e Ad m atian P °ei erer g d d (19 86 (19 88) an F i hm anan Hag ), d s d erty (199 5)) I on t w i thi lteratu w e nc tras th si re, as u e that the med i m f i f ati sm u or norm onex chan e i norm ati , n m on Ac tu l g si f on ot ey aly i nou m od el the trad er w ho rec ei r , vesi f ati norm onrew ard sthe i f ati norm onprovi er b y d d i c l i gan s osn other type ofi f ati Hene w e c oni er °oor-b as s tem sasm ark norm on c sd ed ys ets f trad i gs or n haresan f m to b arter i f ati An d oru norm on other i portan d i m t ®erene i c sthat w e c oni er c om mu i ati sd nc onofi f ati norm ononthe vol m e oflqu d i trad i g e s u i i ty n W how that i m ay b e opti alto `el'(b arter) s ch ani f ati t m sl u norm onan that s esof on u d am en d al n -fn tal i f ati norm onhave ani pac t onmark qu i m et alty T he mod eli es ri ed i sd c b nthe n t s ti Sec ti ex ec on on3 s sthat i c anb e opti alf how t m or °oor b rok ersto s hare i f ati Sec ti norm on on4 an yz esthe i pac t ofi f ati al m norm ons n hari g onvari sm eas resof ou u mark perf ane.Sec ti et orm c on5 c onl d es he proof hi h d o n c u T sw c ot appear i nthe tex are i t nthe Appen i d x T he M od el I f ati norm onShari gAg n reem en ts T he T rad i gCrow d n W e c oni er a m od elof sd trad i gi n nthe m ark f a ri k ec u tyw hi i as onK yl et or s ys ri ch sb ed e (19 85) T he ¯n alval e ofthe s u ty w hi i en u ec ri , ch sd oted v, i orm aly d i tri u ~ sn l s b ted w i th m eanạ an a vari c e ắ v that w e n alze to 1.T hi d an orm i s¯n alval e i u spu lc l reveal at b i y ed d ate T rad i gi thi ec u tytak n n ss ri espl e at d ate 1.At thi ate,i ves ac sd n torss b m i m ark u t et ord ersto b u or to s ls y el haresofthe s u ty he ex es em an (s ppl i l ec ri T c sd d u y) sc eared at the pri e pos b y a c ompeti ve an ri k eu c ted ti d s -n tralm ark m ak et er T he trad i g \c row d " f the s u ty i om pos ofN + °oor b rok At ti e n or ec ri sc ed ers7 m 1, there are tw o typesof°oor b rok : (i N fn am en pec u atorsan (i ) on n ers ) ud tals l d i e on fn pec u ator, B Fu d am en pec u atorshave i f ati ud tals l n tals l norm ononthe ¯n alval e u of s u ty the ec ri For smplc i ,asi yl (19 85),w e asu e that theyperf tl s i i ty nK e sm ec yob erve thi s ¯n alval e,j s b ef s b m i n u ut ore u tti gthei ord ersat d ate 1.B rok B , the n -fn am en r er on u d tal q o e spr a s o t e° o rc n bel r e rsmal rt a in t ea t ma e t a in u td e d n h o a a g ro le h n h u o t d r d gsy e st m,d pe d e n ingo n t est c h r c e ist s.O na e a et eq o e h o kc a a t r ic v r g h u t dspr a s a ee u l h is c n e tw ho rr su tt a e d r q a.T is o sist n it u e l h t t eimpa to f r t n sha in n ma k td pt is a h c fin o ma io r go re e h mbig o s uu Th ema k tma e a asobec n e e a be ga° o rbr k rw oh s n f r t n r e - k rc n l o sid r d s in o o e h a oin o ma io s u ator, rec ei pec l vesord ersf rom lqu d i trad ers W e d en xB the totalqu ti that i i ty ote ~ an ty b rok B m u t ex u onb ehal er s ec te foflqu d i trad ers i i ty Asa w hol lqu d i trad ershave e, i i ty a n d em an equ et d alto x = x0 + xB s ~ ~ ~ hares W e as u e that x0 an xB are n aly sm ~ d ~ orm l an i d epen en y d i tri u d n d tl s b ted w i m ean an vari c es¾ 02 an ¾ B res ti y W e th s0 d an d pec vel n alze the vari c e ofthe ord er °ow d u to lqu d i trad i g x,to 1,i e orm i an e i i ty n ,¾ : 2 ¾ x = ¾ B + ¾ 02 = 1: I nthi ay B c anb e i terpreted asb rok B ' ark s sw ,¾ n er sm et hare of totalord er °ow f the rom lqu d i trad ers he rem i gpart of ord er °ow c anb e s i i ty T nn the eenasb ei gi term ed i n n ated b y °oor b rok ersw ho d o n trad e f thei ow nac c ou t or asb ei g rou el tronc aly ot or r n n ted ec i l to the °oor., B oth typesofs u atorsc anen ag i pec l g e npropri etary trad i g I n nparti u ar b rok B c l er c anac t b oth as anag t (s c han el a f ti en he n s rac onoflqu d i trad ers ord ers an as i i ty ' ) d a pri c i n pal(s s b mi he u tsord ersf her ow nac c ou t) T hi or n sprac ti e i n nas` u c sk ow d al trad i g an i n ' d sau thori ed i ec u ti z ns ri esm ark ets(s Chak ee ravarty an Sark (2 0 ) f a d ar or d i c u son 10 M od el i d u -trad i gi c l d e R Äel (19 90 ),Sark (199 5) or F i hm anan s si ) sw th al n nu o l ar s d Lon s (199 ).I g ta® nthes m od el,asi the pres t arti l rok e s n en c e,b ersen ag i u -trad i g g ed nd al n ex oi thei ab i i to ob s pl t r lty erve ord erss b m i u tted b y u i f ed (lqu d i trad ers11 Non nnorm i i ty) e ofthes mod el e shasc oni ered i f ati sd norm ons n offn am en hari g ud talan n u d am en d onfn tal i f ati norm onamon b rok , how ever.O u pu g ers r rpos i e sto s d y the e®ec tsofthi ti ty tu sac vi Asarg ed i u nthe i trod u ti ,thi n c on stype of norm ati if onex chan e i d i ti c ti f re of g sa s n ve eatu °oor mark T he s u atorsw i fn am en norm ati ets pec l th u d tali f onc anb e s eenasb rok ersw ho ex l svel trad e f thei ow nac c ou t (lk s al c ui y or r n i e c persan l al nd eri vesm ark ) d oc si vati ets T hey c ou d alo b e s l s eenasb rok ersw ho have n c u tom ers ord ersto ex u at d ate o s ' ec te I i t sreas ab l to as u that the ord er °ow f on e s me rom lqu d i trad ersi n epen en i i ty si d d t ac ros rok (f i s c e b rok have d i sb ers or ntan ers ®eren c len ) I on t, sg al onthe t i ts nc tras i n s fn ud talval e ofthe s u ty are c orrel For thes reas s w e asu ed that u ec ri ated e on, sm ony on °oor b rok ob s l e er ervesthe n -fn am en norm ati , xB ,w hereass on u d tali f on ~ everal°oor In t U f l in br k r g h u s e g g in pr pr t r t a in a t it s.D isc u tbr k r d he S, u ll e o e a e o se n a e o ie a y r d g c iv ie o n oes o n t o e e o ,h w v r F rin a e o t e N Y SE,o d r c n r a h ama k tma e h o g ° o rbr k r o l c r n al o st nc , n h res a e c r e - k rt r u h o o e s re e t o ic ly t r u h asy e c le Su r o ho g st m al d pe D t 10 F rin a c , Ch k a a t a d Sa k r( 0 )o r e t a o st n e a r v ry n r a 20 bse v h tin t e N Y SE po e t ldu lt a e s a e h t n ia a rd r r n t n lf l in a io a u ll ebr k r g o se a o e a eh u s ndt ein e me tba k h v st n n s 11 Se aso M a r a 996) W e bo r wt e d in t n be w e `u d me t l s `o - u d me t l e l d ig l( r o h ist c io t e n f n a n a'v n n f n a n a' spe ul t r f o t a t o c a o s r m his u h r b rok ersob s erve the fn i f ati ,~.W e have an yz ed the m od el hen ud tal norm on v al w there i s m ore than e n -fn am en b rok (w i i d epen en ord er °ow ) an b rok on on u d tal er th n d t d ersperf tl ec y s hare i f ati norm on(i f ati hari gi es ri ed b el ).T he pres tati norm ons n sd c b ow en onof m od el the i ore c om pl b u the c onl son sm ex t c u i sare qu i vel sm i ar to thos w e ob tai nthe altati y i l e ni c as w i ony on n -fn e th l e on u d talb rok er O n reas or w hi the m od eli more e onf ch s c om pl sthat the n mb er of lqu exi u c i es(g psof red b rok rou pai ersw i d i ti c t i f ati ) th s n norm on i sen og ou I d en s nequ lb ri m, thi u b er c anb e s aler thanthe m axmu pos i l ii u sn m m l i m sb e n mb er ofc lqu For i s c e i u i es ntan fthere i sanequ u er, N , offn am en aln mb ud talan n d on fn ud talb rok , the n mb er ofc lqu anb e s aler thanN I ers u i esc m l nparti u ar, w i c l th perf t i f ec normati ons n , thi sn es ari y the c as w hen¾ = I hari g si ec s l e nthi as the sc e, ag reg ord er °ow c han el b y the n -fn am en g ate n ed on u d talb rok ersw ho are n a± lated to ot i a c lqu pl i e aysthe rol of~0 i e x nthe pres t arti l en c e I f ati norm onShari g n W e mod eli f ati norm ons n asf l s e as u e that the n u d am en pec hari g olow W sm onfn tals u ator, B , hasanag l reem en to s t hare i f ati norm onw i on fn am en pec u ator, S th e u d tals l Ac c ord i g to thi n sag reem en b ef trad i g at d ate 1, the n -fn am en pec u ator t, ore n on u d tals l s d sa sg al en in x = xB + ´ ; ^ ~ ~ to the fn am en pec u ator.I c han e,the fn am en pec u ator s d sa sg al ud tals l nex g ud tals l en in v = v + "; ^ ~ ~ to the n u d pec u ator.T he ran om vari l ´ an " are i d epen en y an onfn tals l d ab es~ d ~ n d tl d n alyd i tri u w i meanzero an vari c es¾ ´ an ¾ ",res ti y e ref to the orm l s b ted th d an d pec vel.W er i vers of ´ (res " ) asthe prec i ionof sg als t b y b rok B (S).T he l er i ´ n e ¾2 p.¾ s the i n en er arg s¾ 2 (¾ "),the l sprec i e i es s sthe sg als t b y s u ator B (s u ator S) an hene the l er i n en pec l pec l d c ow i tsi f ati val e.T w o pol c as si norm ve u ar esare ofparti u ar i teres c l n t.F i t there i rs sperf t ec 2 if normati s n f ´ = ¾ " = Sec on there i o inorm ations on hari gi ¾ d sn f harin f ´ = ¾ " = gi ¾ I -b etw eenthes tw o c as , there i norm ati n e es si f ons n b u i i m perf t (at l t on hari g t t si ec eas e s u ator d oesn perf tl d i c l e hi pec l ot ec y s os sor her i f ati ) T he i f ati norm on norm ons etsof s u atorsB an S at d ate are d en pec l d oted y = (~B ; x;^ an y = (~; x;^ x ^ v) d S v ^ v),res ti y pec vel B I nrealty °oor b rok i ersare lk y to ex i el chan e i f ati g norm onw i the b rok th ersw i w hom th they have en u n rel onhi I d ri g ati s ps nthi as thei d ec i i sc e r sonto s hare i f ati norm onw i a th gvenb rok mu t b e b as onthe l g i er s ed on -term (averag b en e) e¯tsofnorm ati hari g if ons n For thi sreas ,w e asu e that the s u atorsd ec i e to s on sm pec l d hare i f ati norm onb yc om pari gthei n r ex te (i e.pri to rec ei n norm ati ) ex ted pro¯tsw i an w i -an or vi gi f on pec th d thou i f ati t norm on s n W e s that i f ati hari gi hari g ay norm ons n sposi l i there ex s pai (¾ ´ , ") s ch that sb e f i tsa r ¾2 u the ex ted pro¯tsofs u ator S an B are l er w henthere i norm ati pec pec l d arg si f ons n hari g I ec ti ns on3 e i en f param eters val esf w hi i f ati ,w d ti y ' u or ch norm ons n spos i l hari gi sb e R em ark s I i orth s sn t sw tres i gthat w e f u oc son pos i i i the sb ltyof norm ati s n reem en ani f on hari gag t b u n oni mpl t ot tsi emen on I tati nparti u ar, w e d o n ad d res f em en i s es I c l ot senorc t su n that, w e f l the lteratu oni f ati olow i re norm ons esw here the qu i ofthe i f ati al alty norm on w hi h i ol i s m ed to b e c on ti l 12 W e alo as u e that the i f ati hari g c ss d sasu trac b e s sm norm ons n 2 ag reem en an i c harac teri ti s (¾ ´ ;¾ ") are k ow nb y al t d ts sc n lparti i ts (i c l d i g the c pan n u n m ark et-m ak er).T hi om monk ow l g asu pti salo s d ard i sc n ed e s m oni s tan nthe lteratu on i re i f ati norm ons es al 2 T he equ l ri m of ii u b the F l M ark oor et I nthi ec ti ,w e d eri the equ lb ri m ofthe trad i gs e at d ate 1,gventhe c harac ss on ve ii u n tag i teri ti sof i f sc the normati hari gag ons n reem en b etw eens u atorsB an S.T hen nthe t pec l d ,i n t s ti ,w e an yze w hether or n i i ex ec on al ot t sopti alf B an Sto ex m or d chan e i f ati g norm on W e d en b y Q S(y ) an Q B (y ), the ord erss b m i ote d u tted b y s u atorsS an B , repec l d S B s ti y nthe s offn pec u ators w e asi ni d ex1 to s u ator S.An pec vel.I et ud tals l , sg n pec l ord er s b mi u tted b y the other fn pec u atorsi= ;:: i en ud tals l :;N sd oted Q i(~).T he v totalex es eman that mu t b e c l c sd d s eared b y the c om peti ve m ark m ak i ti et er stheref ore O = i N = X Q i(~) + Q S(y ) + Q B (y ) + x: v ~ S B i = Asthe mark mak i s med to b e c om peti ve,he s et er sas u ti etsa pri e p(O ) equ c alto the as et s 12 Se A d t a d P ° e r r( 986)( 988) e ma i n ide e ,1 So pa r h v sh w h win e t e c n r c s c n be me pe s a e o n o c n iv s o t a t a u d t d c a in o ma io pr v e ot u h u l r v a he q ait o is sig a se A le ( 990 )o se oin u e n f r t n o id rt r t f ly e e lt u l y fh n l( e l n r B h t a h r a a d P ° e e e 985) R e t t n e e t ma aso h l t su a in o ma io sh r g at c ay n id r r( ) pu a io ® c s y l e p o st in f r t n a in a r e n s ( eB e a u a dL a o u ( 992) g e me t se n bo n rq e ) ex ted val e c on i on pec u d ti alonthe n ord er °ow ,i e et p(O ) = E(~ jO ): v (1) Anequ lb ri m c oni tsoftrad i g s ii u ss n trategesQ S(: Q B (: Q i(: i ), ), );i= ;:: an a c om :;N d peti ve pri e fnti ti c u c onp(: s c h that (i eac h trad er' ) u ) strad i gs n trateg i b es res s to y sa t pone 13 other trad ers s ' trategesan (i ) the d eal sb i d i g s i d i er' d n trateg i i y sgvenb y E qu on(1) ati For gvenc harac teri ti s (¾ ´ ;¾ " ), ofani f ati i sc , 2 norm ons n ag hari g reem en the n t l t, ex emm a d es ri esthe u i e ln equ lb ri m ofthe trad i gg e c b nqu i ear ii u n am L m a : T he trad i gs e hasa u iqu l ear equ ibriu w hic h isg em n tag n e in il m ivenby p(O ) = Q S(y ) = S Q i(~) = v Q B (y ) = B ¹ + áO ; (2 ) a (~ Ăạ) + a (^¡¹) + a 3x; v v ^ (3) a v ¡¹);i= ;:: (~ :;N (4 ) b1 xB + b2 x + b3(^¡¹), ~ ^ v (5) w here coe± c i tsa 1;a ;a 3;a0 ;b2 ;b3 an ¸ are en ;b d a1 = a2 = a3 = a0 = b1 = b2 = b3 = 13 2 3(¾ v + ¾ ") ; 2 ¸ (2 (N + )¾ v + 3(N + 1)¾ " ) ¾v Ă ; 2 (2 (N + )ắ v + 3(N + 1)¾ " ) ¾2 ¡ ¡ B Â; ắB + ắ 2 ¾v + " ¾2 ; 2 ¸ (2 (N + )¾ v + 3(N + 1)¾ " ) Ă ; ắ2 Ă2 B Â; ¾B + ¾´ 2 ¾v ; 2 ¸ (2 (N + )¾ v + 3(N + 1)¾ " ) M o epr c l ,w o sid rt eP e f c a e n Eq il iao h t a in a r e ise y ec n e h re tB y sia u ibr ft e r d gg me Appen i d x P roof L m a of em Step1: T he opti maltrad i g s n trateg f s u ator S.Let y = (~;^ x) b e the y or pec l v v; ^ S i f ati s of pec u ator S he l norm on et s l T atter c hoos eshi ark ord er,Q S,s asto m axm i e sm et o i z hi pec ted pro¯t sex ~ ¼ S(y ) = E(Q S(~ ¡p(O )) jy ): v S S T he ¯rs ord er c on i onyi d s t d ti el h i Pj N i = (~ ¡¹ ) ¡¸ £E Q B (y ) + v Q (~) + x0 + xB jy v ~ ~ B S j = Q S(y ) = : S 2¸ Noti e that c an d (9 ) ¡ ¢ E Q i(~) jy = a v ¡¹); v (~ S ¡ ¢ E Q B (y ) jy = b1 E (~B j^ + b2 x + b3(^¡¹ ); x x) ^ v B S an d E (~B j^) = x x ¾B x ^ : 2 ¾B + ¾´ Su s tu n b ti ti gthes ex son nE qu on(9) yi d s e pres i si ati el S Q (yS) = = à ắB (~ Ăạ ) v ¡ £ (N ¡1)a (~ ¡¹) + b3(^¡¹ ) + (b1 + 1) v v x + b2 x ^ ^ 2á ắB + ắ ả ả (N Ă1)a 1 b3 ¾B ¡ v + b2 x: ^ (~ ¡¹) ¡ (^¡¹ ) ¡ v (b1 + 1) 2 2á 2 ắB + ắ Hene, c a1 = a2 = a3 = ả (N Ă1)a Ă 2á b3 Ă 2à ả ¾B + b2 ¡ (b1 + 1) 2 ¾B + ¾´ Step : T he opti mal trad i gs n trateg f s u ator i = S y or pec l ,i6 27 Spec u ator ic hoos l eshi smark ord er,Q i,s asto m axm i e hi pec ted pro¯t et o i z sex ~ ¼ i(v) = E(Q i(~ ¡p(O )) j~ = v): v v T he ¯rs ord er c on i onyi d s t d ti el Q i(~) = v h S (~ ¡¹ ) ¡¸ £E Q (y ) + v S Pj N = j = ¸ i Q (~) + Q (y ) + x j~ = v v ~ v B j B (10 ) We f u oc sons ymm etri trad i g s c n trategesf al i or lthe s u atorsi S T hi m pos pec l = si es Q j v) = Q i(~); (~ v = i b s tu n j b y Q i i qu on(10 ) yi d s j6 Su ti ti gQ nE ati el Q i(~) = v Fu rthermore an d Ă Ê Ô Ê Ô Â (~ Ăạ ) v ¡ £ E Q S(y ) j~ + E Q B (y ) j~ = v : v v S B N N (11) Ă Â E Q S(y ) j~ = v = (a + a2 )(~ Ăạ); v v S Ă Â E Q B (y ) j~ = v = b3(~ ¡¹ ): v v B Conequ tl s en y i Q (~) = v ả (a + a + b3) Ă (~ Ăạ ): v N N (12 ) W e d ed u e that c a0= (a1 + a + b3) ¡ : N ¸ N (13) Step T he opti al : m trad i gs n trateg f s u ator B W e d en y = (~B ;^ x), y or pec l ote B x v; ^ the i f normati et of pec u ator B She choos ons s l esher m ark ord er,Q B ,s asto m axm i e et o i z ~ ¼ B (y ) = E(Q B (~ ¡p(O )) jy ): v B B 28 T he ¯rs ord er c on i onyi d s t d ti el Q B (y ) = B = ¸ · N P i P S E (~ j^ Ăạ Ăá ÊE Q (y ) + v v) Q (~) + v xi jy S B i = Ê 2á Ô E (~ j^ Ăạ Ăá ÊE Q S(y ) + (N ¡1)Q i(~) + x jy v v) v ~ B S : 2¸ W e n c e that oti ¡ ¢ E Q S(y ) jy = a 1E (~ ¡¹ j^ + a (^¡¹) + a 3x; v v) v ^ S B an d ¡ ¢ E Q i(~) jy = a (~ ¡¹ j^ v E v v); B an that d E (~ ¡¹ j^ = v v) ắv (^Ăạ ): v 2 ắv + ắ " Su s tu n thes ex son nthe ¯rs ord er c on i onf Spec u ator B yi d s(af b ti ti g e pres i si t d ti or l el ter s ome aleb ra) g xB ~ a3 Q (yB ) = ¡ ¡ x + ^ 2 2¸ B ắv ắ2 Ăáa Ăá (a + (N ¡1)a v ) 2 ¾ v + ắ" ắv + ắ " ả (^Ăạ ): v (14 ) Hene, c b1 = b2 = b3 = Ă ; a3 Ă ; 2à ả 2 ắv ắv Ăáa Ăá (a1 + (N ¡1)a ) : 2 2 ¸ ¾v + ¾" ¾v + ¾ " Steps1 to gve u equ on i u k ow n , a etc ) Sol n thi ys i s9 ati sw th n n s(a vi g ss tem of 29 equ on el ati syi d a1 = a2 = a3 = a0 = b1 = b2 = b3 = 2 3(ắ v + ắ ") ; 2 (2 (N + )¾ v + 3(N + 1)¾ " ) ắv Ă 2 (2 (N + )¾ v + 3(N + 1)¾ " ) ¾2 Ă Ă B Â; ắB + ắ 2 ¾v + " ¾2 ; 2 ¸ (2 (N + )¾ v + 3(N + 1)¾ " ) ¡ ; ¾2 ¡ B Â; ắB + ắ 2 ắv ; 2 (2 (N + )ắ v + 3(N + 1)¾ " ) Step Com pu onof R ec al tati ¸ lthat p(O ) = E(~ jO = O ): v ~ G i vens u ators trad i gru es pec l ' n l, O = Q S (y ) + (N ¡1)Q i(~) + Q B (y ) + x v ~ S B = (a1 + (N ¡1)a0 v ¡¹ ) + (a2 + b3)(^¡¹ ) + (a3 + b2 ) x + (b1 + 1) xB + x0 : )(~ v ^ ~ ~ Hene O i orm aly d i tri u ,w i m eanzero.Conequ tl c ~ sn l s b ted th s en y p(O ) = + áO ; wi th á= C ov(~; O ) v ~ : ~ V ar(O ) (15) Now ³ ´ ~ = (a + (N ¡1)a 0+ a2 + b3)¾ v = c v; O ov ~ 30 2 (2 ¾ v (N + 1) + 3N ¾ ")¾ v ; 2 (2 (N + )ắ v + 3(N + 1)ắ ") (16) an d Ă2  2 (a + (N ¡1)a ¾ v + (a2 + b3)2 ¾ v + ¾ " + (a + (N ¡1)a0 + b3)¾ v ) )(a ¡2 ¢ 2 + (a + b2 )2 ¾ B + ¾ ´ + (b1 + 1)2 ¾ B + (a3 + b2 )(b1 + 1)ắ B + ắ 02 ả ả2 ³a ´2 a3 2 b3 2 2 ¾" + ¾B + ¾ ´ + ¾ 02 + (a + (N ¡1)a + a + b3) ¾ v + 2 2 V ar (O ) = = 2 ¾v = W e d ed u e that c 2 (2 ¾ v (N + 1) + ¾ ") + ¾ v ¾ " N 5¾ ¾2 ¡ ¡ B ¢+ B + ¾ 02 : 2 36 ¾ B + ¾ (á (2 (N + )ắ v + 3(N + 1)ắ " ))2 q Ă Â 2 2 ¾ v ¾ B + ¾ ´ (4 (N + 1)¾ v + (12 N + 5)¾ v ¾ " + 9N ¾ " ) qĂ Ă á=  à Â:  2 + 3(N + 1)¾ ) + ¾2 + ¾2 (2 (N + )¾ v ¾ B ¾ B + 9¾ ´ 6¾ B " ´ (17 ) P roof L m a of em W e w ri the equ lb ri m val e of¸ i te ii u u nthe f l i gw ay: olow n q p 2 ¾B + ¾ ´ (4 (N + 1)¾ + (12 N + 5)¾ ¾ + 9N ¾ ) ¾v v v " " £q ¡ ¢ ¡2 2 (2 (N + )¾ v + 3(N + 1)¾ ") 2 2 ¾ B ¾ B + 9¾ ´ + 36 02 ắ B + ắ ắ Ă 2 à 2 6Êá ắ " Êá ắ : Ă 2 ắ " ;ắ = = I f l sthat t olow ¡ 2¢ ¡ 2 à  @á ắ @á ¾ ";¾ ´ = 6£¸ ¾ " £ 2 @ắ @ắ = Ă 2 @á ¾ ";¾ ´ = @¾ " = ¡q 2 15Êá (ắ ") Ê(ắ B ) Ă2 ¡ ¢ ¡2 ¢ < 0; ¢ 2 2 ¾ B + ¾ ´ ¾ B ¾ B + 9¾ ´ + 36 02 ¾ B + ắ ắ Ă Â @á (ắ " ) 6Êá ắ Ê @ắ " Ă 2 2 6Êá ¾ ´ £¾ v (3(7 ¡5)¾ " + (5N ¡2 )¾ v ) N p > 0: 2 2 (2 (N + )¾ v + 3(N + 1)¾ " )2 ¾ v (4 (N + 1)¾ v + (12 N + 5)¾ v ¾ " + 9N ¾ " ) 31 W e alo ob s s erve that ¡ 2¢ lm ắ ";ắ = i ắ "! Ă 2 lm ắ ";ắ = i ¾´ ! Conequ tl, s en y p N ắv ; (N + 1) Ă 2 : Êá ắ " Ê p ắ B + ắ 02 Ă 2 6Êá ¾ ´ £ p ¡ 2¢ ¡ 2¢ N ắv p Ê lm ¾ " = i : ¸(1 ;1 ) = lm ¸ ¾ ";¾ ´ = 6£ p i 2 ¾ "! ¾´ ! 9¾ B + 36 02 ¾ (N + 1) ¾ B + ¾ 02 ¾ "! (18) P roof L m a of em W e d en b y ¼ j j s u ator jsex ted pro¯t gvenhi norm ati ote (y ), pec l ' pec i si f ons y pri to et j or 2 trad i gat d ate an b yƯ j(ắ ;ắ " ;N ),hi -an ex ted pro¯t,that i ef ob s n n d sex te pec sb ore ervi g i f ati Noti e that norm on c ¼ j(y ) = Q j£E(~ Ăạ Ăá x ĂáQ ĂjĂáQ j jy ): v ~ j j (19 ) T he ¯rs ord er c on i onf s u ator ji pos t d ti or pec l m es(s the proofofLem m a 1) that ee Q j = E(~ Ăạ ¡¸ x ¡¸Q ¡j jy ): v ~ j Hene,w e d ed u e f E qu on c c rom ati s(19 ) an (2 ) that ¼ j i) = ¸(Q j I f l sthat d (y ) t olow Ư j = E(ẳ j(y )) = ¸ £V ar(Q j): j T hi m plesthat si i Ă Â 2 Ư S(ắ ;ắ " ;N ) = a21 V ar~ + a23V ar^1 + a22 V ar^+ a a2 c (~;^ ; v x v ov v v) w hi h yi d (u i gthe ex son or a1 ,a2 an a 3) c el sn pres i sf d 2 Ư S(ắ ;ắ " ;N ) = à ! 2 2 ¾ v (¾ v + ¾ ")(4 ¾ v + 9ắ " ) áắ B  : + Ă2 2 ắB + ắ (2 (N + )¾ v + 3(N + 1)¾ ")2 32 (2 ) W e d en e áắ Ă B Â; ắ B + ắ d ef ¦S = nf an d ¦ S d ef f = 2 2 ắ v (¾ v + ¾ ")(4 ¾ v + ¾ ") 2 ¸ (2 (N + )¾ v + 3(N + 1)ắ " )2 ả : W e proc eed ex tl i ac y nthe s ame w ay f s u ator B or pec l P roof P roposti of i on2 T he f l i gl m a i s u or the proof olow n em su eflf L ma : I em nabs ce ofi f en normations n, s uator S hasa l er ex ted pro¯ t hari g pec l arg pec thans uator B (¦ B (1 ;1 ;N ) ·¦ S(1 ;1 ;N )) iđ pec l ắ 02 (N Ă1) : + (N ¡1) P roof W e have : á(1 ;1 )ắ B ; (2 1) : ¸(1 ;1 )(N + 1)2 (2 ) ¦ B (1 ;1 ;N ) = an d ¦ S(1 ;1 ;N ) = U sn qu on(18) (proof Lem ma ) w e ob tai i gE ati of nthat ¦ B (1 ;1 ;N ) ·¦ S(1 ;1 ;N ) i đ ắ0 > (N ¡1)¾ B : T henthe res l f l sf the f t that ¾ B = ¡¾ 02 u t olow rom ac W henthere i sperf t i f ati hari g pec u atorsB an Shave the s e ex ted ec norm ons n ,s l d am pec 33 pro¯tsgvenb y i ¦ S(0 ;0 ;N ) = ¦ B (0 ;0 ;N ) = á(0 ;0 )ắ B + : ¸(0 ;0 )(N + )2 (2 3) I f l sthat perf t i f ati t olow ec norm ons n spos i l i hari gi sb e đ (0 ;0 )ắ B + áM axf B (1 ;1 ;N );Ư S(1 ;1 ;N )g: Ư á(0 ;0 )(N + ) (2 ) (N nthi as sn sc e,u i gLem m a ,w e c anrew ri Con i on(2 ) as te d ti Cas 1.ắ 02 + (NĂ1) :I e Ă1) á(0 ;0 )ắ B + áƯ S(1 ;1 ;N ); ¸(0 ;0 )(N + )2 w hi h yi d s(u i gE qu on(2 )), c el sn ati ¸(0 ;0 )ắ B + á (0 ;0 )(N + ) ¸(1 ;1 )(N + 1)2 (2 5) I f l sf the ex sonof¸ (i t olow rom pres i nthe proofofLem m a ) that p N + p ; ¸(0 ;0 ) = (N + ) ¾ B + 9ắ 02 an á(1 ;1 ) i i d sgvenb y equ on(18).U sn thes ex son d the f t that ¾ B = ati ig e pres i san ac ¡¾ 02 ,w e rew ri (af s e aleb ra) E qu on(2 5) as te ter om g ati µ wi th G (N ;ắ 02 )= " ả (N + 1) G (N ;ắ 02 ) á0 ; (N + ) # p N + (N + 1)2 (1 ¡¾ ) (N + 1)(1 + 3¾ 02 ) p + ¡ : N + (N + )(1 + 8¾ ) N (1 + 8¾ 02 ) Noti e that G (N ;: d ec reas i ¾ 02 Fu c ) esw th rtherm ore G i tri tl posti f ¾ 02 = ss c y i ve or (N ¡1) + (N ¡1) (N u an n ati f ¾ 02 = 1.W e c onl d e that there exs c u d eg ve or c u i tsa tođ ắ 0Ô2 (N ) (4 + (N¡1) ;1) s ch ¡1) that Con i on(2 5) i ati ed i ắ 02 à ắ 0Ô2 (N ) T hi u d ti ss s ® sc to® i m plc i y d e¯n asthe si i tl ed s u onof ol ti G (N ;¾ 02 ) = : 34 (2 6) AsG (: ) i c reas ;: n esw i N an d ec reas th d esw i ¾ , w e d ed u e that ¾ Ô(N ) i c reas th c n esw i th N Cas ¾ 02 < e (N ¡1) :I nthi as sn sc e,u i gLem m a + (N ¡1) ,w e c anrew ri Con i on(2 ) as te d ti á(0 ;0 )ắ B + áƯ B (1 ;1 ;N ); ¸ (0 ;0 )(N + ) w hi h yi d s(u i gE qu on(2 1)), c el sn ati 2 á(0 ;0 )ắ B á(1 ;1 )ắ B + ¸ ¸ (0 ;0 )(N + )2 (2 ) U sn the ex son or ¸ (0 ;0 ) an ¸(1 ;1 ), af s e m anpu ati s w e rew ri the ig presi sf d ter om i l on, te previ sc on i onas ou d ti d ef F (N ;¾ 02 ) = à ! p + ) N (1 + 8¾ 02 ) (N (1 ¡¾ 02 ) p ¡(N + 1) ¡1 ·0 : + 8¾ 02 (N + 1)(1 + ) ¾2 W e ob s erve that F (N ;: d ec reas ) esw i ¾ Fu th rtherm ore F > f ¾ 02 = an F < or d f ¾ 02 = or (N ¡1) I t + (N ¡1) (N f l sthat there exs olow i tsa c u tođ ắ 20 (N ) (0 ; + (N¡1) ) s ch that f u or Ă1) ắ 02 áắ 20 (N ),Con i on(2 ) i ati ¯ed T hi u d ti ss s sc to® i m plc i y d e¯n asthe s u onof si i tl ed ol ti F (N ;¾ 02 ) = : AsF (: ) i c reas ;: n esw i N an d ec reas th d esw i ¾ 02 , w e d ed u e that ¾ 20 (N ) i c reas th c n esw i th N Fu rtherm ore w e have < ¾ 20 (N ) < (N ¡1) < ắ 02 Ô(N ) < 1: + (N Ă1) P roof P roposti of i on3 Step 1: P ri esare m ore i f ati w henthere i norm ati c norm ve si f ons hari g ec al n R l that v an p are n ~ d ~ ormaly d i tri u an that p (O ) = + áO T heref l s b ted d ~ ore V ar(~ j~(O ) = p) = ¾ v ¡ v p C ov2 (~; O ) v ~ : ~ V ar(O ) U sn qu on i gE ati s(15) an (16 w hi h appear i d ) c nthe proofofLem m a 1,w e ob tai nthat 35 2 V ar(~ j~(O ) = p) = ¾ v Ăá C ov(~; O ) = ắ v Ă v p v ~ 2 (2 ¾ v (N + 1) + 3N ¾ ")¾ v : 2 (2 (N + )¾ v + 3(N + 1)¾ ") I i mm ed i that V ar(~ j~(O ) = p) i c reas i ¾ " an d oesn d epen on¾ ´ T hi t si ate v p n esw th d ot d s m ean sthat i f ati norm ons n (a d ec reas i " an ¾ ´ ) m ak hari g e n¾ d esequ lb ri m pri esm ore ii u c i f ati norm ve Step : P ri esare l svol l w henthere i norm ati c es ati e si f ons hari g n O bs erve that V ar(~ ¡p) = E(E((~ ¡p)2 j~ = p)): v v p As~ = E(~ j~),the previ sequ i i plesthat p v p ou alty m i V ar(~ ¡p) = E(V ar(~ j~ = p)): v v p F i aly sne v an p are n aly d i tri u ,V ar(~ j~ = p) i ontan s that nl ic ~ d ~ orm l s b ted v p sc s t o V ar(~ ¡p) = V ar(~ j~ = p): v v p Hene pri esare l svol l w henthere i norm ati c c es ati e si f ons n sne pri esare m ore i hari g i c c n f ati i orm ve nthi as sc e P roof P roposti of i on4 Coni er the f l i grati sd olow n o H (N ;¾ 02 ) = ¸(0 ;0 ) : ¸(1 ;1 ) P erf t i f ati ec norm ons n mprovesm ark lqu d i i d ony i hari gi et i i ty fan l f H (N ;¾ 02 ) < 1: U sn i gthe ex sonf ¸ gveni pres i or i nthe proofofLem m a ,w e ob tai n H (N ;¾ 02 p 3(N + 1) (N + 1)(1 + 3¾ 02 ) p )= : (N + ) N (1 + 8¾ 02 ) 36 I i m m ed i that H (N ;: d ec reas i ¾ Fu t si ate ) esw th rtherm ore H (N ;1) < an H (N ;0 ) > d T heref there ex s ore i tsa thres d ¾ (N ) s c h that H < i ¾ 02 > ¾ 02 (N ).T hi hol ạ2 u đ sthres d hol s ves ol H (N ;¾ 02 ) = 1: Sol n vi gthi sequ on e d ed u e that ati ,w c ¾ 02 (N ) = ¹ w here h (N ) = p (N + ) N p + 1) N + 1) (N ¡h (N ) ; 8h (N ) ¡3 < 1.Ash (N ) ¸2 =3 e have ắ 02 < ,w P roof P roposti of i on5 T he ex ted trad i gc os or the lqu d i trad ersw henthere i norm ati pec n tsf i i ty si f ons n hari gare E (C T e ) = Ă2  à  ! 6¾ 02 ¾ B + ¾ ´ + ¾ B + ´ ¾ B ¾2 ¡2 ¢ : ¾ B + ¾´ U sn i gthe ex sonf ¸,w e rew ri thi pres i or te sequ onas ati p ¡ ¡2 ¢ ¡ ¢ ¢ 2 2 ¾ v (4 (N + 1)¾ v + (12 N + 5)¾ v ¾ " + 9N ¾ " ) 02 ¾ B + ¾ ´ + ¾ B + ´ ¾ B ¾ ¾2 e q¡ E (C T ) = ¢£ ¡  à  : Ô 2 + 3(N + 1)¾ ) 2 + ¾2 + ¾2 (2 (N + )¾ v ¾ B + ¾´ ¾B ¾ B + ¾ ´ 6¾ B " ´ W henthe b rok ersd o n s ot hare thei i f ati ,then r norm on ne E (C T ) = = ne E [(P(O ) ¡~) £~]= ¸ v x p 2 ¾ vN (2 ¾ + ¾ B ) £ p 02 : (N + 1) ¾ B + ắ 02 ả 2 ắ + ắB W e d en â the d i ote ®erene b etw een ex ted trad i gc os hen c the pec n tsw there i norm ati si f on s n d w henthere i o i f hari gan sn normati ons n Hene hari g c Ă Â 2 â N ;¾ " ;¾ ´ = E (C T e ) ¡E (C T ne ) 37 Strai htf ard m anpu ati ss g orw i l on how that p p 2 2 ¾ vN ¾ v (4 (N + 1)¾ v + (12 N + 5)¾ v ¾ " + 9N ¾ " ) < 2 (2 (N + )¾ v + 3(N + 1)¾ " ) (N + 1) (2 8) Now c oni er the f l i gfnti sd olow n u c on ¡ 2¡2 ¢ ¡ ¢ ¢ 2 2 ¡ 2¢ ¾ B + ¾ ´ + ¾ B + ´ ¾B ¾ ¾2 (2 ¾ 02 + ¾ B ) ¢£ ¡ ¢ ¡ ¢ Ă Ô : ắ = Ă 2 2 ¾ B + ¾ 02 ¾ B + ¾ ´ ¾ B ¾ B + 9¾ ´ + 36 ¾ B + ¾ ´ ¾2 2 As¾ 02 = ¡¾ B ,w e rew ri the previ sequ onas te ou ati ¡¡2 ¢ ¡ ¢2 ¢ 2 2 Ă 2 ắ B + ắ Ăắ B ¾ B + 3¾ ´ (2 ¡¾ B ) Ă2 ÂÊ Ă Â Ă Â Ă Ô Ã ¾´ = : 2 2 ¡3¾ B ¾ B + ¾ ´ ¾ B + ¾ ´ ¡¾ B ¾ B + ´ ¾2 O bs erve that à (0 ) = an d 2 ¾ B (¡ 11¾ B ¡4 ¾ B ) 7+ < ,sne ¾ B [0 ;1] ic 2 (9 ¡8¾ B )(4 Ă3 B ) ắ2 Ă 2 lm ¾ ´ = : i ¾´ ! an d ¡ ¡ ¢ ¡ ¢ ¡ ¢ ¢ 4 ¡ ¢ ¾ B 17 ¾ B + 14 ¾ B ¾ ´ ¡3 ¡7 ¾ B ¾ ´ 5¾ ´ ¡7 + ¾ B 13¾ ´ ¡88¾ ´ + + 52 ¾ ´ 68 2 ắ = Ă2 ÂÊ Ă Â Ă Â2 Ô 2 ắ + ¾ ¡¾ ¾B + ¾´ B ¾2 ´ B ¾B + ´ (2 ) Ă 2 à 2Â Ê Ô d ore f Now w e rem ark that i B ; 22 , thenà ¾ ´ > an theref à ¾ ´ < I fắ 2 Ê1 Ô 2 ắ B 22 ;1 ,thenthere i u i e val e of¾ ´ s ch that à = T hi sa nqu u u sval e i u s ắ = ạ2 2 ắ B (2 ¾ B ¡2 1) : 3(14 ¡13 B ) ¾2 Hene à hasony on ex c l e tremu an thi tremu i m i i u sne m d sex m sa nm m i c 38 ¡ ¢ 7(14 ¡13 B )4 ¾2 à ắ = > 0; 5ắ B (2 ắ B Ă1) Ă 2 W e d ed u e that ´ an B ,à ¾ ´ < W e c onl d e that c ắ2 d ắ2 cu Ă 2Ă2  à  2 2 ắ B + ¾ ´ + ¾ B + 3¾ ´ ¾ B ¾ (2 ¾ + ¾ B ) q¡ < p 02 : ¢£ ¡  Ă2 Â Ô 2 2 ắ B + ¾ 02 ¾ B + ¾ ´ ¾ B ¾ B + 9¾ ´ + 36 02 ¾ B + ¾ ´ ¾ (3 ) ¡ ¢ 2 U sn n alty(2 8) an I equ i i gI equ i d n alty(3 ),w e d ed u e that â N ;ắ " ;¾ ´ < w hi m ean c ch s that the ex ted trad i gc os pec n tsare al aysl er w henthere i norm ati w ow si f ons n hari g 39 FIGURE 1: Is Perfect Information Sharing Possible? NO σ20*(N) 0,9 YES 0,8 0,7 s0 0,6 0,5 σ20(N) 0,4 NO 0,3 0,2 0,1 10 15 20 25 30 35 Number of Fundamental Speculators 40 45 50 FIGURE 2: Does Perfect Information Sharing Improve Liquidity? σ20*(N) 0,9 YES 0,8 0,7 σ20(N) s0 0,6 0,5 0,4 NO 0,3 0,2 0,1 10 15 20 25 30 35 Number of Fundamental Speculators 40 45 50 ... Perfect Information Sharing Possible? NO σ20*(N) 0,9 YES 0,8 0,7 s0 0,6 0,5 σ20(N) 0,4 NO 0,3 0,2 0,1 10 15 20 25 30 35 Number of Fundamental Speculators 40 45 50 FIGURE 2: Does Perfect Information. .. on s n hari gor 2 2 Ư B f(ắ ;ắ ";N ) + Ư Sf(ắ ;ắ " ;N ) ĂƯ B (1 ;1 ;N ) á0 ; n n nf for ¾´ < and ¾" < O bs erve that thi an c u onyw hen norm ati s n m pai sc oc r l if on hari gi rsm ark d... d d c a nou m od elthey opti aly r m l trad e the s e qu ti am an ty).Conequ tl, on s u ator c and etec t cheati g b y the other s en y e pec l n s u ator b y ob s n sor her trad e sze pec l