1 MARKET MAKERS S UPPLY AND PRICING OF FINANCIAL MARKET LIQUIDITY Pu Shen and Ro M. Sarr NOVEMBER 000 RWP Reearch Diiion Federa Reere Bank of Kana Ciy Pu Shen i an economi a he Federa Reere Bank of Kana Ciy. Ro M. Sarr i a profeor of economic a he Unieriy of Caifornia, San Diego. The iew expreed are hoe of he auhor and no neceariy hoe of he Federa Reere Bank of Kana Ciy or he Federa Reere Syem. Shen e-mai: Sarr e-mai:
2 Abrac Thi udy mode he bid-ak pread in financia marke a a funcion of ae price ariabiiy and order fow. The marke-maker i characerized a paiey acceping order o buy and o e a ecuriy a he marke' preaiing price (pu or minu haf he bid-ak pread). The bid-ak pread adju o coer marke-maker' aerage co. The bid-ak pread hen arie poiiey wih: he ecuriy' price oaiiy, he oaiiy of order fow, and he aboue aue of he marke-maker' ne inenory poiion. Each of hee ariabe increae aerage co and hence i priced in he bid-ak pread. Thu marke iquidiy (arying inerey wih he bid-ak pread) decine wih increaing price and oume oaiiy and wih increaing ize of marke-maker ne inenory poiion. The mode hence proide a paricuary impe expanaion for decining marke iquidiy during period of arge price moemen and rading imbaance ha increae he ize of marke-maker' ne inenory. JEL Caificaion: G.
3 . Inroducion: Marke Maker and Marke Liquidiy Liquidiy in a financia marke --- he abiiy o aborb moohy he fow of buying and eing order --- come from wo principa ource: from he deph of buyer and eer in he marke and from he acion of marke maker (peciai on he organized exchange). The marke maker' buine i acing a inermediarie: buying from pubic eer and eing o pubic buyer. The marke maker proide marke iquidiy --- immediacy erice --- and make hi profi from he difference beween buying and eing (bid and ak) price. One impe meaure of he iquidiy of a financia marke a any momen i he bid/ak pread. Liquid marke wi be characerized by a narrow pread, iiquid marke by a wide pread. Thi decripion appie boh cro ecionay and oer ime. Reduced iquidiy i ignaed by a broadening of he bid/ak pread. In acua marke, he behaior of he bid/ak pread arie wih marke condiion. Increaingy urbuen marke, uch a hoe around Ocober 9, 987, ae Augu hrough Ocober 998, and Apri 0-4, 000, are accompanied by ignifican expanion of he bid/ak pread. In hi paper we deeop a impe mode of marke maker pricing ha gie an eemenary expanaion of he ariaion in marke iquidiy wih ariaion in marke condiion. I derie he marke maker' bid/ak pread a he reu of he marke maker' opimizing behaior. The bid/ak pread i he price of he marke-maker' erice. The marke maker i characerized a a monopoiic compeior, a profi maximizer ubjec o an (aerage) zero-profi condiion due o he hrea of enry. The marke maker borrow capia on a perfec (or ighy imperfec) capia marke; he face conan (or increaing) margina inere rae a hi deb poiion increae. The principa reu of hi mode hen are ha he bid/ak pread i an increaing funcion of wo ariabe: he ize of he marke maker' ne ong or hor poiion and (in he cae of increaing margina co) marke price oaiiy. Boh of hee ariabe increae he marke maker' aerage co of making a marke. Hence hey are priced in he bid/ak pread. A diincie eemen of he reamen preened here i he impiciy of he marke maker' probem. He ac primariy a a reeer of a good ubjec o he rik of price ariabiiy. We purpoefuy ignore he iue of differenia informaion (informed eru uninformed rader) no becaue we regard hem a unimporan bu raher ha hey may be 3
4 independen of he oume and oaiiy conideraion (paricuary for he marke a a whoe) ha he preen udy emphaize. We uppoe he marke maker o be rik neura; he need no aiude oward rik o deermine a bid/ak pread. We conider wo cae, fir where he capia marke i defau rik aere and econd where he capia marke i defau rik neura. In boh cae, he bigger he inenory of he marke maker, hen he higher wi be he aerage capia co, and he wider wi be he bid/ak pread. In he econd cae, where he capia marke i rik aere, he marke maker face an increaing inere rae o finance inenory a hi expoure o marke rik increae. Therefore, increaing price rik ao increae he marke maker' aerage co, and ha co i priced in he bid/ak pread. The mode hence proide a paricuary impe expanaion for decining marke iquidiy during period of arge price moemen and rading imbaance (eiher for he marke a a whoe or for a inge ecuriy). Thu, for exampe, he Wa Sree Journa of Apri 7, 000 repored "by he end of he day [Apri 4, 000], i eemed a if he ony buyer were NASDAQ marke maker and Big Board foor peciai, who hae o buy wih heir own capia when no oher buyer can be found. Afer he NASDAQ unprecedened Monday-hrough-Friday oing reak, marke maker are hoding hree ime a much ock a uua...he peciai communiy i reched." In he eing depiced in hi commen, he marke maker ha ypicay accumuaed a arge ne poiion in he ecuriy he peciaize in; he marke maker buy when he pubic e. If he price decine coninue, boh curren oaiiy and he marke maker' ne poiion wi hae increaed ignificany from heir pre-crah ee. Thi ead o increaed aerage co which are hen priced in he bid/ak pread. There ha been a arge heoreica (and empirica) ieraure on he deerminan of bid-ak pread. The ieraure argey agree ha he price rik of he underying ae and he effec of aymmeric informaion i imporan in deermining he percenage bid-ak pread (for exampe, Copeand and Gaai, 983; Goen and Migrom, 985; and Eaey and O Hara, 987). The ieraure, howeer, differ a he roe of inenory (Madhaan 000). In many udie where inenory wa one of he conidered ariabe, i ended up a a non-facor in deermining he percenage bid/ak pread (for exampe, So, 978; Ho and So, 98). Garman (976), Amihud and Mendeon (980), and O Hara and Odfied (986) are a few excepion. Boh Garman mode and Amihud and Mendeon mode 4
5 rea he inenory poiion of he marke maker a foowing a birh-and-deah proce wih conan ae price, and marke maker e heir bid/ak price o minimize he poibiiy ha heir inenory poiion wi reach zero or infiniy. Conequeny, he bid and ak price depend on he inenory poiion of he marke maker, and he percenage bid/ak pread increae a he gap beween he acua and preferred poiion grow. The O Hara and Odfied mode, on he oher hand, rea he marke maker a an ineor who maximize hi uiiy, which i a funcion of boh he profi of making he marke and he aue of he inenory. Thi eay differ from hee paper a our marke maker neiher arge a deired inenory poiion nor receie uiiy from he aue of heir inenorie. The marke maker in our mode can away iquidae heir ong poiion a he marke bid price (or coe ou heir hor poiion a he marke ak price). Inenory ener heir choice of bid/ak pread mainy becaue he co of carrying he inenory need o be coered by he reenue from making he marke under he equiibrium condiion of expeced zero profi. I i common in he ieraure o characerize he marke maker a eing a bid and an ak price in ee erm. The reamen here impifie he bid/ak pread deciion o chooing a percenage of he preaiing ae price.. Modeing he Marke Maker We conider he marke maker for a inge ecuriy whoe price a dae i denoed P. The eouion of price oer ime i exogenou. The marke maker e a ymmeric proporiona pread S a dae repreening hi price markup for he ak price and he markdown for he bid price. A dae, he face a (ong, buying) demand oume V and a (hor, eing) uppy oume V. The marke preen he marke-maker wih demand for ecuriie V, ha he marke-maker proide. The marke preen he marke maker wih uppie of he ecuriy V ha he marke maker purchae. The marke maker' poiion (ne hoding) of he ecuriy may be poiie or negaie. A he ar of dae, we denoe he poiion a N. The marke maker' poiion in he ecuriy eoe oer ime. A +, By focuing on boh bid and ak price, he udy rie o oe wo probem in one ho: one i he deerminaion of ecuriy price, and he oher i he deerminaion of he price of he marke maker erice he bid/ak pread. Our mode, by uing he percenage bid/ak pread, eparae he wo iue and focue ony on he aer, which aow u o impify he mode greay and hu gain uefu inigh on he roe of inenory. 5
6 hi poiion i hi poiion a increaed by hi purchae a and reduced by hi ae. Tha i, N N V V. + = + The marke maker paiey accep a order o buy and e. Hi ony raegic acion i o adju he pread S. The marke maker ar period wih a cah poiion M, carried oer from he preiou period. In conducing buine a dae, he marke maker incur co C. We denoe he marke maker' ne ae aue poiion a he ar of dae a Π. Noe ha Π hen repreen a ock ariabe, he inenory auaion. Π doe no repreen a profi fow. The aue of he marke maker poiion a he beginning of hen i Π = PN + M. A +, he aue of he poiion i Π = P N + M The marke maker' cah poiion eoe hen a M = ( + S ) PV ( S ) PV + M C. + + There are wo componen in C + he co of proiding marke-making erice. One i mainy he direc co of rading, uch a order aking, order execuion, and record keeping. The oher componen i he co aociaed wih haing o carry inenorie of he ecuriy in which he marke maker i making he marke. For impiciy, he fir componen i ignored here. Tha i, ordinary margina co of record keeping and execuion are aken a conan and do no repreen a ource of ariaion in co. Since we are inereed in ariaion in pricing he bid/ak pread and ariaion in he marke maker' aerage co, we ignore hi conan co porion. Conceiaby, he co funcion can differ depending on wheher he marke maker carrie a ong or hor poiion hough we do no rea hi poibiiy direcy beow. 3. Marke Maker' Pricing: A Zero-Profi Condiion Wihou modeing expiciy he marke rucure of marke making aciiie, we aume a zero-profi condiion. In he cae of NASDAQ, hi repreen he oucome of eae of enry ino marke making aciiie. On he NYSE, hi may be aken o repreen 6
7 he noion of he co of mainaining an ordery marke --- or a normaie ceiing (no neceariy zero) on peciai profi. The zero-profi condiion impie 3 E [ Π ] = E[ M + P N ] = Π = M + PN On aerage, he marke maker aume no ne poiion on he ecuriie in which he i making a marke. A an equiibrium condiion hen, expeced ae and purchae are on aerage equa. Aume he expeced oume of he buy and e order of he ecuriy are he ame, and denoe i a V 0. Then, E [ M ] = ( + S ) PV ( S ) PV + M E [ C ] = S PV + M E [ C ] Now aume ha he diribuion of price and oume a + are no correaed. Furher, we aume a maringae condiion, ha he expeced mean of he ecuriy price a + i equa o he price a. Then E [ P N ] = E [ P ] E [ N ] = P( N + V V ) = PN Therefore, he zero profi condiion impie E [ M + P N ] = S PV + M E[ C ] + PN = M + PN, or E[ C+ ] = SPV 0. In oher word, a he marke equiibrium, he marke maker wi e he bid-ak pread o ha S E[ C ] =. * + PV 0 Thi expreion i he cornerone of hi ine of reearch. The marke maker adju he bid/ak pread a any momen o coer expeced (ariabe aerage) co a expeced rading oume. The marke maker purue expeced aerage co pricing in a ariabe ochaic enironmen. 4. The Quadraic Co Cae and he Aboue Vaue Co Cae In order o derie a predicion from he pricing mode aboe, we mu pecify he form of he co funcion C +. How doe he ize of he marke maker' ne poiion, in he 3 Thi expreion coud be preened incuding a ime dicoun facor, wihou changing he characer of he reu. 7
8 ecuriy in which he make a marke, affec hi aerage co? Hoding he rading inenory N + require financing, and he aerage co of capia may ary wih he ize of he marke maker' poiion. 4 A rik premium may be added o inere rae on ending o a marke maker whoe poiion i increaingy eeraged. Thu we ugge he pecificaion: Cae : Quadraic Co. C + = a( P+ N+ ). Then + = = + p = ap ( + p)( N + + ) E[ C ] ae[(( P P ) N )] a( P ) E[( N V V )] Thu, a S = ( P + )( N + + V ). * p 0 PV 0 Hence, in he quadraic co cae, we find ha he marke maker' bid/ak pread arie poiiey wih price rik, p, wih rading oume rik,, and he ize of he marke maker' rading inenory expoure N. A marke become more oaie in price or oume, he bid-ak pread expand. A he marke maker' expoure --- embodied in he (quared) ize of hi inenory --- expand, o doe he bid-ak pread. Cae : Aboue aue co. C+ = a P+ N+. Then, E[ C ] = ae[ P ] E[ N ] = ape[ N ] Howeer, now we need o know he diribuion funcion of buy and e oume o cacuae he mean of N + ince E[ N ] = E[ N V + V ] = ( N V + V ) d d ( N V + V ) d d N V+ + V+ 0 N V+ + V+ < 0 under he condiion ha buy and e oume are independen and idenicay diribued. Furher, aume hey boh foow uniform diribuion. Tha i, 4 The uni ime inera i mo appropriaey conceied a a inge rading day. A he coe of rade, financing co are incurred on he marke maker ne poiion. The marke maker adju he pread during he rading day o aemp o opimize hi abiiy o coer co on he coing inenory poiion. 8
9 f() = in he range of +, and zero oherwie. 5 Then he reaionhip beween E[ N + ] and N i noninear when he aue of N i reaiey ma. 6 Howeer, for arge enough N, (when N + ), i i proed in Appendix ha E[ N+ ] = N. 7 Tha i, * a S = N. V 0 In he aboue aue aerage co cae we find ha he bid-ak pread arie direcy wih he exen of he marke-maker rading inenory expoure o marke rik. The bigger he marke-maker' poiion (in aboue aue), he bigger he pread Concuion A marke maker' expeced aerage co increae wih he ize (aboue aue) of hi inenory poiion; he co of financing inenory increae wih i ize. Furher, in he cae where he marke maker face increaing margina financing co, expeced aerage co increae wih ecuriy price oaiiy. Thee co, in equiibrium, mu be coered ou of he bid/ak pread. Conequeny, he bid/ak pread arie direcy wih ecuriy price oaiiy and wih he ize of he marke maker' rading inenory poiion. Since an increaingy urbuen ae marke i characerized by imbaance of rading demand and uppy and by increaing price oaiiy, hi mode ugge ha (and proide an expanaion why) uch a marke i ikey o be accompanied by a deerioraion in marke iquidiy. 5 The mean of he diribuion i V 0 and he ariance i 3. 6 See Appendix for exampe. 7 Our eing hu ead o he concuion ha when he iniia inenory poiion i arge, he ime erie of inenory poiion ha a uni roo, which may or may no be conien wih he empirica eidence (Habrouck and Sofiano, 993). Howeer, hi reu houd no be aken ieray, a he focu of hi paper i no he aiica propery of he inenory and hu, he mode i highy yized. 8 Aowing he correaion beween oume and price diribuion i ikey o renghen he poiie reaion beween he inenory poiion and bid/ak pread a a udden urge of inenory houd igna negaie price change are more ikey, which make he exiing inenory een more coy o he marke maker. 9
10 Appendix : The Proof of Cae. In hi appendix, we proe ha in ecion 4, cae, E[ N+ ] = N for ufficieny arge iniia inenory poiion, i.e., N +. The zero-profi condiion impie: S E[ C ] =. * + PV 0 If he co increae wih he aboue aue of he inenory, i.e., C+ = a P+ N+, hen under he aumpion ha price and ranacion oume are independeny diribued, E[ C ] = ae[ P ] E[ N ] = ape[ N ] a price i away poiie. Now aume ha buy and e oume are independen and idenicay diribued, hen E[ N ] = E[ N V + V ] = ( N + ) d d ( N + ) d d N N < 0 Furher aume hey boh foow uniform diribuion: f() = in he range of +, and zero oherwie. A he oume canno be negaie, i i neceary ha. Then (uppre he ubcrip in + and + ) E[ N ] + = d ( N + ) d + 4 N+ 0, + N+, + N+ < 0, + 4 N+ 0, + > N+, + N + < 0, + d 4 (0) d d ( N + ) d d ( N ) d. 0
11 Tha i, E [ N ] = d ( N + ) d + 4 N+ 0, + N+, + d ( N + ) d 4 N+ 0, + > N+, + N+ < 0, + + d ( N + 4 ) d. Thi inegra wi be referred a he baic inegra in he re of he appendix. The baic inegra i difficu o eauae a i depend on he aue of iniia inenory poiion, N, nonineary. Howeer, for arge enough N, (when N ( + ) 0or N V + ), he reaionhip become inear, a i can be proen ha 0 E[ N+ ] = N. Range : N +. A N i poiie, hi i he cae ha he iniia inenory poiion i ong on he ecuriy. Wih uch a arge poiie inenory, N + i away aified, hu he econd and hird erm anih in he baic inegra a neiher condiion N + > nor N + <0 are aified. Therefore, E [ N ] = d ( N + ) d + 4 N+ 0, + N+, ( = V + ) = N + d 4 = + = ( N + + ) ( N + ) d = ( N + 4 ) ( N) ( N) ( N ) + = 3 N ( ) 3 ( ). + N = N 4
12 Range : N ( + ). A N i negaie here, hi i he cae ha iniia inenory poiion i hor on he ecuriie. The ery arge negaie inenory impie ha N + 0 canno be aified wih ric inequaiy, hence he fir wo erm in he baic inegra anih. Conequeny, + [ + ] = ( ) + 4 N+ < 0, V 0 + E N d N d + + d ( N ) d 4 = + + = ( N + ) ( N + ) = ( N + V ) d = ( N 4 = + = = 4 + ( N ) ( N - ) N N. V 0 0) = V + = Therefore, for arge iniia inenory poiion uch ha N +, he equiibrium percenage pread grow ineary wih he ize of he inenory.
13 Appendix : More abou Cae. In cae in ecion 4, he focu i when he iniia inenory poiion i reaiey arge, a which ime he expeced end of period inenory grow ineary wih he iniia inenory. For iniia inenory poiion ha are more moderae, he expeced end of period inenory i grow wih he iniia inenory, bu no neceariy ineary. The reaionhip i more compicaed, and depend on he reaie magniude of he parameer. Thi appendix demonrae he noninear reaionhip wih wo exampe. Again, under he aumpion ha price and ranacion oume are independeny diribued, buy and e oume are independen and idenicay diribued, and hey boh foow uniform diribuion: f() = in he range of +, and zero oherwie. A he oume canno be negaie, i i neceary ha. Then (uppre he ubcrip in + and + ), E [ N ] = d ( N + ) d + 4 N+ 0, + N+, + d ( N + ) d 4 N+ 0, + > N+, + N+ < 0, + + d ( N + 4 ) d. Thi inegra wi be referred a he baic inegra in he re of he appendix. The baic inegra i difficu o eauae a i depend on he aue of iniia inenory poiion, nonineary, in he range of, < N < +. Furher, in order o inegrae he baic inegra, he reaie magniude of,, and 3 need o be pecified. For exampe, if we furher aume ha, hen we can proe he foowing: N, 3
14 . For N 0, E N + + N [ ] [ ( ) ] 3.. For - N < 0, E N + + Proof: N [ ] [ ( ) ] 6. N 0 : Then N + V < 0 wi neer be aified, hu he hird erm of he baic inegra anihe. A a reu, E [ N ] = d ( N + ) d + 4 V 0+ N+, N+ + d ( N + ) d 4 V 0+ N+ +, N+ +.. Tha i, + N N+ + N + [ + ] = ( ) ( ) V 0 N + E N d N d d N d + N [ ( N ) ( N - ) ] d = N ( N) ( N) = + = [( ) ( N ) 0 ( N ) ] 3 N Noice ha N 0 0, he aboe expreion impie E N N 8 N N N + = + [ ] ( ) ( ) ( ) ( ) 3 4 N 8 + = ( ) ( ) [ ( ) ]. N 4
15 . - N < 0 : Then N ( ) ince, which impie ha N + < 0 i neer aified. In oher word, he hird erm of he baic inegra equa o zero. Therefore, E[ N+ ] = d ( N ) + d 4 V 0+ N+, + d ( N + ) d 4 V 0+ > N+, + + N+ + d ( N ) d 4 4 N N = + + = [( N + V + ) + ( N + V ) ] d 8 N ( + ) N + d N d [( ) ( ) 0 ( ) ] N N N N N. 3 4 = + + = N Noice ha - N < 0 0, herefore, - N N E N + = + N [ ] ( ) [ ( )] 3 4 N + N ( ) = [ + ( ) ]
16 REFERENCES Amihud, Yako, and Haim Mendeon 980, Deaerhip Marke: Marke Making wih Inenory. Journa of Financia Economic, o. 8, Copeand, Thoma E., and Dan Gaai 983, Informaion Effec on he Bid-Ak Spread. The Journa of Finance, o. 38, Eaey, Daid, and Maureen O Hara 987, Price, Trade Size, and Informaion in Securiie Marke. Journa of Financia Economic, o. 9, no., Garman, Mark B. 976, Marke Microrucure. The Journa of Finance, o. 3, no., Goen, Lawrence R., and Pau R. Migrom 985, Bid, Ak, and Tranacion Price in a Speciai Marke wih Heerogeneouy Informed Trader. Journa of Financia Economic, o. 4, no., Habrouck, Joe, and George Sofiano 993, The Trade of Marke Maker: An Empirica Anayi of NYSE Speciai. The Journa of Finance, o. 48, no.5, Ho, Thoma. S.Y., and Han R. So 98, Opima Deaer Pricing under Tranacion and Reurn Uncerainy. Journa of Financia Economic, o. 9, no., Madhaan, Ananh 000, Marke Microrucure: A Surey. Journa of Financia Marke, o. 3, O Hara, Maureen, and George S. Odfied 986, The Microeconomic of Marke Making. Journa of Financia and Quaniaie Anayi, o., no. 4, So, Han R. Sep. 978, The Suppy of Deaer Serice in Securiie Marke. The Journa of Finance, o. 33, no. 4,