Transparency and liquidity in securities markets*
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- Αμφιτρίτη Καλογιάννης
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1 Trasarcy ad lqudty scurts marts Taash U Isttut for Motary ad Ecoomc Studs Ba of Jaa (E-mal: taashu@bojorj Abstract Ths ar rods a framwor whch dals wth arous tys of trasarcy cocr th comosto of ordr flow Us ths framwor w study th rlatosh btw trasarcy ad rc olatlty as a masur of lqudty W dr codtos udr whch cras trasarcy rducs rc olatlty dmostrat that crasd trasarcy dos ot always mly lss olatlty Th ws xrssd hr ar thos of th author Thy do ot rflct thos of th Ba of Jaa
2 Itroducto Mart trasarcy s dfd as th ablty of mart artcats to obsr th formato o th trad rocss by O Hara (995 Trasarcy has may dmsos bcaus a mart has may ds of artcats ad may tys of formato I ths ar w focus o a scal ty of trasarcy that cocrs th comosto of ordr flow scally lqudty-motatd ordr flow W should ot that t stll has multl dmsos Madhaa (996 cosdrd a mart whch all tradrs obsr th tr lqudty-motatd ordr flow It s trasart o ss Röll (99 cosdrd a mart whch bror-dalrs trad basd o rat formato rard ordr flow by thr lqudty-motatd customrs It s also trasart aothr ss It should b otd that wh w cosdr trasarcy cocr th comosto of ordr flow w must ay attto to how much of what art of ad by whom th ordr flow s obsrd Ths faturs dstctos ha rarly b thortcally dscussd th ltratur Extd th modls of Kyl (989 Röll (99 ad Madhaa (996 ths ar rods a modl whch a art of ordr flow s dsclosd to th ublc art of t s obsrd by a art of tradrs ad art of t s ot obsrd by ayo Us th modl w study th rlatosh btw trasarcy ad rc olatlty as a masur of lqudty Mor rcsly w stat th otmal ll of trasarcy that mmss rc olatlty W cosdr two tys of trasarcy O s trasarcy for ublc formato Ths cocrs th stuato whch all tradrs commoly obsr th sam ordr flow whch s smlar to that Madhaa (996 Th othr s trasarcy for rat formato Ths cocrs th stuato whch dffrt tradrs obsr dffrt ad ddt ordr flow whch s smlar to that Röll (99 Accord to th ma rsults w ow th follow I th cas of trasarcy for ublc formato wh th arac of ordr flow s lar ouh ad th mart s ot trasart cras trasarcy rducs rc olatlty Too much trasarcy howr crass rc olatlty I th cas of trasarcy for rat formato wh th arac of ordr flow s lar ouh cras trasarcy rducs rc olatlty ad th most trasart marts joy th last rc olatlty Ths ar s orasd as follows Scto troducs th modl Scto shows th xstc of lar symmtrc qulbra of th modl whch w coctrat o Scto 4 studs trasarcy for ublc formato Scto 5 studs trasarcy for rat formato Scto 6 cocluds th ar A modl I our modl a sl rsy asst s tradd a aucto mart Th lqudato alu of th asst s dotd by whch s a radom arabl ormally dstrbutd wth ma ad arac Th ralsd alu of s doatd by I th rmadr of th ar a arabl wth a tld dots a radom arabl ad that wthout a tld dots ts ralsd alu Prc olatlty susts th dr of mart trasarcy at last drctly thouh t may ot cssarly b a drct masur of mart lqudty Madhaa (996 assumd that all ordr flow s obsrd by tradrs Our modl oly allows for art of t to b obsrd W ar trstd th amout of ordr flow that mmss rc olatlty
3 Thr ar formd tradrs ach of whom s dxd by Tradr rcs a rat formato sal cocr whch s a radom arabl ε ε ε ar ddtly dtcally ad ormally dstrbutd wth ma ad arac os tradrs ordr arat dotd by s ormally dstrbutd wth ma ad arac W assum that thr ar radom arabls such that whr s ddtly ad ormally dstrbutd wth ma ad arac Thus W also assum that for s a art of whch s ublcly obsrd by ryo For for s a art of whch s oly by tradr artly obsrd s a art of whch s ot obsrd by ayo Thus t ca b trrtd that larr / ad / or smallr / mly mor trasarcy / cocrs trasarcy for ublc formato / cocrs trasarcy for rat formato Th modls of Kyl (989 Madhaa (996 ad Röll (99 ca b cosdrd as scal cass of th abo modl th follow ss Kyl (989 studd marts wth / ad / Madhaa (996 studd marts wth / ad / Röll (99 studd marts wth / / ad / W stat th otmal dr of trasarcy by cosdr trmdat cass ad coduct comarat statcs of rc olatlty wth rsct to or for Aftr rc sals ad tradr crats a dmad schdul X ( ; Th ctor of dmad schduls ar dotd by X ( X X Th mart clar rc s dtrmd by th follow quotato X ( To mhass th ddc o X w dot th mart clar rc ad th quatty tradd by tradr as ( X ad x x ( X rsctly Each formd tradr has xotal utlty wth rs arso coffct Each formd tradr has a o-stochastc tal dowmt whch s ormalsd to ro Thus th utlty fucto of tradr ca b wrtt as u ( π x( π whr π ( x ad x s th quatty tradd
4 Dfto Th Baysa-ash qulbrum of th am s a ctor of strats X such that Eu υ X x X Eu υ X X x X X ( for ay dmad schdul X ad Symmtrc lar qulbra Follow Kyl (989 w focus o a symmtrc lar qulbrum whch lar fucto of ad for W wrt th qulbrum straty ( ; X X s a dtcal X as ( Thorm Thr xsts a symmtrc lar qulbrum f > Th aramtrs ( ar dtrmd by th follow quatos: ( ϕ ( whr ( ϕ ( ( ϕ ϕ ( ( ( ϕ ( ( ϕ ( ( ( ( (4 (5 Th roof whch s basd uo th tchqu dlod by Kyl (989 s rodd th adx I th rmadr of ths ar w assum that Ths mls that formd tradrs ha a ror dstrbuto for that s last format Ths assumto smlfs our aalyss du to th follow lmma I Baysa statstcs t s oft sustd to us th last format rors
5 Lmma If th W cosdr th rlatosh btw trasarcy ad a arac of th mart clar rc codtoal o whch s by th follow lmma Lmma If th ( [( ( ] V 4 Trasarcy for ublc sals Ths scto studs th cas whr / ad / For s (almost always qual to ad coys o formato I othr words ry tradr rcs a ublc sal but has o rat formato cocr Thorm Suos s lar For > ad ( ( I addto thr xsts thr xsts uqu ( [ such that Th roof s rodd th adx ( Suos > f such that > I ths cas cras trasarcy > ad Icras mor tha ( ( rods th otmal ll of th trasarcy ot that ( f rducs rc olatlty as far as howr crass rc olatlty Thus < Suos I ths cas cras trasarcy always crass th rc olatlty 5 Trasarcy for rat sals Ths scto xams th cass whr / ad / s (almost always qual to ad coys o formato I othr words tradr ows hs rat formato but dos ot ha ay ublc formato cocr 4
6 Thorm Suos s lar For > ad ( / ( I addto thr xst thr xsts uqu ( [ ] ( < / ad < f < Th roof s rodd th adx such that / such that ( < ad f f / Suos I ths cas cras trasarcy always rducs rc olatlty Suos < < I ths cas cras trasarcy rducs rc olatlty as far as ( Icras mor tha ( rods th otmal ll of trasarcy ot that ( howr crass rc olatlty Thus < / Suos I ths cas cras trasarcy always crass rc olatlty 6 Coclud rmars Ths ar has rodd a framwor whch ca dal wth arous tys of trasarcy cocr th comosto of ordr flow Us ths framwor w study th rlatosh btw trasarcy ad rc olatlty W dr som codtos udr whch cras trasarcy rducs rc olatlty dmostrat that crasd trasarcy dos ot always mly lss rc olatlty Possbl tocs for futur rsarch would b to obta that mmss rc olatlty wthout ay rstrctos as thos our thorms ad to study th rlatosh btw ad othr masurs of mart lqudty or wlfar of tradrs 5
7 Adx Proof of Thorm Lt χ b th qulbrum straty whr χ for For th mart clar rc w wrt x χ ( ; Th x Sol for λ x (6 whr ( λ λ ( ot that x s th otmal amout of trad codtoal o ad Thus x maxmss E [ u (( x ] Howr du to (6 ( x uquly dtrms Also ( x uquly dtrms Ths mls that x s th otmal amout of trad codtoal o ad Thus x maxmss E u ( x [ u ( x ] Wh u s xotal utlty wth rs arso coffct t s ow that maxms E s qualt to maxms [( x ] V ( [ x ] E Rwrt (7 E [ ] ( x λx x V[ ] x (7 (8 Th frst ordr codto for maxmsato of (8 wth rsct to x s [ ] E λx x whr / V [ ] ( < λ Th scod ordr codto s (9 6
8 λ x ad E [ ] E[ ] [ ] λ x x Bcaus E Sol ths for x x E [ ] [ E ] λ (9 s rwrtt as ( ot that all th radom arabls o ar jotly ormal Thus s a costat ad E [ ] s lar wth rsct to o Ths mls that χ ( ; s fact a lar fucto Th xt st s to dr I ordr to do so w calculat E [ ] ad / V [ ] χ ; for assum that ( Th mart clar codto s Thus ( ( ( ( ( [( ] ( ( ( [( ] ( Lt µ b th lft had sd of ths quato ot th follow: ; µ ( s statstcally qualt to ( ( s ddt of ad of ( µ Ths mls that E [ ] E [ ] ( E [( µ ] ( µ µ s ormally dstrbutd wth ma ctor ( 7
9 8 ad coarac matrx V µ whr Ths drctly rod E ad Th rsult of calculato s: ϕ ( whr ϕ ad E µ ϕ ϕ ϕ ϕ ( Plu ( ad ( to ( w ha E x ϕ ϕ ϕ ϕ ϕ ϕ
10 9 I a symmtrc qulbrum x Thus w ha ϕ ( ϕ (4 ϕ (5 ϕ (6 Lt Du to ( ad (6 ϕ Thus f (7 ( f has a soluto bcaus ( < f ad ( > f If ( f has multl solutos lt b th larst o satsfy > f Th bcaus f f < Du to ( ad (4 f (8
11 f has a uqu soluto ( bcaus ( f < > f ad > f for ay > Du to Browr s fxd ot thorm th ma has a fxd ot Ths fxd ot rods th alus of ad whch th dtrms by (4 Du to ( ad (5 (9 Ths ad th alus of ad dtrm Proof of Lmma Plu ϕ ϕ to ( ad (4 w ca show that th rht had sd of ( ad (4 ar th sam Proof of Lmma Du to th mart clar codto ad Thus V Proof of Thorm Wh ad
12 ot that s dtrmd oly by f whch s ddt of Rwrt ths: It s strahtforward to s that > for ay W also ow that < Ths lads to / Thus th s for s th oost of th s for W aluat stad of Hr w troduc w arabls ω ad t such that ω ad t Th ths lads to th calculato rsult: B A / whr A ( [ ω t t 4 ( ] 4 4 ω ω ω t t t
13 ad B ( ( ( ( ω ( 6 ω B > bcaus ω < Thus th s for Cosdr ξ ( ω t Bcaus Α 4 ( ( t ( tω ( t ( t < ξ ξ s th sam as that of Α 4 4 ( tω ( t t ω ( ω ( t 6( ( 4( ( ( 4 t > ( t ξ ω ( t 4 ξ 4 ω 6 ( t > 4 ( < Thr xsts uqu ω ( t ( O such that ξ ( ω t > f ( ω ( t ξ ( ω t < f ω ( ω ( t Lt t ( ω Th w ow th follow: If t t ω ( t ω ( ω ( ( ω > ( ( t ω < t ( ω t or < ( ω ( > or ω ( If t ( Wh ad thus ω ξ ( ω t ( ω η( ω ω ω t th t th tas th larst alu ad t t ( ω ω ad I ths cas
14 whr η ( ω 4 ( ( ω ( 4 ω 4 ω ω Bcaus ( 4 < η ( 8( ( ( 4 > η ( ( ( > η ( ( ( ( 8 ( 4 < η thr xsts ω such that η ( ω > ad ξ( ω t ( ω > for ay ( ω η( ω < ad ξ( ω t ( ω < for ay ω ( ω Ths mls that ( ω ω ad t Proof of Thorm Wh ( ( ( ( [ ] Sol for Th ( ( ( ( ( < bcaus > ad < Ths mls that th s for bcaus s th oost of th s for / W aluat stad of
15 ot that [( ] [( ] / Ths lads to th calculato rsult: [( ] A / B whr t tω ( 4t 9 ( 4tω ω Α ω ad ( ω ( ω ( ω B B > Th s of Cosdr ξ ( ωt Α s th sam as that of Α ω ω ( ω t ( ω t t 4 tω 6 tω 4 9 4t ω 6 t ω ω ξ Bcaus ( ω t ξ t < ξ t > > thr xsts uqu ( t ( ω ω t < ξ ( ω t > f ω ( ω ( t ad ξ f ( ω ( t ot that ( ω ( t t ξ( ω ( t t ω ( t ξ( ω ( t t dξ dt Thus ω t ω t ( t ξ ( ω ( t t ξ( ω ( t t t A sml calculato shows that / ω ξ ( ω ( t t t t ω ξ < ad ( ω ( t t ω ω such that > Thus ω t ( t > 4
16 Th w ow th follow: If t t ( ω ( t If t > t ( ω ( t ad th < or < ( ω ( t t th ( ( ω ( ( ω ( < t ( ω ( ( ( ω ( / < If t > t ( ω ( t ad t ( ω ( ( ω ( th ( ot that / ( Thus W ca calculat that ω ( Thrfor ( ω ( ω ( 4 ad ( ( ω 4 Rwrt ths shows that ( ( ω 4 t ω t t Wh ( ω ω tas ts larst alu ad t t ( ω ( ω t ( ω ω ξ η ( ω( ( ω I ths cas 5
17 whr η ( ω ( 8 4 ω ( 4 ω ( 6 9 ω ( 5 6 ω 4 ( ω 5 6 η Ths mls that thr xsts A sml calculato shows that ( < η( > η( ω > ω such that η ( ω > ad ξ( ω t ( ω > for ay ω ( ω ad ( ω < ξ( ω t ( ω < for ay ω ( ω Ths mls that t ( ω 4( Wh / ω tas ts smallst alu ad t t ( ω I ths cas ( ω ω ω ( ω t ( ω η ( ω ξ whr η 4 ( ω 6 ( 8 9 ω ( 4 ω ( 5 ω ω η ad " A sml calculato shows that ( < η( > η( ω < ω such that η ( ω > ad ξ( ω t ( ω > for ay ω ( ω ad ( ω < ( ω ( ω < ω ω Ths mls that ( ω ξ t for ay η Ths mls that thr xsts t η ad 6
18 Rfrcs Kyl A (989: Iformd Sculato wth Imrfct comtto Rw of Ecoomc Studs Madhaa A (996: Scurty Prcs ad Trasarcy Joural of Facal Itrmdato O Hara M (995: Mart Mcrostructur Thory Cambrd MA: Blacwll Röll A (99: Dual-Caacty Trad ad th Qualty of th Mart Joural of Facal Itrmdato 5-4 7
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