Call Auction Markets with Risk-Averse Specialists

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1 Theoretc Economc Letter Puhe Onne My 0 ( C Aucton Mret wth R-Aere Spect Poo Vte Deprtment of Economc Unerty of Pecr Ve Pnro Ity Em: pte@uncht Recee Decemer 8 0; ree Jnury 30 0; ccepte Ferury 0 0 ABSTRACT We tuy generzton of Kye (985) moe to the ce n whch the pect r-ere n oe not et the trncton prce ccorng to em-trong form effcency We ee tht Kye c ucton mret no onger rout mret tructure ner Byen equr o not et rrepecte of funment uch gent nformton enowment n preference Th reut ho oth when cutomer cn umt ony mret orer n when mt orer re owe too Keywor: R-Aeron; C Aucton Mret; Lner Byen Equr Introucton In ucton mret uch the NYSE pect f trncton prce fter cutomer he fe ther orer In Pete Kye formuton of c ucton mret [] thee gent repreent pe gent who et trncton prce ccorng to em-trong effcency conton In th wy t poe to entfy ner equr n tuy the chrctertc of the mret Howeer pect on the NYSE re not mere eecutor of cent orer Ther ehor ncte y eer emprc tue [] cte n nfuence y r mngement conerton In prtcur there eence tht they re r-ere they et trncton prce to contro ther nentore n r epoure We how tht when pect re r-ere the mpe ner equr of Kye ny pper Th ho rrepecte of preference n enowment of mret prtcpnt or of the egree of nformton ymmetry mong the pect n the nforme trer It the conequence of the prtcur protoco of trng mpoe y Kye frmewor pect ct fter ther cutomer To outne the frgty of uch frmewor mportnt gen the mpe ttenton t h recee n the terture Lner Equr wth Informte Mret Orer In Kye ucton mret pect tre wth group of cutomer numerre whch py certn return (normze to 0) for ry et wth n uncertn quton ue whch normy trute wth prmeter n A we coner the ttc formu- ton of Kye moe n pero 0 c ucton run whe n pero the quton ue of the ry et pucy nnounce The protoco of trng goernng the mret correpon to tht of c ucton Thu n pero 0 the pect cutomer pce ther mret orer for the ry et Cutomer compre poputon of quty trer whch coectey pce n unnformte mret orer u n n ner who umt n nformte mret orer By conenton when n u re pote (negte) the pect cent purche (e) the ry et When umtte orer re tche together n pe to the pect Th men tht the pect ony oere n ggregte mret orer u for the ry ecurty On the of the nformton contne n the ggregte mret orer the pect et trncton prce for the r et p t whch nu mret orer re eecute Th prce tht we ncte wth p meure the numer of unt of the r-free et requre to purche one unt of the ry one The oer mret orer of the poputon of quty trer normy trute wth prmeter 0 n u A uu th orer orthogon to other rnom re On the contrry the ner oere noy gn on the quton ue of the ry et efore ny trng te pce Such gn equ to n 0 Th mpe tht the conton quton ue of the ry et gen the ner prte nformton Δ Copyrght 0 ScRe

2 76 P VITALE n Δ In Kye orgn formuton the ner epot her nformton ntge to gn pecute proft Howeer hown y Surhmnym [3] the mret equrum cn e ey chrcterze een when he enowe wth tnr contnt oute r-eron (CARA) utty functon wth coeffcent of oute r eron We ow for oth pecfcton our mn reut ho rrepecte of the ner tttue towr r In Kye orgn formuton the pect eong to poputon of r-neutr mret mer Bertrn competton n the mret mng nutry nuce the pect to re-een n to et the trncton prce for the ry et ccorng to em-trong form effcency We ume nte tht the pect rere n mmze the epecte utty of h fn weth We ume tht h utty functon CARA The pe- wth coeffcent of oute r-eron ct fn weth epen on h enowment of the numerre m n ry et Becue the pect not prce-ter th enowment conton h choce of the trncton prce p The ner trng trtegy X efne her mret orer gen the nformton he poee X The pect prcng rue P functon of the orer fow he oere p P The pyer fn weth w n w functon of thee two trtege w w X P n w w X P Ther utte re functon of ther trtege n ther egree of r eron VW X P n V W X P Snce cutomer cn ony pce mret orer f ome retrcton not mpoe the mmzton proem of the pect unoune Then whe the prtcpton contrnt for the pect EV W X P 0 we ume tht compete- ton n the mret mng nutry re uch epecte ue to zero Gen th pecfcton for the two gent trtege we mofy the equrum concept put forwr y Kye Defnton A Byen (Nh) equrum for the c ucton mret wth the r-ere ner n pect pr of trtege (X P) uch tht: ) The ner mmze her conton epecte utty: X n The pect utty functon unque up to n ffne trnformton o tht the choce of the ower oun n the prtcpton contrnt rtrry EV W X P EV W X P ) The pect et the trncton prce to rech the ower oun for the epecte ue of h utty: P n () EV W X P 0 () Lner Byen equr repect the foowng Defnton Defnton A ner Byen (Nh) equrum pr of trtege (X P) tht tfe Defnton uch tht X ner n the ner gn n P ner n the mret orer the pect oere We re now rey to tte our mn reut Propoton A ner Byen equrum for the c ucton mret wth r-ere pect oe not et Proof Aume tht p K the contnt K epen on the pect nt enowment of the ry et K Aume the ner r-ere n chooe her mret orer mmzng the epecte utty he otn from her fn weth Wthout o of generty ume he oe not poe ny unt of the two et o tht he mmze the epecte utty of her trng proft Thu f her utty functon CARA wth coeffcent of r-eron he oe the foowng proem m π π π n re π the conton men n rnce of her proft π π u B Her optm mret orer u n B A mmum otne for 0 Th the ce f 0 Suppoe then tht the pect et the trncton prce umng tht the mret orer pce y the ner repect Equton (3) wth 0 Hence the epecte men n rnce of h fn weth w re m p n repectey w w ( ) ppyng the projecton theo- (3) Copyrght 0 ScRe

3 P VITALE 77 rem for Norm rnom re we now tht B wth Gen h utty u functon n the contrn n () the pect et p y mpong the foowng conton 0 Wthout o of generty up- w w poe tht m 0 Th jut mpfe our ger ut t nconequent for the ty of our mn reut Then t mmete to ee tht the trncton prce fe y the pect p Suttutng the epreon for n the trncton prce p nce the poteror rnce trcty pote 0 In ton 0 gen the efnton of λ we ee tht 0 0 n 0 Thee nequte mpy tht 0 Snce n o not epen on t not poe tht 0 o tht n the prcng functon there term n the nere of the ggregte mret orer The economc ntuton for the non-etence of ner equr mpe: ecue of h r-eron the pect tre to mnge h nentory of the ry et n hence ceter pru chrge trncton cot whch re not proporton to the ze of h cutomer orer Interetngy our reut ffer from the ny of Surhmnym [3] who coner the ce of rere ner n r-ere pect wth zero enowment of the ry et n how tht ner equrum nee et The proof of Propoton me cer tht no ner equr cn et when the pect poee non-zero enowment of the ry et o tht Propoton how how Surhmnym reut omehow pec Wth CARA utty functon the we nown reut tht the nt enowment (or weth) of n netor oe not nfuence her optm portfoo ho f n ony f uch netor prce-ter Such conton oe ppy to the pect o t not urprng to ee tht fferent concuon re reche when the pect poee or o not poe n nt enowment of the ry et A proem wth th ny of the conequence of r-eron on the prt of the pect tht the umpton tht cutomer cn ony pce mret orer hr to mntn n the current formuton When the ner fce pect who free to et the trncton prce he w e wng to tre ony f he cn conton her orer on the trncton prce Inee n the NYSE pect ccept oth mret n mt orer We now coner the ce n whch cutomer cn umt mt orer we 3 Lner Equr wth Informte Lmt Orer Accorng to th mofe protoco of trng cutomer t pce ther orer for the ry et t the egnnng of pero 0 efore the pect et the trncton prce Howeer whe the quty trer pce coecte mret orer u the ner umt mt orer e emn cheue p Now the pect oere n ggregte emn functon p pu on mt orer oo Conequenty we nee to mofy the trtegy pce n the equrum concept we empoy Uner the new trng protoco trng trtegy for the ner X efne her emn cheue functon of her prte gn ( p) X Smry the prcng rue of the pect P functon of the ggregte emn cheue he oere p P p The ner n the pect t mmze the epecte ue of ther utte Howeer nce the ner cn pce mt orer the pect optmzton proem we efne n t mt mmum We cn then efne proper equrum concept een f there no competton n the mret mng nutry In th ecton we proe tht ner Byen equrum oe not et when the pect free to mmze h epecte utty A fortor the me reut ho wth n upper oun on h epecte utty To ccommote the new trng protoco we mofy the equrum concept foow Defnton 3 When mt orer re owe Byen (Nh) equrum for the c ucton mret wth the r-ere ner n pect pr of trtege X P uch tht the foowng two conton ho: ) The ner mmze her conton epecte utty: X n Copyrght 0 ScRe

4 78 P VITALE E V W X P E V W X P ) The pect mmze her conton epecte utty: n p P p p EV W X P ' p p E V W X P when mt orer re owe ner Byen equr repect the foowng Defnton Defnton 4 When mt orer re owe ner Byen (Nh) equrum pr of trtege X P tht tfe Defnton 3 uch tht X ner n the prte gn of the ner n P ner n the ggregte emn cheue the pect oere p We re now rey to confrm our mn reut Propoton When the ner cn pce mt orer ner Byen equrum for the c ucton mret wth r-ere pect oe not et Proof Suppoe tht the pect fe the trncton prce umng tht the ner emn cheue n ne wth Defnton 4 ner n the ujecte mprcng of the ry et p p (4) (5) (6) ome contnt Then the conton men n rnce of the pect fn weth w re n w ( p) ( p) m p p p p w p p ppyng the projecton theorem for Norm rnom re we now tht u ( p) n ( p) wth p p u The pect fe p ong the foowng progrm: p rgm w p w p It foow tht p K p p n For pote the econ orer conton of the mmzton progrm of the pect tfe K p p p Snce p p the trncton prce cn e wrtten contenty wth Defnton 4 ner functon of the ggregte emn cheue p p K p p (7) p n p To chec f the emn cheue ntrouce n Equton (6) optm gen the preference of the ner coner tht the conton men n rnce of her π proft re π pk p π n Gen her zero enowment n her CARA utty functon he chooe her emn cheue ong the foowng progrm: p prgm π π Conerng tht the ner cn conton on the trncton prce n hence on the rezton of we fn tht p pk pu p Howeer gen the prcng rue (7) the ner emn cheue cn e epcty wrtten contenty wth Defnton 4 n Equton () p To he mmum the econ orer conton p 0 mut e tfe To chec tht t ho epct outon for n p mut e otne They oe the foowng non-ner ytem p ( p) p Suttutng nto p the reutng epreon nto tht for n rerrngng t turn out tht mut e root of the foowng equton 4 u 0 Th mght he ether one root or three root Bee 0 there mght e two etr negte root Anyhow t mut e tht p negte p n pote Hence n ce p negte o tht we he mnmum rther thn Copyrght 0 ScRe

5 P VITALE 79 mmum In ton for negte the econ orer conton of the pect progrm ote uggetng tht the ner fn optm to etze the mret Thu ner Byen equrum cnnot et! Propoton remncent of Kye [4] (Theorem 5) who how tht ner equrum oe not et n mret wth one nforme n one unnforme netor In h formuton thee two r-ere gent ct noncompettey n mutneouy umt emn cheue whch re cere n equrum In the current pecfcton nte the pect ct fter the ner to f trncton prce n cer the mret In prctce Propoton mproe on Kye ny n tht t how how h reut rout to the orer of pyer moe Wheneer the pect not force to et the trncton prce ccorng to em-trong form effcency Kye ner equrum re own Th concuon ho whteer the ner preference n enowment n whteer the quty of her gn Our ny ncte tht the mret tructure n Kye moe not rout n tht t t protoco of trng rther thn funment uch the octon of nformton n et or the preference of the gent tht me the mret unte 4 Acnowegement I wh to thn the Etor n n nonymou referee for ther comment n uggeton I o thn Luc Pnccone n prtcpnt t emnr t the Unerty of Tor Vergt n Lu Unerty REFERENCES [] A S Kye Contnuou Aucton n Iner Trng Econometrc Vo 53 No pp o:0307/930 [] J Hrouc Meurng the Informton Content of Stoc Tre Journ of Fnnce Vo 46 No 99 pp [3] A Surhmnym R Aeron Mret Lquty n Prce Effcency Reew of Fnnc Stue Vo 4 No 3 99 pp o:0093/rf/4347 [4] A S Kye Informe Specuton wth Imperfect Competton Reew of Economc Stue Vo 56 No pp o:0307/9755 Copyrght 0 ScRe