Toxic Arbitrage

42 Pages·2017·0.55 MB·English

Toxic Arbitrage
ThierryFoucault
HEC,Paris
RomanKozhan
WarwickBusinessSchool,UniversityofWarwick
WingWahTham
SchoolofBankingandFinance,UniversityofNewSouthWales
D
o
w
n
Short-livedarbitrageopportunitiesarisewhenpricesadjustwithalagtonewinformation. lo
a
Theyaretoxicbecausetheyexposedealerstotheriskoftradingatstalequotes.Hence, d
e
tthheeoseryopimpoprltiuensittiheastsmhoourledfirmeqpuaiernltiqtuoixdiictya.rWbietrapgroevoidpeposurtpupnoitriteinsgaenvdidfeanstceerurseisnpgodnasetasotno d fro
m
triangulararbitrage.Aspredicted,illiquidityishigherondayswhenthefractionoftoxic h
arbitrageopportunitiesandarbitrageurs’relativespeedarehigher.Overall,ourfindings ttp
s
sinucgrgeeasstedthaadtvtehresepsreicleecetifofincireisnkc.y (gJaEinLoDf5h0i,gFh3-1fr,eGqu1e0n)cy arbitrage comes at the cost of ://ac
a
d
e
m
ReceivedMarch25,2015;editorialdecisionSeptember21,2016byEditorAndrewKarolyi. ic
.o
u
p
.c
o
Arbitrageursplayacentralroleinfinancialmarkets.Whenthelawofoneprice m
(LOP)breaksdown,arbitrageursstepintobuycheapassetsandsellexpensive /rfs
/a
ones.InthiswaytheyenforcetheLOPandmakemarketsmorepriceefficient.In rtic
theory,arbitrageopportunitiesshoulddisappearinstantaneously.Inreality,they le
-a
b
s
tra
c
Wearegratefultotheeditor,AndrewKarolyi,andtwoanonymousrefereeswhosecommentshelpedtoimprove t/3
thepaper.WealsothankMarkVanAchter,YacineAït-Sahalia,HankBessembinder,GeirBjønnes,Michael 0
BCraernonleanG,rAeslsaei,nTCerhraybHouedn,dePrisehrroett,CJoolhliann-DHuofmrebsenrets,,BJoebanJ-aErrdoowu,aFrdraCnkoldliearJdo,nMg,aPtethteijsKFyllee,isGchraecr,eAXriinegGHouz,luOkllgua, /4/10
Lebedeva,BruceLehmann,KatyaMalinova,AlbertMenkveld,MichaelMoore,PamelaMoulton,Maureen 53
O’Hara,MarcoPagano,AndreasPark,JoëlPeress,AngeloRanaldo,VikasRaman,DagfinnRime,FabriceRiva, /2
GideonSaar,MarkSalmon,DanielSchmidt,ElviraSojli,ClaraVega,KumarVenkataraman,ChenYao,and 75
MaoYe.WearegratefultoseminarandconferenceparticipantsatBathUniversity,CornellUniversity,EIEFand 8
6
Consob,theNorwegianSchoolofBusiness,VrijeUniversity,ManchesterUniversity,KULeuven,theAutorité 3
5
desMarchésFinanciers,the2015AmericanFinanceAssociationmeetings,the8thconferenceofthePaulWooley b
CenterforthestudyofCapitalMarketsDysfunctionnality,theworkshoponmarketmicrostructuretheoryand y
applicationsatCambridge,the9thCentralBankWorkshopontheMarketMicrostructureofFinancialMarkets, gu
theBIRSworkshoponmodelinghigh-frequencytradingactivity,the6thErasmusLiquidityConference,the es
ConferenceonLiquidityandArbitrageTradinginGeneva,andtheCityUFinanceConferenceinHongKong. t o
ThierryFoucaultacknowledgesfinancialsupportfromtheInvestissementsd’AvenirLabex(ANR-11-IDEX- n
0003/LabexEcodec/ANR-11-LABX-0047). 03
SupplementarydatacanbefoundonTheReviewofFinancialStudieswebsite.SendcorrespondencetoThierry A
Foucault,HEC,Paris,1ruedelaLibération,Jouy-en-Josas,78351,France;telephone:[phone].E-mail: p
[email protected]. ril 2
0
1
9
©TheAuthor2016.PublishedbyOxfordUniversityPressonbehalfofTheSocietyforFinancialStudies.
Allrightsreserved.ForPermissions,pleasee-mail:[email protected].
doi:10.1093/rfs/hhw103 AdvanceAccesspublicationDecember30,2016
TheReviewofFinancialStudies/v30n42017
donotbecausearbitrageisnotfrictionless.AsDuffie(2010)pointedout:“The
arrivalofnewcapitaltoaninvestmentopportunitycanbedelayedbyfractions
ofasecondinsomemarkets,forexampleanelectroniclimitorder-bookmarket
forequities,orbymonthsinothermarkets[...].”
Variousfrictions(e.g.,short-sellingcosts,fundingconstraints,idiosyncratic
risks) explain why some arbitrage opportunities persist (see Gromb and
Vayanos2010).Forshort-livedarbitrageopportunities–thoselastingfractions
of a second– attention costs and technological constraints are the main
impediments to a seamless law of one price. These barriers are falling
as high-frequency arbitrageurs massively invest to detect and exploit ever
fasterarbitrageopportunities.Returnsonhigh-speedarbitragearesubstantial
because arbitrage opportunities are frequent at the timescale of milliseconds D
o
w
(see Budish, Cramton, and Shim 2015).This evolution has triggered debates n
lo
aboutthesocialvalueofhigh-speedarbitrageandinparticularaboutwhether a
d
arbitrage strategies “benefit or harm the interests of long-term investors and ed
market quality [...]” (U.S. Securities and Exchange Commission (2010), fro
m
SectionB,51). h
Arbitrageurs can be beneficial or harmful for other investors, depending ttp
s
on the cause of arbitrage opportunities. When these opportunities are due to ://a
c
transientdemandorsupplyshocks(“pricepressures”),arbitrageursimplicitly a
d
e
act as liquidity providers in exploiting them (see, for instance, Holden 1995; m
GrombandVayanos2002,2010).Inthiscase,tradesbetweenarbitrageursand ic.o
their counterparties are mutually beneficial.1 However, short-lived arbitrage up
.c
opportunitiesalsoareduetoasynchronousadjustmentsinassetpricesfollowing o
m
information arrival. Arbitrageurs’ profits in these trades are obtained at the /rfs
expense of dealers with stale quotes.2 Thus, asynchronous price adjustments /a
to information in asset pairs generate “toxic” arbitrage opportunities, that is, rticle
opportunitiesinwhichdealersareatriskofbeingadverselyselected.3 High- -a
b
s
speed arbitrageurs can harm market liquidity through this channel because tra
dealerschargelargerbid-askspreadstocovertheriskoftradingatstalequotes ct/3
(CopelandandGalai1983). 0
/4
Our contribution is to model this channel and provide evidence of its /1
0
importanceforliquidity.Toourknowledge,ourpaperisthefirsttodoso.Thisis 53
/2
importantforatleasttworeasons.First,arbitrageisacentralnotioninfinance. 7
5
Thus, understanding how it generally affects market quality, not just pricing 86
3
5
b
y
g
u
1 Forinstance,GrombandVayanos(2002,362)write:“Inourmodel,arbitrageactivitybenefitsallinvestors.This e
s
isbecausethroughtheirtrading,arbitrageursbringpricesclosertofundamentalsandsupplyliquiditytothe t o
market.” n
0
2 Thisproblemisnotnew.Forinstance,inthe90s,professionaldaytraders(so-called“SOESbandits”)were 3 A
pickingoffNasdaqdealerswithstalequotesbyusingNasdaq’sSmallOrderExecutionSystem(asystemthat p
gSucahrualntzte(e1d99au7t)oamndatiFcoeuxceacuultt,ioRnöoefllmanadrkSeatnodrdåesr(s2u0p03to).acertainsizeatNasdaqdealers’quotes).SeeHarrisand ril 2
0
1
3 OurdefinitionofatoxictradefollowsEasley,LopezdePrado,andO’Hara(2012,1,458):“Orderflowisregarded 9
astoxicwhenitadverselyselectsmarketmakerswhomaybeunawarethattheyareprovidingliquidityataloss.”
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ToxicArbitrage
efficiency,isofbroadinterest.Second,recentproposalsadvocateslowingthe
pace of trading precisely on the grounds that high-speed arbitrageurs raise
dealers’risk of trading at stale quotes (see, e.g., Budish, Cramton, and Shim
2015).However,thereisyetnoevidenceonwhetherarbitrageurs’contribution
to this risk is significant. Measuring this contribution is not straightforward
because it is not the level of arbitrage activity per se that should affect
dealers’risk of trading at stale quotes (and therefore their spreads). Rather,
as shown by our model, this risk is determined both by the “arbitrage mix”
(i.e.,theproportionoftoxicarbitrageopportunitiesinthepoolofallarbitrage
opportunities)andbyarbitrageurs’relativespeed ofreactiontotoxicarbitrage
opportunities.Specifically,illiquidityshouldbehigherinperiods(orforasset
pairs) in which (a) the fraction of toxic arbitrage opportunities is higher D
o
w
(the arbitrage mix is more toxic) or (b) the likelihood that a toxic arbitrage n
lo
opportunity terminates with an arbitrageur’s trade is higher (arbitrageurs are a
d
relativelyfaster). ed
Thesetwopredictionsfollowfromanewmodelofcross-marketarbitragethat fro
m
wedevelopinthefirstpartofourpaper.Inthemodel,arbitrageopportunities h
canbeeithertoxic(duetoasynchronouspriceadjustmentstonews)ornontoxic ttp
s
(duetoliquidityshocks).Asinreality,anarbitrageopportunityterminateseither ://a
c
withanarbitrageur’stradeoradealer’squoteupdate,dependingonwhoever a
d
e
observestheopportunityfirst.Wesolveforequilibriumbid-askspreadsineach m
assetandtraders’optimalspeedofreactiontoarbitrageopportunities.Thus,in ic.o
u
equilibrium,illiquidityandthedurationofarbitrageopportunities(ameasure p
.c
ofpricingefficiency)arejointlydetermined. o
m
The model generates the above predictions and two additional predictions /rfs
about the durations of arbitrage opportunities. First, when the arbitrage mix /a
becomes more toxic, arbitrage opportunities should be shorter, even though rticle
bid-ask spread costs of arbitrage are higher. The reason is that dealers react -a
b
s
faster to arbitrage opportunities (by updating their quotes) when they expect tra
moreofthemtobetoxic.Thiseffectinducesarbitrageurstobefasteraswell ct/3
and, as a result, arbitrage opportunities are more short lived. Second, by a 0
/4
similarlogic,atechnologicalchangethatmakesarbitrageursrelativelyfaster /1
0
shouldreducethedurationofarbitrageopportunities,eventhoughitincreases 53
/2
illiquidity. 7
5
Wetestthesepredictionsusingdataontriangulararbitrageopportunitiesfor 86
3
three currency pairs (dollar-euro, dollar-pound, and pound-euro).4 Although 5
b
ourpredictionsandmethodologyapplytoanytypeofhigh-frequencyarbitrage y g
u
opportunities, we focus on triangular arbitrage opportunities for a couple of e
s
reasons. t o
n
0
3
A
p
ril 2
4 Onecanbuyeuroswithdollars,exchangetheeurosagainstpounds,andthenexchangepoundsagainstdollars.If 01
thisstrategyisprofitablethenatriangulararbitrageopportunityexists.Wedefinetriangulararbitrageopportunities 9
formallyinSection2.2.
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TheReviewofFinancialStudies/v30n42017
The first one is practical. For our tests, we must accurately measure when
an arbitrage begins, when it terminates, and how it terminates (with a trade
oraquoteupdate),andwemusttrackpricesafterthearbitrageterminates(to
identifytoxicarbitrageopportunities;seebelow).Ourdatahavetherequired
granularity for this analysis: we observe all orders and trades for currency
pairsinoursamplefromJanuary2003toDecember2004inReutersD-3000
(oneofthetwomajorinterdealertradingplatformsusedbyforeignexchange
dealing banks) with a time stamp accuracy of 10 milliseconds.5 Moreover,
asynchronicities in price reporting for different assets are not an issue in our
databecausealldataaregeneratedbythesametradingplatform.
Second, strategies exploiting triangular arbitrage opportunities are not
hindered by taxes, short selling, or funding constraints, and the risk of D
o
w
these strategies is limited. Hence, standard limits to arbitrage cannot explain n
lo
whytriangulararbitrageopportunitiesarenoteliminatedinstantaneously(see a
d
Pasquariello 2014).6 The most likely explanation is that, as in our model, ed
technological constraints limit the speed at which traders react to arbitrage fro
m
opportunities.Thus,triangulararbitrageopportunitiesaresimilartootherhigh- h
speed opportunities: (1) they are frequent (we observe more than 172,044 ttp
s
in our sample), (2) they are short lived (in our sample, 25% of all arbitrage ://a
c
opportunitieslastlessthanhalfasecond),(3)theyaremoreefficientlyexploited a
d
e
bymachinesthanbyhumans,and(4)theydeliverthinprofitsperopportunity m
(0.6to0.7basispointsinoursample).7 ic.o
u
Asanyotherarbitrageopportunities,triangulararbitrageopportunitiesarise p
.c
fortworeasons:(1)asynchronouspriceadjustmentsofdifferentcurrencypairs o
m
tonewinformationor(2)pricepressureseffects.Pricepressureeffectsgenerate /rfs
price reversals, and asynchronous price adjustments to information generate /a
staggeredpricemovementsinthesamedirectionforrelatedassets.8 Thus,as rticle
in Schultz and Shive (2010), we use price patterns following the occurrence -a
b
s
ofarbitrageopportunitiestosortthemintotwogroups:toxic(characterizedby tra
staggeredpricemovementsinthesamedirectionfollowingtheoccurrenceof ct/3
a triangular arbitrage opportunity) and nontoxic (characterized by a reversal 0
/4
/1
0
5
3
/2
5 KozhanandTham(2012)usedthesamedatatomeasuretheprofitabilityoftriangulararbitrageopportunities. 75
8
6 Pasquariello(2014)foundthatnoneoftheusualproxiesforlimitstoarbitrageexplainthesizeoftriangular 63
5
arbitrageopportunities(Pasquariello2014,seeTable2).Forotherrelativelyshort-livedopportunities,theselimits b
canbemoreimportant.Forinstance,GagnonandKarolyi(2010)foundthatholdingcosts(e.g.,idiosyncratic y
risk)explainthesizeofarbitrageopportunitiesbetweenhomeandU.S.stockpricesforstockscross-listedinthe gu
UnitedStates. e
s
7 Arbitrageopportunitiesintheforeignexchangemarket(eitherviolationsofcoveredinterestparityortriangular t on
arbitrage)arewelldocumented.See,forinstance,Akram,Rime,andSarno(2008),Fong,Valente,andFung 0
(2008),Fennetal.(2009),Mancini-GriffoliandRanaldo(2011),Marshall,Treepongkaruna,andYoung(2008), 3 A
KozhanandTham(2012),Itoetal.(2013),Chaboudetal.(2014),andPasquariello(2014). p
8 Forinstance,forcross-listedstocks,GagnonandKarolyi(2009)showedthatthereisnegativeautocorrelationin ril 2
homeandforeignreturnsatthedailyfrequency.Thisnegativeautocorrelation,however,isweakerforstocksin 01
whichinformedtradingismoreintense.Thisisconsistentwiththeideathatdelaysinadjustmenttoinformation 9
forassetswithcorrelatedpayoffs(e.g.,cross-listedstocks)inducepositivespilloversinpricechanges.
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ToxicArbitrage
intherateofthecurrencypairthattriggersthearbitrageopportunity).9 With
this approach, we obtain 83,488 toxic arbitrage opportunities (about 112 per
day),thatis,about48%ofallarbitrageopportunitiesinthesample.Onaverage
(acrossalldaysinoursample),wefindthattheseopportunitiesterminatewith
anarbitrageurs’tradeinabouttwo-thirdsofthecases.Thus,arbitrageursoften
arefasterthandealersinoursample.
Arbitrageurs’relativespeedisendogenoustoilliquiditybecausearbitrageurs
havelessincentivetoquicklydetectarbitrageopportunitieswhentransactions
costs are high. To account for this in our tests, we use an instrument for
arbitrageurs’relative speed (measured by the frequency with which a toxic
arbitrageopportunityterminateswithatrade).UntilJuly2003,tradersmanually
submitted their orders to Reuters D-3000. In July 2003, Reuters introduced D
o
w
the “AutoQuote API” functionality (API means “Application Programming n
lo
Interface”).Tradersusingthisfunctionalitycanautomateordersubmissionby a
d
directlyfeedingtheiralgorithmstoReutersD-3000.Asaresult,theirmonitoring ed
costsarereduced.ArbitrageurswereamongthefirsttouseAutoquoteAPI(see fro
m
Chaboudetal.2014),andthissuggeststhatthereductioninmonitoringcosts h
mainlyaccruedtothem.10Inourmodel,thisimpliesthatarbitrageurs’relative ttp
s
speedshouldincreasefollowingtheintroductionofAutoquoteAPI.Thus,we ://a
c
instrument arbitrageurs’relative speed withAutoQuoteAPI. In line with our a
d
e
conjecture, the first stage of the IVregression shows that the introduction of m
AutoquoteAPI had a significant positive effect on the likelihood that a toxic ic.o
u
arbitrageopportunityterminateswithatrade. p
.c
Moreimportantly,thesecondstageshowsthat,aspredicted,thelikelihood o
m
thatatoxicarbitrageopportunityterminateswithatradehasapositiveeffecton /rfs
illiquidity.Forinstance,a1%increaseinthislikelihoodinadayisassociated /a
with a 0.063-bps increase in quoted bid-ask spreads in this day (2.3% to 5% rticle
oftheaveragebid-askspreaddependingonthecurrencypair).Theeconomic -a
b
s
sizeofthiseffectissignificantgiventhedailytradingvolumeforthecurrency tra
pairs in our sample (we estimate that a 0.063 basis points increase in quoted ct/3
spreadraisesthetotalcostoftradingforthecurrencypairsinoursampleby 0
/4
/1
0
5
3
9 Supposethateuro/dollardealersreceiveinformationthatcallsforanappreciationoftheeuroandraisetheir /27
bidandaskquotes(expressedindollarspereuro).Ifthisappreciationislargeenoughanddealersin,say,the 58
dollar/poundmarketareslowtoadjusttheirquotes,thereisatriangulararbitrageopportunity:onecanindirectly 63
buydollarswitheurosatapricelessthanthecurrentbidpriceinthedollar/euromarket.Thistoxicarbitrage 5
opportunityvanisheswhendealersinthedollar/poundmarketraisetheirquotesorarbitrageurshitstalequotes by
inthismarket.Ineithercase,therateinthedollar/poundmarketadjustsinthedirectionoftheshiftinthe g
u
euro/dollarmarket.Alternatively,ifeuro/dollardealerstemporarilyaccumulatealargeshortpositionineuro, e
tIhfethyiwspilrlimceaprkreuspsuthreeevfafleucetoisfltahregeeuernooauggahin,satnthoentdooxlilcartrtioaantgtrualcatrsaerlbleitrrsagoefeouprpoosratunnditryedaurciseetsh.eAirsrdisekaleexrps’osshuorert. st on
positiondecreases,theirquoteswillrevert(see,forinstance,GrossmanandMiller1988). 0
3
10 Hendershott,Jones,andMenkveld(2011)usetheimplementationoftheNYSE“autoquote”softwarein2003as A
p
AanPIinbsetrcuamuseentthfeorfoarlmgoerritahumtoimctartaedsinthge.dTihseseNmYinSaEtioanutoofquupodteatfeusnicntiboensatliqtuyoitsesdiffoferrNenYtSfrEomstoRckesutwerhsilAeuthtoeQlautoteter ril 2
automatesorderentry.Automationoforderentryclearlyacceleratesthespeedatwhichtradersreacttomarket 01
events.WediscussthedifferencesbetweenourfindingsandthoseinHendershott,Jones,andMenkveld(2011) 9
inSection3.1.
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TheReviewofFinancialStudies/v30n42017
about$131,319perday).Wefindsimilareffectswhenwemeasureilliquidity
witheffectivespreads,theslopeoflimitorderbooks,orameasureofadverse
selectioncostsfordealers.
Moreover, consistent with our first prediction, we also find a positive and
significantrelationbetweenthedailyfractionoftoxicarbitrageopportunities
andilliquidity.Specifically,ondaysforwhichthisfractionishigher,illiquidity
ishigher,aftercontrollingforthenumberofarbitrageopportunities(scaledby
thenumberoftrades)andstandarddeterminantsofilliquidity.Forinstance,a
one-standard-deviationincreaseinthefractionoftoxicarbitrageopportunities
ononedayisassociatedwitha2.3%increaseintheaveragequotedspreadfor
thecurrenciesinoursampleonthesameday.Thus,thearbitragemixmatters:
illiquidity is higher when arbitrage opportunities are more frequently due to D
o
w
asynchronouspriceadjustmentsthantopricepressures. n
lo
Insum,consistentwithourpredictions,illiquidityispositivelyrelatedto(1) a
d
thefractionoftoxicarbitrageopportunitiesand(2)arbitrageurs’relativespeed. ed
Additionalpredictionsofourmodelaresupportedbythedataaswell:(a)the fro
m
durationofarbitrageopportunitiesisshorterondaysinwhichthefractionof h
toxic arbitrage opportunities is higher and (b) the introduction ofAutoQuote ttp
s
API(anincreaseinarbitrageurs’relativespeed)coincideswitha6.7%(about ://a
c
115milliseconds)decreaseintheaveragedurationofarbitrageopportunities a
d
e
inoursample. m
It is well known that liquidity facilitates arbitrage. The reverse relation– ic.o
u
theeffectofarbitrageursonliquidity(ourfocushere)–hasreceivedmuchless p
.c
attention.11 Kumar and Seppi (1994) modeled cross-market arbitrageurs as o
m
informedtradersandshowedthattheeffectofthenumberofarbitrageurson /rfs
liquidity is nonmonotonic. They did not study how the arbitrage mix affects /a
illiquidity (all arbitrage opportunities are due to stale quotes in their model) rticle
andtraders’speedisnotachoicevariableintheirmodel.Hence,theydidnot -a
b
s
derivethepredictionsthatwetestinthispaper. tra
Severalpapershavearguedthatfasttradersraiseadverseselectioncostsfor ct/3
slow traders.12 Our empirical findings about the effect of arbitrageurs’speed 0
/4
areconsistentwiththisview.Themainmessageofourpaper,however,isnot /1
0
thatadverseselectionisasourceofilliquidity.This,ofcourse,iswellknown. 53
/2
Instead,ourcontributionistoshowthathigh-speedarbitragecanbeasourceof 7
5
adverseselectionandthat,forthisreason,the“arbitragemix”inanassetpairis 86
3
adeterminantofitsliquidity.Thesenewfindingscontributetotheburgeoning 5
b
y
g
u
e
11 Rbaoslils,aSncdhwstaorctkz,maanrdkeStulbiqrauhidmitayn.yInampa(r2ti0c0u7la)r,shaogwreeadtetrhiantdtehxerfeuteuxriesstbtwasois-wGaryanrgeelar-ticoanussebsegtwreeaetnerisntdoecxkfmuaturkreest st on
illiquidity.Roll,Schwartz,andSubrahmanyam(2007)arguedthatthiseffectcouldbeduetoarbitrageurs’s 0
tradesbutdonotspecificallyshowthatthesetradesexplaintherelation.Rosch(2014)usedthesizeofarbitrage 3 A
opportunitiesinDepositaryReceiptsasaninverseproxyforarbitrageactivityandfindsapositiveassociation p
betweenarbitrageactivityandliquidity. ril 2
12 See,forinstance,Biais,Foucault,andMoinas(2015),Brogaardetal.(2015)Budish,Cramton,andShim(2015), 01
Foucault,RöellandSandås(2003),Foucault,Hombert,andRosu(2016),GarveyandWu(2010),Hendershott 9
andMoulton(2011),Hoffmann(2014),JovanovicandMenkveld(2012),orMenkveldandZoican(2014).
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literatureonshort-livedarbitrageopportunitiesinvariousassetpairs,suchas
currencies,ETFs,cross-listedstocks,anddualclassshares.(see,forinstance,
Akram, Rime, and Sarno 2008; Ben-David, Franzoni, and Moussawi 2012;
GagnonandKarolyi2010;orSchultzandShive2010).
1. Illiquidity,ArbitrageMix,andArbitrageurs’RelativeSpeed
Inthissection,wepresentthemodelofcross-marketarbitragethatguidesour
empiricalanalysis.Beforedescribingitformally,itisworthoutliningitsmain
ingredients and why these are required for our analysis. The model has two
assetswithidenticalpayoffs.Quotesforeachassetarepostedbytwodifferent
marketmakers.Togeneratearbitrageopportunities,weassumethatthemarket D
o
makerinoneassetcanreceivearandomshocktohisvaluationforthisasset, w
n
eitherduetonewsarrivalorliquidityneeds.Thisfeatureenablesustostudyhow loa
d
thelikelihoodthatanarbitrageistoxic(i.e.,duetonewsarrival)affectsmarket e
d
makers’bid-askspreads(illiquidity).Moreover,weallowforheterogeneityin fro
m
traders’speedsofreactiontoarbitrageopportunities.Thisfeatureisrequired h
foranalyzinghowarbitrageurs’relativespeedaffectsilliquidity.Importantly, ttp
s
weendogenizetraders’speedsbecause,inreality,theirincentivetoeliminate ://a
arbitrageopportunitiesquicklyisendogenoustoilliquidity.Finally,whenthere ca
d
isnoshocktomarketmakers’valuation,weassumethatmarketmakerstrade e
m
with liquidity traders to capture the fact that, in practice, arbitrageurs’trades ic
.o
accountforafractionoftotaltradingvolume. up
.c
o
m
1.1 Model /rfs
Themodelhastwoassets,XandY,threedates(t∈{0,1,2}),twomarketmakers, /a
and one arbitrageur. At date t=2, the payoffs of the assets, represented by rtic
le
θX for X and θY for Y, are realized. These payoffs are identical and given -a
b
by θY=θX=μ+ε, where ε=σ/2 or ε=−σ/2 with equal probabilities, where stra
σ>0.13 c
t/3
0
/4
1.1.1 Market makers. Market maker j∈{X,Y} is specialized in asset j. /1
0
As in other models of multi-asset trading (e.g., Boulatov, Hendershott, and 53
Livdan2013orPasquariello2016),marketsforassetsXandY aresegmented /27
5
because market makers in each asset are different and, for this reason, 86
3
information available to one market maker (e.g., from news or past trades) 5
b
is not instantaneously available to the other.14 Thus, short-lived arbitrage y g
u
e
s
t o
n
13 Forinstance,assetsXandYmightbetwoderivativesonthesameunderlyingasset(e.g.,theE-miniS&P500 0
Fthuetusraemse(EpSay)oafnfdatshaesSsePtDYR.S&P500ExchangeTradedFunds(SPY)),orassetXmightbeasyntheticassetwith 3 Ap
14 Inreality,informationflowsbetweenmarketscannotbeinstantaneousandmarketmakersarespecialized.For ril 2
instance,marketmakersinequitiesmarketsspecializeinafewindividualstocksanddonotshareinformation 01
inrealtime,evenwhentheybelongtothesametradingdesk(seeNaikandYadav2003).Thisisalsothecasein 9
currencymarketsinwhichmarketmakersoftenspecializeinonecurrencypair(seeBjønnesandRime2005).
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opportunitiesbetweenmarketsXandY canhappen.Inthebaselineversionof
themodel,wefocusonthecaseinwhichshockstomarketmakerY’svaluation
(due to information arrival or liquidity needs) cause these opportunities (see
below).Thus,assetY “leads”assetX.
Atdatet=1,marketmakerssimultaneouslypostanaskprice,aj,andabid
price,bj,forj∈{X,Y}suchthat:
S S
aj=vj+ j, and bj=vj− j, (1)
2 2
wherevj ismarketmakerj’svaluationforassetj andSj isthebid-askspread
forassetj.Quotesareforafixednumberofshares,normalizedto1,ofeach D
o
asset. w
n
Market makers’ valuations for assets X and Y are determined at date 0. lo
a
Market maker X derives a utility θX per share of asset X owned at date 2 de
d
andhasnoinformationaboutthepayoffofassetX.Thus,priortotrading,his fro
valWuaitthiopnriosbvaXbi=liEty(θ(1X−)=αμ),.marketmakerY alsoderivesautilityθY pershare m http
oInfathssisetcYaseo,whniesdvaatludaattieon2iasnvdYh=asEn(θoYi)n=foμrm.Aatlitoenrnaabtiovuetlyth,ewpitahyopfrfoboafbaislsiteytYα,. s://aca
thereisashocktomarketmakerY’svaluation.Thisshockcanbedueeither de
m
toinformationarrival(withprobabilityϕ)orliquidityneeds(withprobability ic
(1−ϕ)).Incaseofinformationarrival,marketmakerY privatelyobservesε .ou
p
and therefore his valuation for asset Y becomes vY=E(θY|ε)=μ+ε. In case .co
oshfaareliqoufiadsistyetnYeeodw,mneadrkaettdmataek2e,rwYh’seruetiδlitiysfeoqrutahletoasσs/e2tboerc−omσe/s2(wθYit+hδe)qpuearl m/rfs/a
probabilitiesandisindependentfromε.Thus,δ isaprivatevaluationshock, rtic
le
which represents, for instance, the hedging value of the asset for the market -a
b
maker(asinDuffie,Garleanu,andPedersen2005).Afterreceivingthisshock, s
marketmakerY’svaluationfortheassetbecomesvY=E(θY)+δ=μ+δ. trac
t/3
0
/4
1.1.2Arbitrageopportunities. AsassetsXandY haveidenticalpayoffs,the /10
5
arbitrageurcantakeadvantageofadivergenceinmarketmakers’valuations. 3/2
For instance, suppose that vY>vX. If the arbitrageur buys asset X (at ask 75
8
price aX) and sells asset Y (at bid price bY), she locks in a sure profit of 63
5
bY−aX=(vY−vX)−(SY+SX)/2sinceassetsXandY haveidenticalpayoffs. b
y
By symmetry, if vX>vY, the arbitrageur’s profit is (vX−vY)−(SY+SX)/2. gu
Thus,ifshetrades,thearbitrageur’sprofitis: es
t o
n
Arbprofit=(cid:8)XY−(SY+SX)/2, (2) 03
A
p
where (cid:8)XY=|vY−vX|. The arbitrageur’s profit is positive if (cid:8)XY>(SY+ ril 2
SX)/2, that is, if the difference in market makers’valuations is large enough 019
relativetothebid-askspreadcostbornebythearbitrageur.
1060
ToxicArbitrage
Givenourassumptions,(cid:8)XY=σ/2whenthereisashocktomarketmaker
Y’svaluation,whetherthisshockisduetonewsarrivaloraliquidityneed.15
Thus,inthesecases,thearbitrageur’sexpectedprofit(Equation(2))isstrictly
positivewhen
SX+SY<σ. (3)
Thisconditionalwayswillbesatisfiedinequilibrium(seebelow).
If there is no shock to market maker Y’s valuation (probability (1−α)),
then(cid:8)XY=0,andthereisnoprofitabletradeforthearbitrageur.Inthiscase,a
liquiditytraderarrivesinthemarkettobuyorselloneshareofassetX orY,
withequalprobabilities.
D
1.1.3Toxicandnontoxicarbitrageopportunities. IfmarketmakerXtrades ow
n
withaliquiditytrader,heearnshalfhisbid-askspread.Incontrast,ifhetrades lo
a
d
with the arbitrageur, his expected profit depends on the type of shock that e
d
triggersthearbitrageopportunity. fro
Forinstance,considerapositiveshocktomarketmakerY’svaluation.Inthis m
h
case,ifthearbitrageurtrades,hebuysassetX.IftheshocktoY’svaluationis ttp
duetonews,thenassetX’sexpectedpayoffisE(θX|ε=σ/2)=μ+σ/2.Thus, s://a
if market maker X sells the asset to the arbitrageur, he earns an expected c
a
profitofaX−(μ+σ/2)=(SX−σ)/2,whichisnegativeifthearbitrageurfindsit dem
profitabletotrade(i.e.,ifCondition(3)issatisfied).Inotherwords,anarbitrage ic
opportunityduetothearrivalofnewsaboutassetY istoxicformarketmaker .ou
p
Xbecauseheisexposedtotheriskoftradingatalosswiththearbitrageur. .c
o
theInfitnhseteeaxdp,etchteedshpoacykoftfoomfaarskseettXmaikseEr(YθX’s|vδa=luσa/ti2o)n=iμsd.Tuehutos,aiflimquairdkietytmshaokcekr m/rfs/a
Xtradeswiththearbitrageur,heobtainsanexpectedprofitofaX−μ=SX,as rtic
2 le
ifheweretradingwithaliquiditytrader.Thus,anarbitrageopportunitydueto -a
aliquidityshockformarketmakerY isnontoxicformarketmakerX. bs
MarketmakerX canavoidtoxictradesifhecancelshisquotesbeforethe trac
arbitrageurhitsthem.Wedenoteby(1−π)theprobabilitythatXisfastenough t/30
todoso.Thus,whenatoxicarbitrageopportunityoccurs,thearbitrageurcan /4/1
actually trade on it with probability π. In contrast, if a nontoxic arbitrage 05
3
opportunity happens, there is no reason for market maker X to cancel his /2
7
quotessincehemakesaprofitwhenhetradeswiththearbitrageur.Thus,we 58
6
assumethatthearbitrageurcanexploitanontoxicarbitrageopportunitywith 3
5
certainty. by
g
Table1givestheexpectedpayoffsofthearbitrageurandeachmarketmaker u
e
foreachpossibleeventatdate1:aliquiditytraderarrives(probability(1−α)); st o
a toxic arbitrage happens (probability αϕ); a nontoxic arbitrage happens n
0
(probabilityα(1−ϕ)).Thus,parameterαcontrolsthelikelihoodofoccurrence 3 A
p
ril 2
0
1
15 Forinstance,ifthisshockispositive,wehavevY=μ+σ/2.Accordingly,inthiscase,(cid:8)XY=(vY−vX)=μ+ 9
σ/2−μ=σ/2.
1061
TheReviewofFinancialStudies/v30n42017
Table1
Traders’expectedpayoffs
Liq.trader Atoxic Anontoxic
arrives arbitragehappens arbitragehappens
(withprob1−α) (withprobαϕ) (withprobα(1−ϕ))
↓ (cid:4) (cid:5) (cid:4) (cid:5)
Liq.trader Arb.trades Xcancels Arb.trades Xcancels
Termination trades (withprobπ) (withprob1−π) (withprob1) (withprob0)
Arb.’sexpectedpayoff 0 σ−(SX+SY) 0 σ−(SX+SY) 0
2 2
X’sexpectedpayoff SX −(σ−SX) 0 SX 0
4 2 2
Y’sexpectedpayoff SY SY 0 SY 0
4 2 2
Aggregateexpectedpayoff SX+SY 0 0 σ 0
(Arb.+Marketmakers) 4 2 D
o
w
n
lo
a
d
e
of an arbitrage opportunity while parameter ϕ controls the likelihood that d
an arbitrage is toxic conditional on an arbitrage opportunity occurring. We fro
m
thereforecallitthe“arbitragemix”. h
MarketmakerY’squotesalwaysreflectallavailableinformationaboutthe ttps
payoffofassetY.Thus,incontrasttomarketmakerX,heearnshalfhisbid- ://a
c
a
askspreadinallcases.Whenatoxicarbitrageopportunityhappens,totalgains d
e
fromtradebetweenthearbitrageurandmarketmakers(lastlineofTable1)are m
ic
zero:thelossofmarketmakerXifhetradeswiththearbitrageurisjustequalto .o
u
theprofitsofthearbitrageurandmarketmakerY.Incontrast,whenanontoxic p.c
o
arbitrageopportunityhappens,totalgainsfromtradearestrictlypositiveand m
equal to σ/2, the difference between market makers’valuations. Indeed, in /rfs
/a
thiscase,thisdifferencereflectstruegainsfromtradebetweenmarketmakers. rtic
By trading across the two markets, the arbitrageur enables market makers to le
achievethesegains. -ab
s
tra
c
t/3
1.1.4 Speed. In reality, the probability, π, that an arbitrageur can hit stale 0/4
quotes before they are cancelled depends on her speed of reaction to market /10
5
events relative to liquidity providers. Speed is a choice variable for traders. 3/2
We model it as in Foucault, Röell and Sandås (2003). Specifically, when 75
8
an arbitrage opportunity occurs, it takes DA and DX units of time for the 63
arbitrageurandmarketmakerX,respectively,tospotit(subscriptArefersto 5 b
thearbitrageur),whereDAandDXareexponentiallydistributedwithintensity y g
u
γ andλ,respectively.TherandomnessofDAandDX capturesthefactthat,in es
practice,traders’responsetimestomarketevents(e.g.,anarbitrageopportunity) t o
n
depend on a myriad of random factors (e.g., time required by platforms to 03
processorders)thatcannotbefullycontrolledbytraders. Ap
IfDA<DX,thearbitrageurisfirsttoobservethearbitrageopportunityand ril 2
sheexploitsit.OtherwisemarketmakerXisfirsttoobservetheopportunityand 01
9
cancelshisquotes.Thus,thelikelihood,π,thatatoxicarbitrageopportunity
1062