A Parimutuel Maret Microtructure for Contingent Claim effre Lange* and Nichola Economide** Revied Augut Atract Parimutuel rincile are idel ued a an alternative to fied odd gamling in hich a oomaer act a a dealer quoting fied rate of return on ecified ager. A arimutuel game i conducted a a call auction in hich odd are alloed to fluctuate during the etting eriod until the etting eriod i cloed or the auction called. The rice or odd of ager are et aed uon the relative amount agered on each ri outcome. In financial microtructure term trading under arimutuel rincile i characterized call auction non-continuou trading; rile funding of claim aout uing the amount aid for all of the claim during the auction; ecial equilirium ricing condition requiring the relative rice of contingent claim equal the relative aggregate amount agered on uch claim; 4 endogenou determination of unique tate rice; and 5 higher efficienc. Recentl a numer of large invetment an have adoted a arimutuel mechanim for offering contingent claim on variou economic indice uch a the U.. Nonfarm aroll reort and Eurozone Harmonized inflation. Our aer ho ho the maret microtructure incororating arimutuel rincile for contingent claim hich allo for notional tranaction limit order and undling of claim acro tate i contructed. We rove the eitence of a unique rice equilirium for uch a maret and ugget an algorithm for comuting the equilirium. We alo ugget that for a road cla of contingent claim that the arimutuel microtructure recentl deloed offer man advantage over the dominant dealer and echange continuou time mechanim. Keord: maret microtructure contingent claim echange Parimutuel EL Claification numer: G G G4 * Longitude Inc. 468-859 468-859fa email: lange@longitude.com. : htt://.longitude.com/. ** tern chool of uine Ne Yor Univerit 998-864 fa 995-48 email: neconomi@tern.nu.edu : htt://.tern.nu.edu/netor/ and Director NET Intitute htt://.netint.org. Acnoledgement: We ould lie to than Ken aron of Longitude Darrell Duffie of tanford Univerit and Michael Overton of the Courant Intitute at NYU for roviding critical inight and uort. Thi article i aed on Lange and Economide.
Content I. Introduction... II. Parimutuel Microtructure and Maret Game... 4 III. Parimutuel Microtructure for Contingent Claim... A. Develoment of the Parimutuel Microtructure: Definition and etu.... Parimutuel Equilirium Pricing Condition... 7 C. Parimutuel Limit Order oo Equilirium.... Limit Order oo Equilirium.... An Eamle of Limit Order oo Equilirium... III. Parimutuel Microtructure: Aritrage and Efficienc Conideration... 5 A. Ri Neutralit... 6. Aritrage-free Claim... 7 C. Efficienc of Parimutuel Price Dicover... 8 D. Price Uniquene... 9 E. Multilateral Order-Matching... F. Information Production... IV. Concluion... V. Reference... 4 VI. Aendi... 7
A Parimutuel Maret Microtructure for Contingent Claim I. Introduction Parimutuel rincile ere invented in late 9 th centur France Pierre Oller a an alternative to the oomaer ndicate that dominated French gaming at the time. The arimutuel mechanim ulanted oomaer hore racing in the United tate eginning in the 9 and 9 facilitated in large art the invention of the automatic odd calculator or totelizator Harr trau. Recentl a numer of large invetment an have adoted a arimutuel mechanim for offering contingent claim on variou economic indice uch a the U.. Nonfarm aroll reort Eurozone Harmonized inflation and Fannie Mae mortgage ool reament eed. The arimutuel mechanim emloed i a call auction lating aout one hour for claim on the underling inde hich include a variet of tandard and eotic derivative including vanilla call and ut otion forard digital otion range inar otion and lined u/ell otion uch a ri reveral. A unique feature of the microtructure i that all of the claim offered are riced in equilirium aed uon an imlementation of arimutuel mechanim rincile. Our aim i to formalize thee rincile and oint out ome of the inherent advantage of the mechanim a alied to the recent auction. A a maret microtructure the arimutuel mechanim ha four ditinguihing feature: the arimutuel mechanim i a call auction maret rather than a continuou auction; relative rice of contingent claim are equal to the relative aggregate cot of uch claim; the total amount aid for the contingent claim i eactl ufficient to a for the contingent claim having a oitive return that i the mechanim i elffunding and ri-neutral in the ene that the total remium aid for contingent claim i equal to the tate contingent aout for all contingent claim eiring in-the-mone ; 4 a unique et of endogenoul determined rice i dicovered; and 5 higher efficienc than other trading mechanim. Coniderale emirical or ha een done on the efficienc and information characteritic of arimutuel agering. ee Hauh Lo and Ziema 994. In thi aer e ignore tranaction cot hich can e quite ignificant in arimutuel gamling contet.
Our aroach i to formall rovide a foundation for the arimutuel mechanim and then decrie in detail the mechanim recentl emloed in the caital maret. Our firt te then i hoing a foundational connection eteen arimutuel rincile and the theor of maret game. In ection II e ho that a arimutuel contingent claim maret i a natural etenion of a hale-hui maret game for contingent claim. Thu e connect the arimutuel mechanim to the ell-develoed maret game literature and ho that a arimutuel mechanim i a viale mechanim for a contingent claim maret ith endogenou rice formation. In ection III e dicu in detail the arimutuel maret microtructure recentl emloed to offer contingent claim on the Eurozone inflation inde U.. economic tatitic uch a the nonfarm aroll releae Fannie Mae mortgage ool reament eed and other indice. We ho that the arimutuel microtructure ith notional claim limit order and claim undling acro tate ha a unique rice equilirium. We alo reent a theorem hich ho that all arimutuel mechanim can e ereed a a olution to a general eigenvalue rolem. ection IV dicue the efficienc and no-aritrage characteritic of the arimutuel microtructure a alied to the caital maret. In articular e ho that the liquidit aggregation feature of the arimutuel microtructure oth acro time in a call auction and acro diarate te of contingent claim can reduce the amount of noie around the fair rice of uch claim. ection V conclude. II. Parimutuel Microtructure and Maret Game Parimutuel rincile are idel ued a an alternative to fied odd gamling in hich a oomaer act a a dealer quoting fied rate of return on ecified ager. A arimutuel game i conducted a a call auction in hich odd are alloed to fluctuate during the etting eriod until the etting eriod i cloed or the auction called. The rice or odd of ager are et aed uon the relative amount agered on each ri outcome. In microtructure term agering under arimutuel rincile i characterized call auction non-continuou trading; rile funding of claim aout uing the amount aid for all of the claim during the auction; ecial equilirium ricing condition requiring the relative rice of contingent claim equal the relative aggregate ee hale and hui 977. 4
amount agered on uch claim; and 4 endogenou determination of unique tate rice. When alied to the theor of contingent claim maret the elf-funding and relative ricing feature of a arimutuel tem reult from the guaranteed eitence of a oitive tate rice vector hich eclude aritrage over the tate ace. 4 contain the rice for each elemental tate outcome. The vector We ill ho that the eitence of the oitive tate rice vector comined ith enforcing the equalit of the aggregate aout for each tate are ufficient to guarantee that contingent claim are oth elf-funding and that the relative rice of claim are equal to the relative amount aid for uch claim. Auming no tranaction cot and for uroe of thi dicuion zero interet rate the aence of aritrage require the folloing normalization condition on the tate rice: T e > here i a trictl oitive -dimenional vector of tate rice roailitie e i an -dimenional unit vector and uercrit T i the familiar tranoe oerator. Multiling a vector an -dimenional vector containing the aggregate tate aout for each tate ield the rile condition that all aout are identical acro the tate: T e ince the left-hand ide of i a vector containing the aggregate remium invetment tate that the tate contingent aout of each tate i equal to the aggregate remium invetment i.e. that total amount aid for all of the contingent claim are equal to the total contingent aout. And ince there i no aritrage the ricing tem i linear o that clearl: T T... 4 We emlo the term tate ace to include the uual formalim i.e. a et Ω contain an algera of event F for hich there eit a roailit meaure P: F [ ] atifing PØ and PΩ and for an dioint event A and : P A P A P. The trile Ω F P i called a roailit tate ace or tate ace. ee Duffie 99 Aendi A. 5
here and are the -th element of the vector and reectivel. Thi condition tate that the relative rice of each fundamental tate contingent claim i equal to the aggregate relative amount aid for the reective claim. In addition arimutuel rincile include a maret tructure for arriving at the equilirium rice in hich tate rice are dicovered endogenoul via a call auction roce. It i the endogenou nature of the rice dicover hich rovide a fundamental connection of arimutuel rincile maret game to e dicued net and the contingent claim and maret microtructure reearch. The eminal aer of Arro 964 demontrated the equivalence of a cometitive echange econom for contingent commoditie ith an econom hich ha a comlete and cometitive ecuritie maret and a ot maret in the commoditie. In thi cometitive anali the ecuritie maret ha contingent claim rice hich are fied eogenoul. ince rice are fied each agent demand ha a negligile effect on the rice. uequent reearch ha hon that thi equivalence reult deend cruciall on the cometitive nature of the ecuritie maret. For eamle Pec hell and ear 99 ho that if the ecuritie maret i modeled uing a noncooerative maret game ith endogenou rice formation then the Arro equivalence reult no longer hold. ee alo Weer 999. The maret microtructure literature i largel concerned ith endogenou rice formation here each agent demand ha a otentiall ignificant imact on the maret rice. Outide the finance literature there eit a large od of reearch utilizing the theor of noncooerative maret game to model endogenou rice formation. An influential aer hale and hui 977 introduced a noncooerative maret game for a maret ith commoditie and fiat mone ut ith no uncertaint. In the hale-hui maret game MG each trader conign hi endoment of each commodit to a trading ot dedicated to that commodit. Trade occur ith each trader idding ome of hi fiat mone to each trading ot. When the trading eriod ceae the equilirium rice of each commodit i the um of all the id in fiat mone committed to a trading ot divided the total quantit of commodit conigned to that ot. Each trader receive an amount of good reulting from hi id of fiat mone equal to hi id divided the equilirium rice. hale and hui 977 and uequent aer ho that an interior Nah Equilirium NE ala eit and that the NE converge 6
to a cometitive equilirium a the econom i relicated. ee for eamle Poer hui and Yao 994. The MG frameor ha een alied to maret ith uncertaint Pec hell and ear 99 and Weer 999 a indicated aove. Our intent here i to analze an MG maret adated to contingent claim over a tate ace i.e. e are intereted in the ecuritie maret microtructure hich ma e generall alicale to derivative and other contingent claim maret. We firt ho that the MG maret game ith a credit olic retriction on elling i a arimutuel maret microtructure. The credit olic hich i defined further elo require that elling e done on a ecured or collateralized ai. Prooition : A hale-hui maret game for contingent claim ithin a roailit tate ace ith ecured elling i a arimutuel maret. Proof: The folloing notation i required: agent indeed ; tate indeed ; o initial ealth of agent ; f final ealth of agent in tate ; agent id in dollar for tate contingent claim ; agent offer in dollar for inuring contingent claim ; and rice for tate. Firt e define the hale-hui maret game model. In the MG model each trader mae id and offer to each trading ot here each trading ot correond to a contingent claim ithin a roailit tate ace. A in the claical MG rice are equal to the ratio of total mone id divided total commodit conignment or offer for each trading ot. For a contingent claim maret uing the aove notation endogenou rice formation therefore tae the folloing ell-non functional form 7
. 4 Each tate contingent claim rice i therefore the um total of id in unit of mone e.g. dollar divided offer in unit of mone. The offer can e interreted a ale of the contingent claim or offer to aout unit of tate contingent inurance hould the tate correonding to the trading ot e realized. aed uon the receding notation the udget contraint for agent i therefore f o.... 5 We aume interet rate are zero and there i no roduction. Thu the initial and final ealth in the econom are equal f o a imlied from the definition of rice....... 6 7 We refer to thi condition a the maret clearing condition. umming over ield the initial i.e. at the time of remium ettlement maret clearing condition that total remium aid equal total remium old or:. 8 ince all the tate comrie a tate ace it i required that:. 9 Clearl nothing o far develoed revent eller of claim i.e. eller of inurance from defaulting. To addre the oiilit of default e aume that the maret imoe the folloing credit retriction on offer of notional inurance. 8
Define a credit olic a follo: Total offer of notional inurance for an tate mut e ecured at leat the total remium old for all of the tate i.e. hich ield. utituting from the maret clearing condition ield:....... ince the tate comrie a roailit tate ace Thu it mut e the cae that.... 4 hich tate that the rice of each tate i equal to the total id for that tate divided the total id for all of the tate. Thu the equilirium ricing condition for the hale-hui maret game for contingent claim require the relative rice of contingent claim to equal the relative aggregate id for the reective claim. ince the MG i alo a call auction maret hich i elf-funding ith endogenou rice determination the MG for contingent claim i arimutuel. We can alo interret Prooition in the folloing a. Each trader ho mae an offer for a contingent claim i.e. a ale of notional inurance i required to ot margin. The margin amount i equal to the remium roceed. Thi i a tandard ractice at mot otion echange and i non a remium margin. Prooition require that the total amount of notional inurance on offer for an tate cannot eceed the total remium margin deoited. At mot otion echange an additional amount of margin related to the ri of the otion old i alo required oftentime non a 9
additional margin a i the cae at Eure Clearing A.G. the clearinghoue for the Eure echange. A no additional margin i required Prooition e interret the credit olic to e not overl tight eeciall a comared to eiting margin mechanim in ue. Prooition : The credit olic contraint requiring the total notional offer of inurance for an tate not eceed the total remium old can ala e atified i.e. it i never inding. Proof: It can eail e hon that an notional ale can e relicated through a urchae of comlementar tate ithin the tate ace over hich claim are traded o that Conider a notional ale here. 5 > and for. 6 In thi cae agent ell a claim on tate and on no other tate. We ue the term relicated ale to denote the trateg of idding on the comlementar tate to tate in the folloing a 5 : and for. 7 The id on the -th tate of the relicated ale i herea id on all other tate are non-zero. To enure the relication i availale e allo the trading ot for each tate- contingent claim to oen ith an aritraril mall id and offer i.e. ε ε... 8 here the argument and indicate the mall amount of eiting id and offer allocated to each tate here thee amount are vanihingl mall. 6 In equilirium the rofit of a relicated ale are identical to thoe of the original notional ale 5 We note that idding on all of the tate roortional to the rice achieve the autar trateg of effecting no change in each agent endoment. ee Pec hell and ear 99.
o o f o o f 9 i.e. the final ealth from the relicated ale i identical to the original notional ale for each tate. An notional ale can therefore e relicated into a comlementar id hich atifie the credit olic and therefore relicated ale are aout-achievale. We have et to ho that an equilirium eit ith uch relication going on during the auction. We turn to thi net and ho that an aritrar numer of relication ha a fied-oint equilirium. Prooition : A unique arimutuel equilirium eit ith relicated ale hich are ued to atif the credit olic. Proof: Conider a notional ale here. for and > A indicated aove the relicated ale trateg i for and uch that the trateg id for i a follo:. g 7 the anach Fied Point Theorem there eit a fied oint trateg id for the differentiale function g if there eit a contant z < uch that. z g 8 Differentiation of ield: g. g 9 From the maret clearing condition aove 6 Thee mall liquidit amount tae the lace of the uual MG convention that the quantit / oing to zero id and offer i equal to.
Together ith the oviou. < comlete the roof. Uniquene follo from the contraction roert of the maing that lead to the fied oint. Prooition 4: The MG and a arimutuel maret have equivalent aout and firt-order otimalit condition. Proof: ee the aendi. 7 The firt order condition for the MG equilirium can readil e hon to e: q u q u f f hich ho that the ratio of eected marginal utilitie i equal to the ratio of tate contingent rice i.e. the cometitive Arro-Dereu econom reult multilied the term in racet. Thee are the ame oligool condition derived in hale-hui 977. The oligool condition deend on the ratio of the ize of trader i id in each tate to the total amount of id in that tate ecluding trader i id. The therefore meaure the maret ailit to aor trader i id trateg. A trader i id trateg ecome maller relative to the total id on each tate the maret aroache the cometitivel otimal maret III. Parimutuel Microtructure for Contingent Claim In thi ection e decrie the arimutuel microtructure recentl ued to auction off claim on U.. economic data releae uch a nonfarm aroll retail ale and roduction indice a ell a the Eurozone Harmonized inflation inde e toacco rice and Fannie Mae mortgage ool reament eed. Our goal i to ho that the arimutuel maret ha een deigned in a manner hich ield contingent claim familiar 7 Rather length roof are availale at the Aendi.
to maret articiant in eiting derivative maret. For eamle one feature of the arimutuel maret microtructure i that all trading trategie are imlemented ith id and offer of notional amount of ri claim. In the hale-hui contingent claim maret game of ection II agent imlement trategie ith offer of notional inurance and id of remium dollar. Conventionall hoever derivative contract are aed uon the notional amount to e ought or old and not denominated in remium dollar. The urchaer of an otion a on the dollar-en foreign echange rate ill ecif a deired ize of the oition in notional term e.g. million dollar rather than in term of the amount of deired remium outla. We ho net that the arimutuel microtructure allo trading trategie to e imlemented ith limit order here a trader ma ecif a reervation rice aove elo hich the ecified urchae ale of a given contingent claim ill not e eecuted. Limit rice have heretofore not een ued in arimutuel game. A. Develoment of the Parimutuel Microtructure: Definition and etu In thi ection e develo the concet and mathematical notation needed to adequatel decrie the arimutuel contingent claim microtructure recentl ued in the caital maret in hich trader trategie can e imlemented ith a notional u or ell order; a vector of aout ratio correonding to a range of tate claim undling; and a limit rice. To egin e let U denote the value of an underling variale elected for a arimutuel auction for eamle thi variale ma e the value of an ucoming releae of an economic tatitic uch a Eurozone Harmonized Inflation inde. efore the tart of the auction the trie for the otion to e traded on the underling are determined. The otion trie are et acro the range of liel outcome of the underling to maimize interet in the claim eing offered. Let... - denote the otion trie and let < <... < - auming for imlicit that the underling U cannot tae on an value eteen an to trie. Thee - trie divide U into tate a hon in the firt to column of
Tale. Aociated ith each of thee tate are tate contingent claim that a out if and onl if that articular tate occur. Prior to the oening of the arimutuel call auction the financial intermediar hoting the auction enter order for each of the tate contingent claim. We refer to thee order a the oening order. Let the oening order remium e denoted a for uch that >... 4 Though oening order can e mall relative to the cutomer order oening order enure that the arimutuel equilirium rice are unique. Tale. The tate outcome and tate contingent claim in a PDCA auction. tate Outcome tate Contingent Claim U < Digital ut truc at U Digital range ith trie of and......... - U - Digital range ith trie of - and - U - Digital range ith trie of - and U Digital range ith trie of and......... - U - Digital range ith trie of - and - U - Digital call truc at - In the arimutuel maret recentl run on Eurozone inflation cutomer umitted otion order to u or ell otion folloing tandard otion maret rotocol. For notation aume that cutomer umit a total of order in the auction indeed.... When umitting an order the cutomer requet a ecific numer of contract denoted r. For digital otion e adot the convention that one contract a out $ if the digital otion eire in-the-mone. For vanilla otion e adot the convention that one contract a out $ er oint that the otion i in-the-mone. The arimutuel mechanim i novel ith reect to eiting arimutuel agering cheme in that cutomer can ecif a limit rice for each order a i done at echange including the Ne Yor toc Echange and the Chicago oard of Trade i.e. the limit rice for a urchae of an otion rereent the maimum rice the cutomer i illing to a for the 4
otion ecified. The limit rice for a ell of an otion rereent the minimum rice at hich the cutomer i illing to ell the otion. We ue to denote the limit rice for cutomer order. The arimutuel mechanim relicate each otion uing the auction tate contingent claim. For notation let a rereent the notional aout amount of tate contingent claim ued to relicate cutomer order. Recall for intance from Tale that the firt tate contingent claim i the digital ut truc at. Therefore a i the notional amount of the digital ut truc at ued to relicate order. We require a to e non-negative and e refer to the vector [a a... a ] a order relication eight. The mechanim determine the relication eight to match order aout. For eamle a digital call a out a fied amount if uon eiration U i greater than or equal to it trie denoted a v. If order i a u order for thi otion then the relicating eight are... v v v... a 5 Thi et of tate contingent claim a out if U i greater than or equal to v matching the aout of the digital call. If order i a u of a vanilla call read ith trie v and ith v < then the relicating eight are 8... v a v v v... 6 v... For a ell of digital call otion truc at v the cutomer rofit if U i le than v at eiration. In thi cae 8 If the underling U can tae on value eteen adacent trie then the relicating eight for a vanilla call read ecome a E[ U v U < ] v... v v v...... In thi cae the relicating eight are aed on the conditional eected value of the underling. 5
... v v v... a 7 The mechanim relicate other otion uch a digital ut and vanilla ut read in a imilar fahion. Let denote the equilirium rice of the th tate contingent claim for.... uch that the rice of each tate contingent claim i oitive and that the rice of the tate contingent claim um to unit. Mathematicall... > 8. 9 Note that ha a imle interretation a the imlied roailit that tate occur and the th tate contingent claim eire in-the-mone. Let π denote the equilirium rice for the otion requeted in order. For imlicit of eoition e aume here that the auction onor doe not charge fee. Then π a. Each otion i riced a the um of the roduct of the otion relicating eight and the rice of the tate contingent claim. aed on equation 8 and 9 rice can e hon to e aritrage-free in the ene that it i imoile to comine the otion in uch a a o a to guarantee a rile rofit. 9 Let denote the equilirium numer of filled contract for order. If the cutomer limit rice i elo the arimutuel equilirium rice π then the order id i elo the maret and the order receive no fill o. If the order limit rice i eactl equal to the arimutuel equilirium rice π then the order id i at the maret and the order ma receive a fill o r recall that r denote order requeted 9 ee for eamle theorem of Ingeroll 987 at age 55. 6
numer of contract. If the order limit rice i aove the arimutuel equilirium rice π then the order id i aove the maret and the order i full filled o r. Mathematicall the logic for a u order i a follo < π π > π r. r The logic for a ell order roceed in a imilar manner. A a final iece of notation let M denote the total remium aid in the auction hich i the um of the filled order remium aid lu the um of the oening order. M π. Note that a i the aout order receive if tate occur and define a a. Here i the aggregated cutomer aout aed on the th tate contingent claim. In matri form e can rite: T A. 4. Parimutuel Equilirium Pricing Condition We no roceed to develo the mathematical formulation of the arimutuel equilirium ricing condition. After develoing the necear notation e firt rove that eitence of a unique arimutuel equilirium here all order are maret order i.e. the limit order contraint are non-inding. We then rovide a general arimutuel rereentation theorem hich ho that all arimutuel equiliria in our microtructure 7
are olution to an eigenvalue rolem. In the folloing ection e then rela the retriction on limit order and rove the eitence of a unique equilirium for aritrar limit order. One of the core equilirium condition of the arimutuel mechanim i that the tem contain ufficient remium to eactl fund an tate contingent liailit hich can e ritten a follo M... 5 Here i the total amount of cutomer aout filled for tate and the quantit / i the notional aout amount of the oening order for tate. Thu the left hand ide of equation 5 rereent the total aout that the auction mechanim mut mae if tate occur. The right hand ide i the total remium collected the mechanim. Thu in a arimutuel tem the amount of remium collected i eactl equal to the amount needed to ettle the total of filled requet for ever tate. In thi ene the order in arimutuel equilirium are elf-hedging. In another ene equation 5 relate the aggregate order aout if tate occur and the rice of the th tate contingent claim. For M and fied the greater the aout then the higher and the higher the rice of otion that a out if tate occur. imilarl the loer the aout then the loer and the loer the rice of otion that a out if tate occur. Thu in arimutuel ricing equilirium the aggregate demand for a articular tate i cloel related to the rice for that tate contingent claim. Finall let m denote the total filled remium aociated ith tate contingent claim. Then m and equation 5 imlie triviall that: 8
m m... 6 hich tate that the relative remium demand for to tate i equal to the relative tate rice for thoe tate. We are no in a oition to tate the folloing rooition: Prooition 5: Given demand for order hich are ereed in the form of maret order there eit a unique arimutuel equilirium. Proof: The roof i aed uon a imle alication of a contracting fied oint theorem umming u Equation 5 over all. ee the aendi. We are no in a oition to formulate the folloing theorem regarding the arimutuel maret microtructure: Parimutuel Rereentation Theorem: All arimutuel equiliria are olution to the folloing eigenvalue rolem: H M. 7 Proof: Define the matri H hich ha ro and column here i again the numer of defined contingent tate in the arimutuel auction a follo H M M M L L L L M. H i a quare matri ith each diagonal entr of H i equal to. The off-diagonal entrie for ro are equal to for. Recall that i the vector of length hoe -th element i. Note that 8 9
m m m.................. H. 9 ince the um to unit Eqn. 9 e can rite. 4 m m m... H Reriting Eqn. 5 e have M. 4 The left hand ide of thi ereion i iml the -th ro of H. Thu e can rite H M 4 hich i the matri equivalent to Eqn. 4. The intuition for the eigenvalue rereentation i that a arimutuel ricing vector mut lie in the null ace of the net ri ince in a arimutuel mechanim all claim aout are funded remium aid. The net ri of the arimutuel mechanim i: H MI. 4 Thu a ricing vector hich lie in the null ace of the net ri mean that there eit a olution to I H M 44 hich i the eigenvalue reult. The eigentem rereentation mae it ea to ee that the arimutuel tem ha a unique fied oint equilirium. Michael Overton of the Courant Intitute of Ne Yor Univerit firt uggeted to u that our arimutuel rolem might have an intereting eigentem rereentation.
C. Parimutuel Limit Order oo Equilirium We no introduce limit order into the arimutuel equilirium calculation. Limit order are an imortant feature of the arimutuel microtructure under dicuion. Traditional arimutuel agering method do not allo for either notional trading limit order or undling acro ri tate. Thee deficit render the ra arimutuel tructure ued for agering le than otimal for ue in the caital maret. A revioul mentioned otion future and other derivative contract are aed uon notional contract ize rather than the amount to e inveted in uch contract. Furthermore arimutuel agering maret eoe articiant to an eceive amount of tranaction ri a all ager are eecuted at rice hich var throughout the auction eriod and are not non until all ager have een made. In the caital maret it i cutomar to ue the device of limit rice to limit tranaction ri hich articiant can aure themelve that their order are eecuted onl if the maret rice i more favorale than their indicated limit rice. Finall arimutuel agering i normall conducted in an ad hoc manner in hich liquidit hich could e aggregated ithin the ame tate ace i fragmented into different ool. For eamle ager on et for a hore to in are held in a arimutuel ool hich i earate from ager on a hore to lace. Thi mean that not onl can there e aritrage oortunitie acro the earate ool the ricing ithin each ool i le efficient due to the diaggregation of liquidit. A viale arimutuel microtructure for the caital maret hould aggregate all liquidit ithin a tate ace effectivel alloing for the no-aritrage undling of an te of contingent claim from the fundamental tate claim. In the reviou ection e have hon ho a unique arimutuel equilirium eit here limit rice are not inding i.e. all order are maret order. In thi ection e rove the eitence of a unique arimutuel rice equilirium for limit order ith limit rice that can tae an aritrar value.. Limit Order oo Equilirium We regard limit order a articularl imortant ithin the contet of the arimutuel mechanim for to reaon. Firt the allo mitigation of eecution ri oing to changing contingent claim rice during the auction eriod. In arimutuel agering an earl ettor uect himelf to the ri that the final odd are loer than hen the et a laced. In our microtructure e allo trader to control the eecution rice effectivel utituting a roailit of eecution at the limit rice or etter for the continuou change in odd faced a
arimutuel ettor. econd limit order are a familiar order eecution mechanim in the caital maret hich e elieve hould e incororated into an viale and racticale microtructure for contingent claim. With the introduction of limit order come the requirement of ecifing an oective function for determining uect to the atifaction of the limit rice contraint hich order are eecuted in equilirium. We chooe to maimize the total volume of notional order that can e eecuted uect to the limit rice contraint. We do thi for to reaon. Firt e tae a our definition of liquidit the maimum amount of notional value that can e accommodated in the auction uect to limit rice contraint. Thu the choice of oective function reflect the definition of liquidit hich e are tring to maimize. econd it i anticiated that the onor of the auction ill earn tranaction fee income a a ercentage of notional for each order. Our choice of oective function therefore reflect chooing the et of order that generate maimum fee income. The otimization rolem can therefore e ritten in the folloing form: maimize M uect to < π M a a < π π r > π π M r............... 45 aed uon thi rereentation of the arimutuel equilirium ith limit order the folloing rooition can e tated.
Prooition 6: The arimutuel limit order oo rolem ha a unique rice equilirium in tate rice hen there are non-zero oening order on each tate. Proof: The roof i aed uon fied oint continuation method. ee the aendi. In ractice the onor of the auction can guarantee that there are non-zero oening order on each tate. Prooition 6 etalihe the uniquene of tate rice ut doe not guarantee the uniquene of the eecuted order amount in equilirium. The uniquene of tate rice i aed uon fied oint method hich are indeendent of the maimand in Eqn. 45 the maimand i M the total remium eecuted in equilirium. There are - oile degree of freedom in the eecuted order amount in equilirium meaning that the maimum numer of order hich are artiall eecuted i equal to one le the numer of tate. A in mot microtructure mechanim the allocation of artiall filled order i not unique under equilirium rice ut i intead ticall determined riorit rule uch a time riorit or ro rata allocation. In the maimization of Eqn. 45 the riorit rule ued for the artiall filled order i to allocate them o a to maimize the total rice-eighted volume hich i equal to the otion remium uect to the unique and alread determined equilirium tate rice. ince the maimization for the artiall filled order i undertaen ith reect to fied equilirium tate rice the otimization rolem i a linear rogram. There ma e more than one olution for the artiall filled order under thi linear rogram.. An Eamle of Limit Order oo Equilirium We rovide a imle eamle of the olution of the arimutuel limit order oo rolem. In our eamle e ue the folloing inut data: 5 tate 8 order
4 e r A 5 5 4.75.5.9.9.9.7.8.4 The olution to the otimization rolem i: * * M* 8.57.. 99. 8.57.. 9.56. 8.57 7.4 7.58 7.58 8.57 *..856.69997966.999.987.859 The interretation of thi eamle i a follo. There are 5 contingent tate rereenting the fundamental Arro-Dereu ecuritie. There are 8 umitted order a rereented in the matri A each ro of hich contain a if the order an the tate rereented in the firt column and zero otherie. For eamle the firt ro of A indicate a digital ut otion hich ould a unit er quantit requeted hould either of the firt to tate occur. The quantit requeted or order ize i rereented in the vector r. For eamle the firt ro of r i equal to indicating that the order ize for the digital ut anning the firt to tate i. The limit rice are contained in the vector. For eamle the firt ro of indicated a limit rice of.4 for the firt order Detail on the comuter algorithm ued to olve thi eamle are availale from the author.
a digital ut anning the firt to tate of quantit equal to. The limit rice indicate that the urchaer of thi digital ut ould lie to have hi order eecuted in equilirium at a rice of.4 er unit of claim 4 in total or loer. A earch rocedure i ued to find the equilirium aed uon the otimization in Eqn. 45. The reult of olving the equilirium include the equilirium amount that can e eecuted for each order contained in the vector * the total amount of eecuted fill for each tate the vector * the total amount of remium aid for the eecuted claim in equilirium aed uon their equilirium rice the calar M* and the equilirium rice of the fundamental tate the vector *. The earch rocedure need to run to a high level of tolerance hich i h i reorted to a high level of reciion. The equilirium reult can e undertood eamining the firt three order. The firt order for a digital ut anning the firt to tate for quantit equal to and a limit rice of.4 i full filled a can e een from the firt ro of *. It i full filled in equilirium ince the rice of a digital ut anning the firt to tate i the um of the firt to ro of the equilirium tate rice a hon in * hich i equal to aroimatel.9. ince.9 i le than the limit rice of.4 for thi order the order mut e full filled in equilirium hich i the cae. The econd order an the lat to tate a een from the econd ro of A and therefore hould e interreted a a digital call covering the lat to tate. From the econd ro or r and reectivel the order i one to urchae unit at a rice of.8 or loer. A can e een from the equilirium reult in the econd ro of * the order i artiall filled at 9.56 out of the requeted. The rice of the order i equal to the um of the lat to tate rice in * or.8. ince the order equilirium rice i equal to it limit rice it ma receive a fill anhere eteen and the the requeted amount. Finall order three i a digital range anning the third and fourth tate for unit at a limit rice of.7. A can e een adding the third and fourth ro of * the equilirium rice of thi claim i equal to aroimatel.79. ince thi i higher than the indicated limit rice of.7 the order eecuted amount in equilirium i zero a indicated the third ro of *. III. Parimutuel Microtructure: Aritrage and Efficienc Conideration 5
We elieve the arimutuel microtructure rooed and analzed in ection II comare favoral to other microtructure that ma e ued for contingent claim trading. We thin the arimutuel microtructure under dicuion ma e uerior to dealer-aed and currentl ued echange tructure for a ide variet of ri. We elieve that the arimutuel microtructure decried in thi aer i eeciall uerior for thoe ri hich do not have tradale underling ecuritie or intrument. We organize our dicuion of the enefit of our microtructure into the folloing i area: ri-neutralit; the aence of aritrage; efficienc; 4 rice uniquene; 5 multilateral order matching and; 6 information roduction. A. Ri Neutralit Parimutuel rincile entail a elf-funded auction of contingent claim: all remium collected ecluding tranaction cot i eactl ufficient to a for all tate contingent aout. From a dealer erective the arimutuel microtructure ill e referale to tandard OTC tranaction for certain te of derivative ri. For eamle a dealer in fied income derivative ill liel find the rooed arimutuel microtructure favorale for tranacting otion on the monthl announcement of the level of the Eurozone Harmonized inflation inde ince there i no underling ecurit or hedgeale intrument. The rooed arimutuel microtructure effect an aritrage-free and rile et of contingent claim rice and order eecution. Effectivel the mechanim achieve hat a dealer ould need to do manuall through hedging activit in an underling intrument here availale and through alancing ri aduting rice ith trading counterartie to equilirate net notional tranaction acro tate. We thin thi imlicit and efficac of the arimutuel microtructure a adated to the caital maret i therefore a otentiall ueful comlement to the traditional OTC dealer maret tructure eeciall for te of ri hich have no tradale underling. We alo thin that the rooed arimutuel microtructure i uerior to conventional echange-aed continuou doule auction for ome te of illiquid ri. For eamle for a numer of ear the Chicago oard of Trade COT ha offered ri-neutralit e mean that the arimutuel auction i elf-funding in the ene that remium inut equal tate contingent outut. We do not mean to ugget a connection to the continuou time otion literature hich i focued on ri-neutral ricing. 6
otion on inurance catatrohe loe a meaured indice ulihed the Proert Claim ervice PC. The microtructure ued to tranact thee claim i a conventional continuou doule auction i.e. the ame mechanim that i ued to trade the highl liquid ond future and otion at the COT. While there are erha reaon h the PC contract have failed to attract liquidit hich are unrelated to maret microtructure ee e.g. Cummin and Mahul e elieve that the conventional microtructure ma e a ignificant imediment to liquidit a e dicu further elo.. Aritrage-free Claim A arimutuel tem i aritrage-free in the ene that there eit a oitive tate rice vector hich eclude aritrage. Folloing the tandard definition ee Ingeroll 987. 57 e can define the return tale Z of a arimutuel tate ace a follo: Z Adiag π No it i ell non that if there eit a tate ricing vector uorting the return tale uch that: 46 Z 47 then there eit no aritrage oiilitie in the ene that there eit no invetment η acro the tate hich olve either: e T Z η T η one trictl 48 or T e η < T Z η. In the rooed arimutuel maret microtructure a definition i that all contingent claim rice are linear comination of the tate rice i.e. π A. Multilication of thi definition diagπ - etalihe that there i a uorting tate rice vector and that no aritrage i oile contruction of the arimutuel microtructure. The claim undling feature of our arimutuel microtructure definition rule out aritrage in the aove-defined ene. A maret for tate contingent claim even a ee Ingeroll 987. 54-57 for the elementar roof. 7
call auction lie the arimutuel mechanim under dicuion need not enforce the noaritrage condition elicitl. Namel e can readil enviion a contingent claim maret for a tate ace hich can e modeled ithout uch elicit retriction a follo: * argma i uect to < π i π > π n r r... hich are limit order condition ithout the arimutuel and no-aritrage rice retriction. In uch a maret reumal aritrageur ould devote caital to enuring that aritrage ould e ecluded from the rice. The arimutuel mechanim enforce the normalization of tate rice and the aence of uch aritrage endogenoul ithin the microtructure. C. Efficienc of Parimutuel Price Dicover The enforcement of the no aritrage condition lead naturall to the folloing elfare reult on the efficienc of the arimutuel microtructure comared to a model in hich contingent claim are traded earatel in a call auction over a tate ace the trading ot model. Eentiall the arimutuel maret a imlemented in thi aer lead to more efficient le noi rice ecaue the mechanim utilize information on id and ece demand in all individual maret trading ot. Put it differentl the arimutuel mechanim a imlemented dicover rice that reflect information from all trading ot maret and thi mae the rice reflect more efficientl trading condition in all ot. Prooition 7: A arimutuel microtructure dicover rice for contingent claim uch that the average order tandard deviation around fair value i le than a microtructure ith earate call auction trading ot for each claim. The average order noie aving i equal to 49 here δ ασ M.9 ασ M 5 8
δ aving due to arimutuel microtructure M total remium in tem σ average volatilit of rice error around the true rice noie volatilit α id/offer read aumed roortional to average noie volatilit Proof: ee the aendi. We alo note that the arimutuel mechanim ha an additional efficienc gain over the traditional continuou maret ecaue of the time aggregation of order rovided the call auction itelf. 4 There i uggetive emirical evidence uorting the receding efficienc reult. Gariel and Marden 99 and Gariel and Marden 99 eamine ritih etting on hore in hich arimutuel and oomaer mae rice imultaneoul. The oomaer offer odd on ager uing the tarting rice odd convention here a oomaer tae a et at odd formed a conenu of oomaer ut efore the race i run. Thu oth the arimutuel and tarting rice odd reflect odd ut efore the race i run. On the ame amle of race Gariel and Marden 99 Tale find that arimutuel return on the ame race are aout 8.7% higher almot eactl the amount of efficienc oing to the arimutuel tem redicted in Prooition 7. D. Price Uniquene The arimutuel microtructure oee a unique rice equilirium for a given et of oening order and other order for contingent claim. Not all microtructure of thi cla need oe unique equilirium rice. Conider in thi regard the folloing modified microtructure imilar to the arimutuel dicued in ection II aove: 4 ee Economide and chartz 995. 9
.... for * r r e to uect argma T > < < < π π π K 5 Thi microtructure rolem i otherie identical ith that of Eqn. 45 ecet that the arimutuel contraint ha een relaced ith a eaer contraint in Eqn. 5. The contraint in Eqn. 5 merel require that the tate contingent aout for each tate e equal. Thi microtructure ha ome arimutuel feature in the ene that elemental tate claim are normalized ehiit no aritrage and relative rice are equal to relative remium invetment for each air of tate. Yet there eit no unique et of tate rice hich atif Eqn. 5. To ee thi e conider a tate ace ith three tate. Aume that there are order: a limit u order for notional covering tate at limit rice of. a limit u order for notional covering tate at limit rice of.4 and a limit u order for notional covering tate at limit rice of.5. Clearl an tate roailitie atifing.5.4. i a olution to Eqn. 5 and there are ovioul man uch olution hich ill atif the ri neutralit contraint that all tate aout are equal. For eamle one uch olution i
.5.5.5 contrat the arimutuel microtructure e rooe emodied a the olution to Eqn. 45 oee a unique et of tate rice. In the imle eamle under conideration e aume that there eit oening order on each tate of one unit o that. The unique olution i:..4. ;.8.. E. Multilateral Order-Matching The arimutuel microtructure e rooe i fundamentall a multilateral ordermatching mechanim hich e mean there eit no requirement of a dicrete order match eteen a ingle uer and a ingle eller. Rather the order-matching mechanim i inherentl man-to-one in the ene that an given contingent claim aout i funded multilaterall all of the other order hich are filled in equilirium. We regard thi feature a articularl imortant for claim for hich there i no tradale underling and for hich there i not a natural demand for a continuou time maret. For eamle e regard our maret microtructure to e of otential ue to trade contingent claim on eather economic tatitic releae cororate earning releae and mortgage reament eed. The character of our arimutuel microtructure i influenced greatl the commitment of oening order. For the microtructure reemle a multilateral matching mechanim in hich tate rice are normalized ut are not
necearil unique. For all order hich have limit rice etter than the rice reflected in the oening order ill eecute and ill have no imact on the tate rice. Thu large ill tend to reemle a dealer microtructure in the ene that the dealer ma ear ignificant ri that the ditriution reflected in the oening order ditriution ill deart from the true ditriution. We elieve the arimutuel microtructure e rooe ill tend to e mot attractive at mall value of. We define mall uch that < π < i.e. that the ratio of total remium filled in equilirium to the total amount of oening order i greater than and le than. F. Information Production Our arimutuel microtructure dicover tate rice through a tate ace artition of an underling roailit ditriution. It therefore dicover the roailit denit function imlied actual trading activit in a tranarent and natural a. ome eerimental data ho that tandard arimutuel mechanim have the ailit to aggregate rivate information ee Plott et al. 997 into the maret denit function. We thin the imlied denit roduced in our microtructure ill e an imortant and high qualit informational eternalit to the maret. The qualit of the imlied denit ill e high ince the denit itelf i eing traded iece iece in our microtructure. The denit dicovered on our microtructure i ala enforced to e a roailit tate ace deign. Continuou time otion maret contrat roduce anchronou otion rice at trie hich have varing liquidit and rice noie. A a conequence the traditional technique ued to etract imlied denit function from continuou otion data tend to roduce ver oor information due to data limitation and large noie in continuou time otion rice ee reeden and Litzenerger 978. IV. Concluion A arimutuel maret microtructure for contingent claim recentl ued Goldman ach and Deutche an to offer derivative on Eurozone Harmonized inflation and other economic indice ha een dicued and analzed in thi aer. A
arimutuel microtructure i a call auction maret ith ecial equilirium ricing condition on the relative rice of contingent claim. We have hon that the arimutuel contingent claim mechanim recentl emloed in the caital maret i quite general and ha it root in the maret game literature. We have hon ho the maret microtructure incororating arimutuel rincile for contingent claim hich allo for notional tranaction limit order and undling of claim acro tate i contructed. We have roven the eitence of a unique rice equilirium for uch a maret and ugget an algorithm for comuting the equilirium. We elieve that for a road cla of contingent claim the arimutuel microtructure recentl deloed offer man advantage over the dominant dealer and echange continuou-time mechanim. Firt the arimutuel mechanim doe not require a dicrete order match eteen to counterartie. Intead order are eecuted multilaterall. All eecuted order remium i ued to fund all of the contingent in-themone otion i.e. the aout. econd e elieve the tranarent and traightforard ricing mechanim ill e attractive to maret articiant. We elieve that the ucce of the arimutuel mechanim in the agering maret can ith the modification hich e have made to the mechanim e carried over into the caital maret. Third e elieve that the ri neutral and elf-hedging nature of the arimutuel mechanim from the erective of the roer/dealer or other entit hich hot the auction offer a uerior tradeoff eteen the ri of derivative dealing and the comenation for roviding liquidit for contingent claim. We elieve that the arimutuel microtructure ma in fact avoid altogether ome of ri inherent in derivative maret-maing that eriodicall reult in ell-ulicized diatrou outcome. Fourth e have hon that the Parimutuel mechanim a imlemented in thi aer i more efficient than other trading mechanim. Finall e elieve that the arimutuel microtructure i ideall uited for comleting ome maret here there currentl i an aence of liquidit uch a contingent claim on mortgage reament eed cororate earning eather and economic tatitic uch a the recent Eurozone inflation auction.
V. Reference Arro K. 964 The Role of ecuritie in The Otimal Allocation of Ri earing Revie of Economic tudie 9-96. reeden D. and R. Litzenerger 978 Price of tate Contingent Claim Imlicit in Otion Price ournal of uine 5 6-65. aron K. and. Lange From Hore to Hedging Ri Magazine Feruar. lac F. and chole M. 97 The Pricing of Otion and Cororate Liailitie. Political Econom 8 67-654. Cummin.D. and Mahul O. Managing Catatrohic Ri ith Inurance Contract uect to Default Ri oring aer. Duffie D. 99 Dnamic Aet Pricing Theor Princeton Univerit Pre Princeton N... Duont D.Y. 995 Maret Maing Price and Quantit Limit Woring Paer oard of Governor of the Federal Reerve tem. Economide N. and chartz R.A. 995 Electronic Call Maret Trading ournal of Portfolio Management vol. no.. -8. Gariel P.E. and Marden.R. 99 An Eamination of Maret Efficienc in ritih Racetrac etting. Political Econom 98 874-885. Gariel P.E. and Marden.R. 99 An Eamination of Maret Efficienc in ritih Racetrac etting: Errata and Correction. Political Econom 99 657-659. 4
Gloten L and Milgrom P. 985 id A and Tranaction Price in a ecialit Maret ith Heterogeneoul Informed Trader. Financial Economic 7-. Groh C. 998 equential Move and Comarative tatic in trategic Maret Game Deartment of Economic Univerit of Mannheim oring aer. Handa P. and chartz R.A. 996 Limit Order Trading ournal of Finance 5 85-86. Hauch D. Lo V. and Ziema W. ed. 994 Efficienc of Racetrac etting Maret Academic Pre an Diego CA. Ingeroll. 987 Theor of Financial Deciion Maing Roman & Littlefield avage MD. Kle A.. 985 Continuou Auction and Inider Trading Econometrica 5 5-6. Lange. and Economide N. A Parimutuel Maret Microtructure for Contingent Claim Dicuion Paer no. EC-- tern chool of uine NYU. Lange. and Economide N. A Parimutuel Maret Microtructure for Contingent Claim at htt://.tern.nu.edu/netor/parimutuel.df. Levin N. 994 Otimal et in Parimutuel tem oring aer no. 8/84 The Irael Intitute of uine Reearch in Hauch Lo and Ziema ed. Efficienc of Racetrac etting Maret 9-5 Academic Pre an Diego CA. O Hara M. 995 Maret Microtructure Theor lacell Malden MA. 5
Pec. hell K. and ear. 99 The Maret Game: Eitence and tructure of Equilirium. Math. Econ. 7-99. Plott C.R. Wit. and Yang W.C. 997 Parimutuel etting Maret a Information Aggregation Device CalTech ocial cience Woring Paer 986 California Intitute of Technolog Aril 997. Poer M. hui M. and Yao.994 Inurance Maret Game: cale Effect and Pulic Polic Cole Foundation Dicuion Paer No. 76. hale L. and hui M. 977 Trade Uing One Commodit a a Mean of Pament ournal of Political Econom vol. 85:5 97-968. Weer. 999 Uncertaint and Inurance in trategic Maret Game Economic Theor 4 8-. 6
7 VI. Aendi Proof of Prooition 4: We firt ho that the MG and a arimutuel maret are aout-equivalent. We then ho that the firt order necear condition characterizing the Nah Equilirium are identical in each maret. With reect to aout equivalence the roof of Prooition ho that an trateg vector of remium id and notional offer can e relicated uing a id trateg a follo. Namel an trateg coniting of the folloing vector air K K A hich reult in final ealth for agent i equal to... o f A can e relicated uing a ingle vector trateg in id a follo. K A Noting that K A4 the final tate-contingent ealth for agent i oing to the relicated trateg i equal to.......... o f o f o f A5 Thi ho that the final ealth from the relication trateg emloing no offer i identical to the final ealth to the trateg emloing offer i.e. the MG for contingent claim and the arimutuel maret are aout-equivalent.
8 We no ho the firt order necear condition of the arimutuel maret are equivalent to thoe of the MG a reorted etenivel in the maret game literature. ince the entire trateg ace can e otained uing id the otimization rolem faced agent ma e ritten a... ma... o q to uect o u f u q π A6 here q denote the uective roailit aement of agent for tate. Folloing Levin 994 e find it more convenient to mae the folloing change of variale in the otimization rolem:. A7 Reectivel thee ne variale denote the total id for tate eceting agent id; the total id for tate including agent id; the total id for all tate for all agent; and 4 the total id for all the tate eceting agent id. traightforard utitution of the ne variale into the otimization rolem ield:. ma... o... q to uect o u f u q π A8
traightforard differentiation of the aociated Lagrangean ield the folloing firt order necear condition for an interior otimum 5 q u q u f f. A9 With the folloing definition of the tate contingent claim rice ecluding the effect of agent i trateg A and ith the definition of the tate rice including trateg A the firt order condition ecome q u f q u f A hich i identical to the otimalit condition derived Pec hell and ear 99 Prooition.4 for their imlementation of the hale-hui commoditie maret game. Thu a arimutuel maret i oth aout- and firt-order-condition-equivalent to an MG maret for contingent claim. Thi rovide a connection eteen the etenive maret game literature hale and hui 977 Pec et al. 99 and the maller literature on arimutuel gamling Levin 994. Proof of Prooition 5 6 : We rovide a roof for notional order i.e. thoe ith order amount in term of notional that are indeendent of equilirium rice. From Eqn. 5 and the aumtion that the roailitie of the defined tate mut um to one Eqn. 9 again ignoring an 5 auming an interior olution e aume a oitive id for each tate uch that a no-id trateg correond to a vanihingl mall oitive id for a tate. 6 We ould lie to than Ken aron of Longitude ho contriuted to thi roof. 9
4 interet rate conideration the folloing equation ma e olved to otain the unique et of defined tate rice and the total eecuted remium.. a M A. M Eqn. A contain unnon and equation. The unnon are the and M the total eecuted remium for all of the defined tate. We firt olve for M. Uing Eqn. A and the fact that i greater than and le than one e conclude that < < M for. A4 Thi equation imlie that M > for. A5 Thu ma M > for. A6 o a loer ound for M i equal to ma loer M. A7 here the maimum i taen over. Net e derive an uer ound for M. Uing the definition for M Eqn. and m Eqn. 6. m M A8 Letting m e the maimum value of the M. m m m A9. Thu the uer ound for M i equal to A ma m uer M The olution for the total remium in the defined outcome therefore lie in the range M M loer M uer ] or
ma Let the function f e defined a < M ma f M M Note that FM loer > fm uer <.. A. A A No over the range M M loer M uer ] e can chec that fm i differentiale and trictl monotonicall decreaing. Thu e conclude that there i a unique M in the range uch that fm. A4 Thu M i uniquel determined from the and therefore the demand for order hich rove the rooition. Once M i non e can comute the vector from Eqn. A ince the are non. We no ho ho e can olve iterativel for M uing the. Uing Eqn. A e can rite that f M Thu for M tae for an initial gue For the κ t gue ue df dm M M loer. M. A5 M κ M f M f M κ κ - κ. A6 The olution for fm over the interval M loer M uer ] can therefore e otained uing Neton iteration. Once the olution i otained the value of M can e utituted into each of the equation in Eqn. A to olve for the. Proof of Prooition 6: 4
We ho that there eit a fied oint iteration equence leading to a unique et of rice hich olve the otimization rolem. To rove the eitence and convergence to a unique rice equilirium conider the folloing iterative maing F β * g. A7 Eqn. A7 can e roved to e contraction maing hich for a te ize β indeendent of ill gloall converge to a unique equilirium i.e. it can e roven that Eqn. A7 ha a unique fied oint of the form F * *. A8 To firt ho that F i a contraction maing matri differentiation of Eqn. A7 ield: df I β * D d here D * C * Z C Z * * * M * * i T A9 The matri D of Eqn. A i the matri of order rice firt derivative i.e. the order rice acoian. ell-non rincile Eqn. A9 can e hon to e a contraction if the folloing condition hold df < A d hich i the cae if the folloing condition hold β * ρ D < here ρ D ma λ D i. e. the ectral radiu of D. A the Gerchgorin Circle Theorem the eigenvalue of C are ounded eteen and. The matri Z- i a diagonall dominant matri all ro of hich um to /M. ecaue 4
of the diagonal dominance the other eigenvalue of Z- are clutered around the diagonal element of the matri and are aroimatel equal to /. The larget eigenvalue of Z- i therefore ounded aove /. The ectral radiu of D i therefore ounded eteen and linear comination of a follo: ρ D L L. A here the quantit L a function of the oening order amount can e interreted a the liquidit caacitance of the equilirium mathematicall L i quite imilar to the total caacitance of caacitor in erie. The function F of Equation i therefore a contraction if β < L. A Eqn. A tate that a contraction to the unique rice equilirium can e guaranteed for contraction te ize no larger than L hich i an increaing function of the oening order in the auction. The fied oint iteration of Eqn. A7 converge to *. ince * A T * * can e ued in Eqn. A to comute the fundamental tate rice * and the total quantit of remium inveted M*. If there are linear deendencie in the A matri it ma e oile to reerve * through a different allocation of the correonding to the linearl deendent ro of A. For eamle conider to order and hich an the ame tate and have the ame limit order rice. Aume that r and r and that * * 5 from the fied oint iteration. Then clearl it ould e oile to et and ithout dituring *. For eamle different order riorit rule ma give eecution recedence to the earlier umitted identical order. In an event the fied oint iteration reult in a unique rice equilirium that i unique in. In our current model of the arimutuel limit order oo the riorit rule i the otimization of the total notional order uect to the otimal rice. At the otimal rice the nonlinear rogram in Eqn. 45 ecome the folloing linear rogram: 4
* argma uect to H* M* here * olve the fied oint iteration. A4 Proof of Prooition 7: 7 Aume a maret for m ingle tate claim. We model the maret rice of thee claim a ~ ~ µ f A5 f ~ σ ~ for. In the arimutuel microtructure the um of the forard tate rice are enforced to e one or Therefore Therefore Net let ~. A6 Var ~. A7 ~ ~ ~ Var σ cov f f. A8 i m ~ ~ cov cov f i f m m i i m m utituting and rearranging term ield: σ σ. A9 ~ ~ cov f f σ cov σ σ cov. A4 7 We ould lie to than Ken aron of Longitude for helful dicuion regarding thi roof. 44
45 No e mae a imlifing aumtion of unit variance. Thi ill not affect our anali a e are intereted in relative average noie eteen a arimutuel and tradingot microtructure. Thu the lat equation imlifie to cov. A4 We no analze uing a imle tale the total variance of a contingent claim coniting of tate in the arimutuel microtructure hich imoe a covariance tructure and a non-arimutuel microtructure in hich covariance are zero. Order TM Varianceof Order PM Varianceof Order Numerof tate Numer of - - - M M M M M M M M A4 We no calculate the total variance TV of order in a arimutuel PM microtructure and trading ot microtructure non-arimutuel a follo:
46!!!!!! l PM l TV A4 ince l l Further note that!!!!!!!!! l TP l TV A44 ince. Hence the ratio of trading ot average order noie to arimutuel order noie i l l PM TP TV TV. A45 o the average order noie for a arimutuel tem i half that for the non-arimutuel tem.
Aume that average noie volatilit i % of the rice. If therefore there i million UD in remium million UD i one tandard deviation of noie around the true rice. the reviou reult a arimutuel microtructure ould have 7.7 million in noie million divided quare root of. Therefore if the average id-offer read in a non-arimutuel microtructure i roortional to the noie volatilit of rice the net efficienc of the arimutuel tem can e ritten a: δ ασ M.9 ασ M A46 here δ aving due to arimutuel microtructure M total remium in tem σ average volatilit of rice error around the true rice noie volatilit α id/offer read aumed roortional to average noie volatilit hich i Prooition 7. 47