Howrisklessis“riskless”arbitrage?∗RomanKozhan†WingWahTham‡AbstractInthispaper,wechallengethenotionthatexploiting“riskless”,“executionrisk”,:executionrisk,limittoarbitrage,liquidity,inventorycosts∗WethankKeesBouwman,AlainChaboud,DickvanDijk,MathijsvanDijk,IngolfDittman,MatthijsFleischer,ThierryFoucault,GordonGemmill,AllaudeenHameed,LawrenceHarris,JoelHasbrouck,MarcinKacperczyk,KostasKoufopoulos,BruceLehmann,FrancisLongstaff,AlbertMenkveld,MichaelMoore,An-thonyNeuberger,StijnVanNieuwerburgh,CarolOsler,AndrewOswald,ChristineParlour,PaoloPasquariello,RichardPayne,DagfinnRime,MarkSalmon,AsaniSarkar,LucioSarno,ElviraSojli,PeterSwan,IliasTsiakas,GiorgioValente,ClaraVega,MichelvandeWel,IngridWerner,AviWohl,GuntherWuyts,WeiXiong,BerntArneØdegaardandparticipantsatthe2009EuropeanFinanceAssociationConferenceinBergen,2009Euro-peanEconomicAssociationMeetinginBarcelona,2009NYSEEuronext&TinbergenInstituteWorkshoponLiquidityandVolatilityinAmsterdam,2009SecondErasmusLiquidityConferenceinRotterdam,2010AFFISpringMeetinginSaintMalo,2010AsianFinanceAssociationMeetinginHongKong,2010AnnualCentralBankWorkshopontheMicrostructureofFinancialMarketsinNewYork,seminarparticipantsinCarletonfortheircommentsandsuggestions.†WarwickBusinessSchool,UniversityofWarwick,ScarmanRoad,Coventry,CV47AL,UK;tel:+442476522114;e-mail:@‡EconometricInstitute,ErasmusSchoolofEconomics,ErasmusUniversityRotterdam,,POBox1738,3000DR,Rotterdam,theNetherlands,tel:+31104081424;e-mail:tham@
“riskless”arbitrageandriskyarbitrage,seeMitchell,PulvinoandStafford(2002)andLiuandTimmermann(2009).1Contrarytoariskyarbitragestrategy,“riskless”arbitragedoesnotrelyonconvergencetradingwhereonebetsthatthefuturepricedifferencebetweentwoassetswithsimilar,butnotidentical,,exploiting“riskless”,wechallengethisconventionalbeliefandshowthatexecutionriskcreateslimitsto“riskless”,“riskless”arbitrage,liketriangulararbitrage,put-callparity,cross-listedsecurityparityandcoveredinterestparity,shouldnotpersistandshouldbeimmediatelyeliminatedintoday’,severalrecentpapersshowempiricallythatthereare“riskless”arbitrageopportunities,suchasdevia-tionsfromtriangulararbitrageparity(Marshall,TreepongkarunaandYoung(2008)),covered1“Riskless”arbitrageisdefinedasthesimultaneoussaleandpurchaseofidenticalassets,whichensuresthatarbitrageursrequirenooutlayofpersonalwealth,-listedcompanies(DLCs),,arbitrageurswithwealthconstraintsareconcernedwithnoisetrader,fundamental,,RosenthalandVanDijk(2009)andFrootandDabora(1999)(1992),DeLong,Shleifer,SummersandWaldmann(1990),ShleiferandSummers(1990),AbreuandBrunnermeier(2002),BarberisandThaler(2003),Fleckenstein,LongstaffandLustig(2010),GrombandVayanos(2002)andSchultzandShive(2009)(HFT)strategiesincludecross-assetarbitrage,electronicmarketmaking,liquiditydetection,short-termstatisticalarbitrageandvolatilityarbitrage,seeTabb,IatiandSussman(2009)..(2009).InastudyofalgorithmictradingintheFXmarket,Chaboud,Chiquoine,HjalmarssonandVega(2010)
interestparity(Akram,RimeandSarno(2008),Fong,ValenteandFung(2008)),put-callparity(LamontandThaler(2003)andOfek,RichardsonandWhitelaw(2004)),andcross-listedsecu-rityparity(GagnonandKarolyi(2009)).5Althoughdurationsofthesearbitrageopportunitiesareshort,,whichfocusesonconvergencetrading,doesnotprovidesatisfactoryex-planationstothepersistenceof“riskless”,weproposeanewlimittoarbitrage,,“executionrisk”inarbitrageexploitation,whichdependsonthedegreeofcompetitionamongarbitrageurs,:consider2arbitrageurscompetingtoformalong-shortarbitrageportfoliooftwoidenticalbutmispricedassets,Awith2availableunitsandB,,exploitingthisarbitragecanberiskyasoneofthearbitrageurswillfailatacquiring/shortingAandBataprofitablepriceandmightincurliquiditycosts(walkingupthelimitorderbook)orinventorycosts(unwantedinventoryfromsuccessfullyacquiringAbutunsuccessfulfromshortingB).,-5Explanationsfortheexistenceofthesedeviationsareoftenbasedoninstitutionalfrictionslikeregulatoryandshort-sellingconstraints,taxes,(1998)andEngleandFerstenberg(2006)
viouslimitstoarbitrage,themechanismwepresentdoesnotrelyonconvergencetrading,taxes,regulatory,“riskless”,,takingintoaccountdirecttransactioncosts(quotedbid-askspread),executionrisk,(FX)-bitrage,ourstudyfocusesontriangulararbitrageexploitationinmajorcurrencypairs,becausetriangulararbitrageisanon-convergencetradingbasedarbitrageandisnotsubjecttotaxes,regulatory,,bid-askspread(directtransactioncost)andlatencycost,,theneitheritmightnotbeoptimalforthemtodosoimmediately,,SchwartzandSubrah-manyam(2007),,wearguethattheeconomicreasonbehindthisrelation,inthecaseof“riskless”arbitrage,
evaluationofarbitragingstrategiesandfindthatarbitrageursmightincurlossesintextbook“riskless”,-tradingarbitrageopportunitieshasbeenstudiedrecentlyby,amongothers,Stein(2009),Kon-dor(2009),andOehmke(2009).Kondor(2009)(2009)investigatestheroleofstrategicarbitrageurs,tradingwithliquidityfrictions,,thesepapers,likemanyothersthatexaminelimitstoarbitrage,(DLCs),theydonotexplainthedelayedeliminationof“riskless”(2009)andKondor(2009),(2009),,intheeventwithfundamentalanchor,thepricemechanismmediates7LiuandLongstaff(2004)
congestion,,(1992),KumarandSeppi(1994)andHolden(1995)highlighttheimportanceofexecutionriskinindexarbitrageunderstressfulmarketconditions(crashofOctober1987),whereexecutionriskcanbeaconcernduetotradingonstale(lagged)“paperenvironment”“riskless”,thisisthefirstpapertoaddresstheriskofsupposedly“riskless”,,sincewedemonstrate,byusingthelimitorderbook,-pricingtheory,-basedtheoryforimpedimentstoarbitrage,thussupportingtheempiricalpapersthatrelate“riskless”,canbeconsideredaformofconvergencetrading,whichissusceptibletoshortsellingconstraints,fundamental,noisetrader,,includeBrennanandSchwartz(1988),KumarandSeppi(1994),Sofianos(1993),andNeal(1996).6
focusonthecostofinventoryonmarketmakersbutnotonarbitrageurs,,webelieveourpaperisamongthefirsttoempiricallystudyhighfrequencyarbitrageandtherelationamongarbitragedeviations,,,,,,,wherethereareIassetsindexedbyi∈{1,2,...,I},RP,consistingofallassetsfromtheset{2,...,I},,thisportfolioincludeslongandshortpositionsofoneunitineachassetdenotedbythevector[w2,...,wI].witakesthevalueof1ifitisalongpositionand−1ifitisashort10TheUSSecuritiesandExchangeCommission(SEC)(-61358;-02-10)andCESRproposaltoMIFID(Ref:CESR/10-142).7
(asset1)orindirectly(portfolio),,,traditionalimpedimentstoarbitragelikefundamentalrisk,(2009),weassumethatthereareIgroupsoflocaltraders,(LOB),,
,,weassume:={1,...,k}andthesetofallcompetitorsofarbitragerjforj∈{1,...,k}byK−j=K\{j}.,,(allpricesanddepths),−∆bandthenextbestaskpriceispa+∆aatiiiithesecondlayer.∆band∆,,whichincreasesthecostofliquidity,willonlyexacerbateexecutionrisk9
,portfolioRPconsistsofallassetsfromtheset{2,...,I}.ThebestpriceatwhichoneunitofportfolioRPcanthenbeboughtis∑IPa=wipi(wi),i=2wherepi(wi)=pbifwii=−1andpi(wi)=paifwii=,dividendstreamsandriskexposure,,amispricingoccursif:Pa<pb1orPb>pa1,}A=max{0b,Pb−pa1,p1−(1995)andAbreuandBrunnermeier(2002),−pa1>0orpb1−Pa>0istrueundertheassumptionofpositivebid-askspreads(pa1>pb1andPa>Pb).,Titman(1985)pointsouttherelevanceoftaxclienteleeffect;AmihudandMendelson(1980)providesanavenueofhowinventorymanagementcancausetheexistencearbitrageopportunities;KumarandSeppi(1994)
,weassumethatthereexistsanexcessdemandfortheseassetsamongcompetingarbitrageurssuchthat:Assumpt}{na,nb<kforeachi=1,...,;,,,iftherewerethreearbitrageursvyingforoneavailableunitofassetatthebestavailableprice,,see:HendershottandMoulton(2007)andHendershott,JonesandMenkveld(2007).17Inreality,
eitherpa+∆aforbuytrade(orpb−∆bforselltrades),thepenaltyformissingiiiiabuytradeatthebestpriceinoneofrequiredassetsiis∆,thearbitrageurwillbeleftwithapayoffofA−∆−∑∆i(wi)where∆i(wi)=i=1∆bifwii=−1and∆i(wi)=∆aifwii=,∑IA−∆i(wi)<=1Allarbitrageurshavetwopossiblestrategiesuponobservinganarbitrageopportunity,“totrade”or“nottotrade”.,arbitrageurs’strategies,,arbitrageurswillchoosewhethertoparticipateinexploitingarbitrageopportunitiesofaparticulardeviationsize,,ithenherexpectedpayoffE(Uj)isgivenby()∑IEUj=A−∆()i(wi)1−Pj.(1)ii=,suchthatA=pb1−Pa>0,andanarbitrageurfailingtogetthebestpriceini−=1,thentheprofit−∑Iofthearbitrageurwillbe:pb1wιpι(wι)−paai−∆i−∑wιpι(wι)=ι=2ι=i+1pb1−Pa−∆ai=A−∆
Equation(1)showsthatanarbitrageur’sexpectedpayoffdependsonthepriceslippage∆()(w)and1−Pj,,thearbitrageurfacesalossduetopriceslippageof∆i(wi)sotheterm∆()i(wi)1−∆i(wi),,,theprobabilityoftraderjexecutingamarketorderatthebestpriceforassetiisPji|ni(wi),k=ni,kwhereni(wi)|ni,khighlighttheroleofni(wi),,,theexpectedlossofarbitrageurjis()()∑IE)Lj=∆i(wi)1−ni(wi.(2)ki=1IfA≥E(Lj),itisoptimalforthetradertousethestrategy“trade”,theexpectedpayoffE(Uj)convergestoA−∑I∆i(wi)<0,=
Weassumethatarbitrageursadoptmixedstrategiesintheirarbitragestrategies,∈{1,...,k}bypij∈[0,1].ForamixedstrategyprofileΠ=(pi1,...,pik),weuseastandardnotationΠ−j=(pi1,...,pij−1,pij+1,...,pik),Π−j,inthenotationPji|ni,ktounde,Π−rlineitsdependenceonstrategiesoftraderj’sopponentsanddenotejP¯ji|ni,k=1−Pj,Π−ji|ni,k,Π−.jTheexpectedpayoffofarbitrageurjinthecaseofmixedstrategiesispijE(Uj|Π−j),whereaccordingtoProposition1,()∑IEUj|Π−j=A−∆i(wi)P¯ji|n(3)i,k,Π−ji=1istheexpectedpayoffofarbitrageurjplayingpurestrategy“trade”whileheropponentsusemixedstrategiesΠ−Π=(pi1,...,pik):(i)theprobabilityPjecreasesmoni|ni,k,Π−dotonicallywiththenumberofparticipatingarbi-jtrageursk;(ii)theexpectedprofitpijE(Uj|Π−j),’stheorem,thereexistsamixedstrategyprofileΠ
Π=(pi1,...,pik)withpij∈(0,1)isaNashequilibriumofthegame,thenpij=pij′=piforeachj,j′inKandtheobservedarbitragedeviationisalinearfunctionofthedifferencesbetweenthebestandthesecondbestpricesonthecorrespondingmarkets∑IA=∆i(wi)P¯i|ni,k,pi;(4)i=-neutralarbitrageursdemandanexpectedpayoffofzeroandhaveanidenticalprobabilityofparticipation,pi,,,,,,wepresenttheequilibriumprofitinamoregeneralform∑IA=φi(wi)P¯i|ni,k,pi,(5)i=1whereφ(5)showsthatthemagnitudeofthearbitragedeviationisassociatedwiththe15
,φi(wi),andthefailureprobabilityofexecutingthebestpricemarketorders,P¯i|ni,k,,theoptimalprobabilityofparticipationisalsoafunctionthearbitragedeviation,thebreadthoftheassetsupply,,theprobabilityofparticipationendogenizestheaveragenumberofactivelyparticipatingarbitrageurstopi×,arbitrageursinourmodelbehavestrategicallyandusetheprobabilityofparticipation,takingintoaccountactionsoftheiropponents,,,inordertoeliminatetheobservedarbitragedeviation,itisnecessarytoexecuteinaggregateallavailableunitsofatleastoneoftheassets{1,...,I}.Wedenotetheminimumbreadthofallassetsbyn=min{ni(wi)}.Inequilibrium,ifarbitrageursadoptmixedstrategies(.,thei∈Iarbitrageopportunityisnotlargeenoughtomakethefullparticipationoptimal),theywillparticipatewithprobabilitypi<,theprobabilitythatthearbitrageopportunitydisappearsimmediatelyfromthemarketisequaltotheprobabilitythatnormorearbitrageursoutofkdecidetocompete(whichcanbedescribedbythebinomialdistribution).Thus,,theprobabilitythatthearbitrageopportunitydisappearsimmediatelyis∑kP=kpis(1−pik−s)<1.(6)s=ns16
Thisresultshowsthatundercompetitionforscarcesupplyoftheassetsrequiredforthearbitrageportfolio,,,,(2009)(2009),,,,,,,itisnotguaranteed17
,,inthepresenceofcompetition,,arbitrageursdemandacompensationfortheexecutionriskandwillparticipateinarbitrageactivitieswithcertaintyonlyiftheobservedmispricingexceedstheexpectedlossgiveninEquation(2).Otherwise,arbitrageursdonotemploypurestrategy“trade”.Inthiscase,arbitrageursadoptmixedstrategieswithsomepositiveprobabilityofparticipationandthemispricingmightnotbeexploitedimmediately,,,,,’,
Finally,(.(2007),DevilleandRiva(2007),Akrametal.(2008),Fongetal.(2008),Marshalletal.(2008),etc.).Inequilibrium,,,weestablishthefollowingtestablehypotheses:1.“Riskless”,becausetriangu-lararbitrageisnon-convergencetradingbasedarbitrageandisnotsubjecttotaxes,regulatory,,
,,adirectpriceandanindirectprice(vis-a-visothercurrencies).(A/B)-ratebetweencurrenciesAandBisknownthroughthetwocurrencies’,thetriangularno-arbitrageconditionsare()()()SA/Bask≥SC/Bbid×SA/Cbid,(7)()()()SB/Aask≥SC/Abid×SB/Cbid.(8)AnydeviationfromEquation(7)or(8),Takayasu,MarumoandShimizu(2002)findexploitablearbi-trageopportunitiesthatlastabout90minutesadayintheFXmarketusingtransactiondatabetweenyen-dollar,dollar-euroandyen-eurofortheperiodJanuary25,1999toMarch12,.(2008)findstheexistenceofexploitablearbitrageopportunitiesusingoneyearbindingquotedatafromEBSandarguesthattheseopportunitiesaremoniesleftonthetabletocompensatearbitrageursfortheirserviceinrelievingmarket-maker’
andMoore(2009),,weusetickbytickdatafromtheReuterstradingsystemDealing3000forthreecurrencypairs:USdollarpereuro,USdollarperpoundsterling,poundsterlingpereuro(hereafterEUR/USD,GBP/USD,andEUR/GBP,respectively),2003toDecember30,(BIS,2004)estimatesthattradesinthesecurrenciesconstituteupto60percentoftheFXspottransactions,,,,,(2008),ContinuousLinkedSettlement(CLS)Bankwentintooperation,settlingtransactionsinvolvingsevencurrencies:theUSdollar,euro,yen,poundsterling,Swissfranc,,(2008),Lyons(2001)andRime(2003)
orderbook,,thedatasetreportsthecurrencypair,uniqueorderidentifier,price,orderquantity,hiddenquantity(D3000func-tion),quantitytraded,ordertype,transactionidentifieroforderenteredorremoved,statusofmarketorder,entrytypeoforders,removalreason,,andenablesustotrackalltypesoforderssubmittedthroughoutthedayandtoupdatethelimitorderbookforallentries,removals,amendments,,,
,wereportthepreliminarystatisticsofarbitragedeviationsandclusters(sequences),,thereare40,-triparbitrageopportunityasfollows:(
tradefees).25Furthermore,,(),,,,wetestfortherelationbetweenmarketilliquidity,,whichconfirmsfindingsbyAibaetal.(2002),Aiba,Takayasu,MarumoandShimizu(2003)andMarshalletal.(2008).However,thefinanceliteratureoftenassumesthattheseopportunities25TherearecostsinvolvedinobtainingaReuterstradingsystem,
:“Riskless”,,,(marketorder,limitorderandcancelation),,median,-testthattriangulararbitrage25
,-thousandthofasecond,,toinvestigatetheroleofdatalatency,,’,,
,.(1990),ShleiferandVishny(1992),AbreuandBrunnermeier(2002),GrombandVayanos(2002)andKondor(2009):,(I=3),-nityarises,
dothis,,hencethetotaldemandD=d×,,,eacharbitrageurhasaprobabilityofP=na1ktogetthecurrencyatthebestprice,,,|S|+,(thatislargerthanoneunit).,ifanarbitrageuronlymanagestobuyorsellcurrency1and2atthebestavailablepricesbutmissesoutoncurrency3becauseofexcessdemand,(currency3)atthenextavailablepriceorre-sellandre-buycurrency1and2(losingoutontransactioncosts)(thepayoffcanstillbepositiveifthenextbestpriceofcurrency3stillyieldsapositiveprofit).28
.(2008),,arbitrageopportunities,whicharenotimmediatelyeliminatedfromthemarketorwithadurationofmorethanasecond,,,(seeTable1)andasettingofeightarbitrageurs,eachwithademandofonemillionunit,,they29
(pi<1)mightbepreferredoverapurestrategy(pi=1),=∑∆i(wi)P¯i|ni,k,pi,whereAistheaverageobservedarbitragedeviationi=1duringacluster,∆i(wi)isanaveragedifferencebetweenthebestandthesecondbestpricesofthecorrespondingexchangerateduringthearbitragecluster,ni(wi)¯i|ni,k,piiscomputedaccordingtoEquation(12).Figure2showsthetimeseriesplotsoftheprobabilityofparticipationfortwo,eight,(6):
’,:PL=x0+x1×k.(9),,,costofinventoryandmarketilliquidityInequilibrium,-setprices(see,interalia,Stoll(1978),O’HaraandOldfield(1986),KumarandSeppi(1994),Chordia,RollandSubrahmanyam(2002)andreferencestherein).Morespecifically,
(2007),∆i,λiistheslopeofthecorrespondingsideofthelimitorderbook,,(completionofthethirdleg),shehasanunwantedinventoryandsheclearstheunwantedinventorybytradingitawayatthecheapestwayafter10,,,(negative)meanthatindicatesthatanarbitrageurmakes(loses)moneyacrossoursampleforholdingtheinventoryin10,,,standarddeviationofprofit/lossofthearbitrageurwhosystematicallymissesEUR/
,:=a0+a1×illiqGBP/USD(wGBP/USD)+a2×illiqEUR/USD(wEUR/USD)+a3×illiqEUR/GBP(wEUR/GBP)+b1×ICGBP/USD(wGBP/USD)+b2×ICEUR/USD(wEUR/USD)+b3×ICEUR/GBP(wEUR/GBP)(10)+c1×+c2×TED,whereilliqiisameasureofilliquidityinthemarketi,ICiisameasureofcostofinventoryinthemarketi,TEDisTEDspreadandi=GBP/USD,EUR/USD,EUR/,,asawideningTEDspreadreflectstheriskofdefaultoninterbankloans(counterpartyrisk).Fromourmodel,,morecompetingarbitrageurs,whoexertnegativeexternalitiesoneachother,,,seeCornelliandLi(2002),wecontrolfortheeffectofthenumberofarbitrageursinourregression,,(1976)andGrossmanandStiglitz(1980)ontheriskpremiumpaidtoariskadversearbitrageurfortheuncertaintyof33
(GMM,Hansen(1982))∆-statisticsshowthatalltheestimatedparametersaresignificantlydifferentfromzero,,-statisticsshowthatalltheestimatedparametersarepositiveandsignificantlydifferentfromzero,.(2007)inthatthemoreliquidthemarkets,,wearguethattheeconomicreasonbehindthisrelation,inthecaseofnon-convergencearbitrage,,-statisticsshowthatalltheestimatedparametersaresignificantlydifferentfromzero,,,
,,,,,“riskless”“risklessarbitrage”,,,
TablesandfiguresFigure1:,−∆biandthenextbestaskpriceispai+∆
Figure2:=∑I∆i(wi)P¯i|ni,k,pi,whereAistheaverageobservedarbitragedeviationduringacluster,∆i(wi)isani=1averagedifferencebetweenthebestandthesecondbestpricesofthecorrespondingexchangerateduringthearbitragecluster,ni(wi)¯i|ni,k,piaccordingtoEquation(12).
Figure3:(6)=∑I∆i(wi)P¯i|ni,k,pi,whereAistheaverageobservedarbitragedeviationduringacluster,∆i(wi)isanaveragei=1differencebetweenthebestandthesecondbestpricesofthecorrespondingexchangerateduringthearbitragecluster,andni(wi)¯i|ni,k,piaccordingtoEquation(12).
Figure4:ExpectedProfitComparedtotheNumberofArbitrageursThefigurepresentstherelationbetweenthemeanofarbitrageurs’,2003toDecember30,
Table1:PreliminaryDataAnalysis:LiquidityThetableprovidesdescriptivestatisticsontheaverageinter-limitorderduration(inseconds),averagebid-askspread(inpips),averageslopeofthedemandandsupplyscheduleinbasispointsperbillionofthebasecurrency,theaveragedepth(inmillionofbasecurrency)andtheaveragedifferencebetweenthebestandthesecondbestprices(inpips)fortheEUR/USD,GBP/USDandEUR/,2003toDecember30,
Table3:-portsstatisticsofarbitrageclustersthatareeliminatedbyanynextincomingorder(marketorders,limitordersandcancelation),,,2003toDecember30,,00125,165t-stat(textbookarbitragedur.=riskyarbitragedur.)(textbookarbitragedur.=riskyarbitragedur.):,2003toDecember30,
Table5:LatencyCostsThetablepresentsdescriptivestatisticsandt-statisticsoftheprofitsfromexploitingtriangulararbitragede-viationsbyamonopolisticarbitrageur(nocompetition)..“WithoutLatency”columnreportsstatisticsofarbitrageprofitsunderassumptionsthatallobservedarbitragedeviationsareexploitedimmediately.“WithLatency”,2003toDecember30,,265,,438,-stat(averageprofit=zero),166t-stat(profitwithoutlatency=profitwithlatency):ArbitrageStrategyAverageProfitsThetablepresentsthemeanandstandarddeviation(inparentheses)ofarbitrageurs’,2003toDecember30,==========()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()43
Table7:ArbitrageStrategyAverageProfitsComparedtotheNumberofTradersThetablepresentstheregressionestimatesx1ofEquation(9),PL=x0+x1×-statistics(inparentheses),,2003toDecember30,==========()()()()()()()()()()%%%%%%%%%%Table8:DescriptiveStatisticsforProfit/,,2003toDecember30,
45Table9:TheEffectofInventoryRiskonArbitrageDeviationThetablelistscoefficientestimatesfromregressionofarbitragedeviationonmeasuresofliquidity,,166triangulararbitrageclustersamongEUR/USD,GBP/USDandEUR/GBPexchangeratesappearedbetweenJanuary2,2003toDecember30,∆GBP/USD,∆EUR/USDand∆EUR/GBPareaveragedifferencebetweenthebestandsecondbestpricesofthecorrespondingexchangerateswithineachcluster.λGBP/USD,λEUR/USDandλEUR/(dependsonwhetherapurchaseorsalesofthedirectcurrencypriceisinvolvedinthetransaction).ICGBP/USD,ICEUR/USDandICEUR/GBParestandarddeviationofprofit(orloss)-spread,defineasthespreadbetweenthethree-monthinterestratebankschargeeachother(intheeuro-dollarmarket)overthethree-monthTreasurybills,-statisticsaregiveninparenthesesandareadjustedforautocorrelation.∆GBP/USD∆EUR/USD∆EUR/GBPλGBP/USDλEUR/USDλEUR/GBPICGBP/USDICEUR/USDICEUR/()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()
46Table10:TheEffectofInventoryRiskonArbitrageDeviation:RobusnessCheck(LaggedVariables)Thetablelistscoefficientestimatesfromregressionofarbitragedeviationonmeasuresofliquidity,,166triangulararbitrageclustersamongEUR/USD,GBP/USDandEUR/GBPexchangeratesappearedbetweenJanuary2,2003toDecember30,∆GBP/USD,∆EUR/USDand∆EUR/GBPareaveragedifferencebetweenthebestandsecondbestpricesofthecorrespondingexchangeratesoveranhourprecedingthearbitragecluster.λGBP/USD,λEUR/USDandλEUR/GBParetheaverage(overanhourprecedingthearbitragecluster)(dependsonwhetherapurchaseorsalesofthedirectcurrencypriceisinvolvedinthetransaction).ICGBP/USD,ICEUR/USDandICEUR/GBParestandarddeviationofprofit(orloss)-spread,defineasthespreadbetweenthethree-monthinterestratebankschargeeachother(intheeuro-dollarmarket)overthethree-monthTreasurybills,-statisticsaregiveninparenthesesandareadjustedforautocorrelation.∆GBP/USD∆EUR/USD∆EUR/GBPλGBP/USDλEUR/USDλEUR/GBPICGBP/USDICEUR/USDICEUR/()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()()
=,(inthemarket2)butwinningthebestpriceinthemar()()ket2(inthemarket1)isPj21−Pj1(respectively,Pj11−Pj2).Sheearns,inthiscase,theprofitA−∆1(w1)(A−∆2(w2),respectively).With()()theprobability1−Pj11−Pj2,thearbitrageurfailstogetbestpricesinbothmarketsandearnsA−∆1(w1)−∆2(w2).Itiseasytocheckthattheexpectedprofitofthearbitrageurfromthe“trade”strategyis()()()EUj=APj1Pj2+(A−∆1(w1))Pj21−Pj1+(A−∆2(w2))Pj11−Pj2+(A−∆()()1(w1)−∆2(w2))1−Pj11−Pj2()()=A−∆j1(w1)1−Pj1−∆2(w2)1−−1markets,thatis,IE(Uj)=A−∑−1∆()ji(wi)1−={1,...,I},\Jmarkets,herpayoffwillbeA−∑∆i(wi).Thei∈JprobabilityoftheeventthatthetraderfailsexactlyineachofJmarketsandwinsthebestpricesallothermarketsisequalto∏∏P¯j·∈Ji∈I\J47
∑()E(Uj)=∑∏()∏A−∆i(wi)·1−Pj·PjiιJ∈2Ii∈Ji∈Jι∈I\J∑()()()∏=A∏∏1−Pj·Pj∑ι−∑∏∆i(wii)·1−Pj·PjiιJ∈2Ii∈Jι∈I\JJ∈2Ii∈Ji∈Jι∈I\J∑()()∑∏∏=A−∆i(wi)·1−Pj·Pjiι.(11)J∈2Ii∈Ji∈Jι∈I\JIntheaboveexpressionweusedtheequality∑∏()∏∏I()1−Pj·Pjiι=(1−Pj)+Pj=∈2Ii∈Jι∈I\Ji=1Letusdecomposethelastsumoftheequality(11)intothetermcontaining∆I(wI)andnotcontaining∆I(wI).Thisgives∑(∑)∏()∏Ej(Uj)=A−(1−PI)∆I(wI)+∆i(wi)·1−Pj·PjiιJ∈2I\{I}i∈Ji∈Jι∈I\J∑(−Pj∑)∏()∏wjI∆i(i)·1−Pj·PiιJ∈2I\{I}i∈Ji∈J(ι∈I\J∑)()I−1∑I−1=A−∆PjjI(wI)(1−PjI)−(1−I)∆ji(wi)(1−Pj)−P∆iIi(wi)(1−P)i∑i=1i=1I=A−∆i(wi)(1−Pj).ii=(i)Let2K−jdenotesafamilyofallsubsetsofthesetK−,suchthatS∈2K−j,where|S|∈2K−jmeansthatallopponentsoftraderjfromthesubsetSparticipateinthemarketwithcertaintyandtherestK−j\
TheexpressionfortheprobabilityP¯joffailingtogetthebestpriceinthemarketi|ni,k,Π−∈2K−jeachofwhichmeansthatallopponentsoftraderjfromthesubsetSparticipateinthemarketwithcertaintyandtherestK−j\∏∏P(XS)=pis(1−pis)s∈Ss∈K−j\SandtheprobabilityoffailingtogetthebestpriceinthemarketifortraderjconditionalonXSis0,|S|≤nj=i(wi)−1P¯i|ni,|XS|1−ni(wi)|S|+1,|S|>ni(wi)−1sincethereareonly|S|∑∑∏∏P¯jP¯j·P(Xpii|ni,k=,Π−ji|ni,|S|S)=s(1−pis)P¯ji|ni,|S.(12)|S∈2K−jS∈2K−js∈Ss∈K−j\SLetusnowaddonemorearbitrageurk+1intothemarketwhoplayshermixedstrategypik+′={1,...,k+1}.LetthenewmixedstrategyprofilebeΠ′={pi1,...,pik,pik+1}.Inordertoprovethesecondstatementofthetheorem,weneedtoshowthatP¯ji|ni,k+1,Π′>P¯jachj∈Kandi∈,k,Π−foreI.−ji|nijLetusdecomposethesuminEquation12intothepartwithsubsetsScontainingthearbitrageurk+1andnotcontainingher.∑∏∏∑∏∏P¯j=pipii|ns(1−i,k+1,Π′−s)P¯j+piji|ni,|S|+1s(1−pis)P¯ji|ni,|S|S∈2K−j∑s∈S∪{∏k+1}s∏∈K−j\SS∈2K−js∈S∑s∈K−′∏j∪{k+1}∏\S=pik+1pis(1−pis)P¯j1−pii|nk+1)pis(1−pis)P¯ji,|S|+1+(i|ni,|S|S∈∑2K−js∏∈Ss∈K∏−j\S∑S∈2K−∏js∈Ss∏∈K−′j\S>pik+1pis(1−pis)P¯jpii|ni,|S+(1−pi|k+1)s(1−pis)P¯ji|ni,|S|S∈2K−js∈Ss∈K−j\SS∈2K−js∈Ss∈K−′j\S=P¯ji|ni,k,Π−.j49
ThestrictinequalityappearsduetoAssumption3.(ii)Thestatementisaconsequenceof(i)Π(Uj|Π−j).IfE(Uj|Π−j)>0,itcontradictstothedefinitionofNashequilibriumsincetraderjcanalwayschooseastrategypi′j>(Uj|Π−j)<pi′jE(Uj|Π−j).Ontheotherhand,conditionE(Uj|Π−j)<0cannotbetrueinequilibriumasthestrategy“nottotrade”withpij=,inequilibrium,zeroprofitconditionE(Uj|Π−j)=×2subgameplayedbytwoarbitrarilychosentradersjandj′.ThesubgamehasaformTRADERj′01TRADERj00,00,y1y,0z,zydenotesthepayoffofthetraderchoosing“trade”(pi=1)andtheothertraderinthesubgamechoosing“nottrade”(pi=0)whiletheremainingk−2arbitrageurssticktothemixedstrategyprofileΠ−j,j′.zdenotesthepayoffoftradersjandj′whentheybothchooseto“trade”.AccordingtoProposition2,thepayoffzissmallerthanyasthereisonemoreparticipatingopponentj′.Inequilibrium,eachtradermustbeindifferentbetweenusing“trade”or“nottrade”strategies,sopijz+(1−pij)y=0,pij′z+(1−pij′)y=0whichimplies(pij−pij′)(z−y)=−y>0,wegetpij=pij′.Asarbitrageursjandj′50
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