Detecting abnormal trading activities in option markets Journal of Empirical Finance

72 Pages·2016·1.62 MB·English

by  Marc Chesney

JournalofEmpiricalFinance33(2015)263–275
ContentslistsavailableatScienceDirect
Journal of Empirical Finance
journal homepage: www.elsevier.com/locate/jempfin
Detecting abnormal trading activities in option markets
MarcChesneya,⁎,RemoCramerib,LorianoMancinic
aUniversityofZurich,DepartmentofBankingandFinance,Plattenstrasse32,CH-8032Zurich,Switzerland
bUBSAG,Stockerstrasse64,P.O.Box8092,Zurich,Switzerland
cEPFL,SwissFinanceInstituteatEPFL,QuartierUNIL-Dorigny,Extranef217,CH-1015Lausanne,Switzerland
a r t i c l e i n f o a b s t r a c t
Articlehistory: Wedevelopaneconometricmethodtodetect“abnormaltrades”inoptionmarkets,i.e.,trades
Received31August2012 whicharenotdrivenbyliquiditymotives.Abnormaltradesarecharacterizedbyunusuallylarge
Receivedinrevisedform23January2015 incrementsinopeninterest,tradingvolume,andoptionreturns,andarenotusedforoption
Accepted16March2015 hedgingpurposes.Weuseamultiplehypothesistestingtechniquetocontrolforfalsediscoveries
Availableonline26March2015
inabnormaltrades.Weapplythemethodto9.6millionofdailyoptionprices.
©2015ElsevierB.V.Allrightsreserved.
JELclassification:
G12
G13
G17
Keywords:
Optiontrades
Openinterest
Falsediscoveryrate
Massivedataset
1.Introduction
Animportantdistinctionofoptiontradesisbetweenliquidityandnon-liquiditytrades.Theformerissolelydrivenbyliquidity
shockstooptiontraders.Thelattercanbedrivenbyvariousmotives,includingprivateinformationandhedgingneeds.Disentangling
theseoptiontradescanpotentiallyimproveourunderstandingofthefunctioningofoptionmarkets.
Thispaperdevelopsaneconometricapproachtodetectcertainnon-liquidityoptiontradesthatwecallabnormaltrades.Wedefine
abnormaltradesasunusualtradesinoptioncontractswhichgeneratelargegains,arenotusedforoptionhedgingpurposes,andare
madeafewdaysbeforetheoccurrenceofaspecificevent.
Wedeveloptwostatisticalmethodstodetectabnormaltrades.Thefirstmethodusesonlyex-anteinformationandaimstodetect
abnormaltradesassoonastheytakeplace.Welookforoptiontradescharacterizedbyunusuallylargeincrementsinopeninterest,
i.e.,numberofoutstandingcontracts,whichareclosetodailytradingvolumes.Inthosecasestheoriginatorofsuchtransactionsis
notinterestedinintradayspeculationsbuthasreasonsforkeepingherpositionforalongerperiod.Asitturnsoutinourempirical
study,thehighertheincrementinopeninterestandvolumethehigherthefuturereturnofthecorrespondingoption.Werefine
thefirstmethodusinganonparametrictesttocheckwhetherthoseoptiontradesarehedgedwiththeunderlyingassetorused
foroptionhedgingpurposes.Thesecondmethodusesalsoex-postinformationandencompassesthefirstmethodbyaddingan
additionalcriterion.Anoptiontradeisidentifiedasabnormalwhentheincrementinopeninterestandvolumeisunusual,nothedged
(asinthefirstmethod),andgenerateslargeoptiongains.
⁎ Correspondingauthorat:UniversityofZurich,DepartmentofBankingandFinance,Plattenstrasse32,CH-8032Zurich,Switzerland.Tel.:[phone];
fax:[phone].
E-mailaddresses:[email protected](M.Chesney),[email protected](R.Crameri),loriano.mancini@epfl.ch(L.Mancini).
http://dx.doi.org/10.1016/j.jempfin.2015.03.008
0927-5398/©2015ElsevierB.V.Allrightsreserved.
264 M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275
Ourapproachtodetectoptionabnormaltradinghastwodistinctivefeatures:itcontrolsforfalsediscoveriesinabnormaltrades
andaccountsforoptionhedging.Addressingtheseissuesisachallengingtask.Inanystatisticalmethod,theprobabilitythatany
liquiditytradewillappeartobeabnormalsimplybychanceisnotzero.ThismisclassificationisinducedbytheTypeIerrorin
hypothesistesting,asthetestofabnormaltradeisrepeatedeachday.However,thismisclassificationerrorcanbeformallyquantified
usingmultiplehypothesistestingtechniques.Intuitively,liquiditytradesshouldhavezeroreturnonaverage,whileabnormaltrades
shouldhavestatisticallylargereturns.Underthenullhypothesisthatalltradesareliquiditytrades,theproportionofluckyliquidity
tradesdependsonthesizeofthetestandcanbecalculatedusingoptionreturns.Whenthedifferencebetweentheactualfractionof
largereturns(duetoabnormalandluckytrades)andtheexpectedfractionoflargereturnsduetoluckyliquiditytradesisstatistically
large,thetestrejectsthenullhypothesisthatalltradesareliquiditytrades.
Wedevelopanonparametrictesttoassesswhetheroptionhedgingtakesplaceornot.Forexample,whenstudyinglongpositions
incalloptions,theideaistodecomposetheunderlyingstockseller-initiatedtradingvolumeinthehedgingandnon-hedging
components.Thisdecompositionisachievedusingthetheoreticalamountofstocktradingwhichwouldhavebeengeneratedifno
abnormaltradingwouldhaveoccurred.Thenthetestrejectsthenullhypothesisofabsenceofhedgingwhenthehedgingcomponent
isstatisticallylarge.
Anobviousquestionatthisstageiswhooriginatesabnormaltrades.Althoughinformationontraders'identityisnotavailable,itis
conceivablethatmainlyinformedtradersarebehindabnormaltradesincalloptions.Thisconjecturewouldbeconsistentwiththe
largereturnsgeneratedbycalloptionabnormaltrades.Forabnormaltradesinputoptionsthesituationisdifferent.Informedtraders
and/orcorporateinsidershedgingtheirhumancapitalaremostprobablybehindthosetrades.1Withoutknowingtraderidentities,itis
notpossibletodisentanglewhetherputoptionabnormaltradesareduetoinformedtradersorcorporateinsidershedgingtheir
humancapital.Wedescribethissituationassayingthatwearetestingajointhypothesis.
Weapplythetwostatisticalmethodsto9.6millionofdailyoptionpricesof31selectedcompaniesmainlyfromairline,banking
andinsurancesectors.Severalmillionsofintradaystockpriceandvolumedataarealsoanalyzedtoassesswhetheranoptiontrade
ishedgedornot.Thesampleperiodspans14years,fromJanuary1996toSeptember2009(partofoursampleendsinApril2006),
andouranalysisisatthelevelofindividualoption,ratherthanonthecross-sectionofstockreturns.2
Ourempiricalfindingscanbesummarizedasfollows.First,abnormaltradestendtoclusterpriortocertaineventssuchasmerger
andacquisition(M&A)announcements,quarterlyfinancialorearningrelatedstatements,theterroristattacksofSeptember11th,and
firstannouncementsoffinancialdisruptionsofbankingandinsurancecompaniesduringtheSubprimefinancialcrisis2007–2009.
Second,priortoaparticulareventwhichwillimpactaparticularcompany,abnormaltradescaninvolvemorethanoneoptionbut
rarelythecheapestoption,i.e.,deepout-of-the-moneyandwithshortestmaturity.Third,themajorityofabnormaltradestake
placeinputratherthancalloptions.Fourth,estimatedoptiongainsofabnormaltradeseasilyexceedseveralmillionsforasingle
event.Finally,theunderlyingstockpricedoesnotdisplayanyparticularbehavioronthedayofthedetectedabnormaltrade.Only
somedayslater,forexamplewhenanegativecompanynewsisreleased,thestockpricedropsgeneratinglargegainsinlongput
positions.
Thepaperisorganizedasfollows.Section2reviewstherelatedliterature.Section3presentsourmethodtodetectabnormal
trades.Section4describesthedataset.Section5summarizestheempiricalresults.Section6quantifiesfalsediscoveriesinabnormal
trades.Section7discussesvariousrobustnesschecks.Section8concludes.
2.Relatedliterature
Althoughwearetestingajointhypothesisforputoptions,abnormaltradescanberelatedtoinformedtradeswhichhavebeenthe
subjectofanextensiveliterature;see,e.g.,Hasbrouck(1991),EasleyandO'Hara(1992),Easleyetal.(1998),Poteshman(2006),and
Boulatov et al. (2013). As discussed in Grossman (1977), Diamond and Verrechia (1987), and others, option markets offer
significantadvantagestoinformedtraders.Optionsprovidepotentialdownsideprotection,analternativewayofshortselling
whenshortingstocksisexpensiveorforbidden,additionalleveragewhichmightnotbepossibleinstockorbondmarkets(Biais
andHillion,1994),andpossiblymorediscreetnessfortradingonprivatesignals.Indeed,Caoetal.(2005)showthatcall–volume
imbalancespriortounscheduledtakeoverannouncementsarestronglyrelatedtostockreturnsontheannouncementday.Panand
Poteshman(2006)reportclearevidencethatoptiontradingvolumespredictfuturepricechanges.BaliandHovakimian(2009)
showthatthedifferencebetweenrealizedandimpliedvolatilitiesofindividualstockspredictsthecross-sectionalvariationof
expectedreturns.CremersandWeibaum(2010)findthatdeviationsfromput–callparitycontaininformationaboutfuturestock
returns.Yan(2011)documentsanegativerelationbetweentheslopeofimpliedvolatilitysmileandstockreturn.Inthesestudies
(andothers),theanalysisissystematicallyconductedatanaggregatelevel,e.g.,extractinginformationfromallcurrentoptionprices,
whileweconducttheanalysisatindividualoptioncontracts.
StephanandWhaley(1990),Chanetal.(1993),ManasterandRendleman(1982),andLeeandYi(2001),amongothers,discuss
whyinformedtradersmayconsideroptionsassuperiortradingvehicles.Ourresultsshowthatoptionmarketscanoffersignificant
1 Humancapitalcanbedefinedasthesumofthepresentvalueofthefuturecashincome,shares,stockoptions,etc.,anditrepresentsthemostsignificantriskfacedby
corporateinsidersespeciallyseniormanagers.Totheextentthatitislegal,alongputoptionisprobablytheonlyliquidinstrumentthatcanbeusedbycorporateinsiders
tohedgetheriskattachedtotheirhumancapital.
2 Asmentionedabove,werelyonstatisticalmethodstodetectabnormaltrades.Therefore,thosetradeswillbeabnormalonlywithacertainprobability.Forbrevity,
werefertothosetradessimplyasabnormaltrades.Moreover,detectedabnormaltradesmightormightnotbelegal.Fromalegalviewpointthisstudydoesnotcon-
stituteproofperseofillegalactivities.Legalproofwouldrequiretraderidentitiesandtheirmotivations,informationwhichisnotcontainedinourdataset.
M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275 265
profitstoinformedtraders,lendingempiricalsupporttothesestudies.Chenetal.(2001)showthatassetcrashescanbepredicted
usingsharestradingvolume.Wecomplementthisworkbyshowingthatcertainincrementsinopeninterestandtradingvolume
havepredictivepowerforfuturemovementsintheunderlyingstock.Blumeetal.(1994)andVijh(1990)providerelatedstudies
ontradingvolumeandinformation-relatedtrading.
3.Detectingoptionabnormaltrades
Weproposetwomethodstodetectoptionabnormaltrades.Thefirstmethodreliesonabroaddefinitionofanoptionabnormal
trade,basedonopeninterestandvolume,andmakesuseonlyofex-anteinformation.Thesecondmethodisbasedonamorestringent
definitionofabnormaltradesandusesex-postinformationaswell.
Wenowdescribethesecondmethodwiththefirstmethodbeingaspecialcase.Wedefineanoptionabnormaltradeasfollows:
C1)anunusualtradeinanoptioncontract,C2)whichismadeafewdaysbeforetheoccurrenceofaspecificeventandgenerateslarge
gainsinthefollowingdays,andC )thepositionisnothedgedinthestockmarketandnotusedforoptionhedgingpurposes.These
3
threecharacteristics,Ci,i=1,2,3,leadtothefollowingmethodtodetectabnormaltradingactivitiesinoptionmarkets:firstoneach
daytheoptioncontractwithlargestincrementinopeninterest(i.e.,numberofoutstandingcontracts)andvolumeisidentified,then
therateofreturnanddollargaingeneratedbythistransactionarecalculated,andfinallyitisstudiedwhetheroptionhedgingoccurs.
Optiontradeswhicharedeltahedgedorusedforoptionhedgingpurposesarenotregardedasabnormaltrades.Thefirstmethod
reliesonlyoncharacteristicsC andC ,andtheirpracticalimplementation.Importantly,bothmethodsrequireonlycommonly
1 3
availabledataandthuscanbeeasilyusedtodetectabnormaltradesinvariousoptionmarkets.
Wenowexplainhowtodetectabnormaltradesincalloptions.Theapplicationtoputoptionscaneasilybededuced.Inthe
empiricalsection,weapplybothmethodstoalargedatasetofcallandputoptions.
3.1.Firstcriterion:incrementinopeninterestrelativetovolume
ForeverycalloptionkavailableatdaytwecomputethedifferenceΔOItk:=OItk−OItk−1,whereOItkisitsopeninterestatdayt,and
:=meansdefinedas.Whentheoptiondoesnotexistattimet−1,itsopeninterestissettozero.Sinceweareinterestedinunusual
transactions,onlytheoptionwiththelargestincrementinopeninterestisconsidered
X : maxΔOIk 1
t ¼ k∈Kt t ð Þ
whereK isthesetofallcalloptionsavailableatdayt.Themotivationforusingopeninterestisthefollowing.Largetradingvolumes
t
canemergeundervariousscenariosforexamplewhenthesamecalloptionistradedseveraltimesduringthedayorlargesellorders
areexecuted.Incontrastlargeincrementsinopeninterestareusuallyoriginatedbylargebuyorders.Theseincrementsalsoimplythat
otherlonginvestorsareunwillingtoclosetheirpositionsforcingthedealerormarketmakertoissuenewcalloptions.Consequently,
weuselargeincrementinopeninterestasaproxyforlargebuyorders.
Wefocusontransactionsforwhichthecorrespondingvolumealmostcoincideswiththeincrementinopeninterest.LetV denote
t
thedailytradingvolumecorrespondingtothecalloptionselectedinEq.(1).ThepositivedifferenceZ :=(V −X)providesameasure
t t t
ofhowoftenthenewlyissuedoptionsareexchanged:thesmallertheZ,thelessthenewoptionsaretradedduringthedayonwhich
t
theyarecreated.Inthatcasetheoriginatorofsuchtransactionsisnotinterestedinintradayspeculationsbuthasreasonsforkeeping
herpositionforalongerperiodpossiblywaitingfortherealizationoffutureevents.
Thisfirstcriterionalreadyallowsustoidentifysingletransactionsaspotentialcandidatesforabnormaltrades.Letqtdenotethe
time-tex-antejointhistoricalprobabilityofobservinganunusuallargeincrementinopeninterestclosetothetradingvolume
1 N
q : ℙX≥X ;Z≤Z 1 2
t ¼ ½ t t%¼N fXi≥Xt;Zi≤Ztg ð Þ
i 1
X¼
whereℙdenotestheempiricalprobability,Nthelengthoftheestimationwindow,e.g.,N=500tradingdays,and1 =1when
{A}
eventAoccurs,andzerootherwise.Byconstruction,lowvaluesofq suggestthatthesetransactionswereunusual.Forexample
t
whenq =1/N,itmeansthatwhatoccurredondaythasnoprecedentsintheprevioustwoyears.
t
3.2.Secondcriterion:relativereturnandrealizedgain
Thesecondcriteriontakesintoconsiderationex-postoptionreturnsandrealizedgains.Foreachdayttheoptiontradewiththelowest
ex-anteprobabilityq isconsidered.Letrmaxdenotethemaximumoptionreturngeneratedinthefollowingtwotradingweeks
t t
P −P
rmtax:¼j¼m1;…ax;10 tþPjt t ð3Þ
whereP denotesthemid-quotepriceoftheselectedcalloptionatdayt.Whenrmaxisunusuallyhigh,anunusualeventoccursduring
t t
thefollowingtwotradingweeks.
266 M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275
Forthecomputationofrealizedgains,weconsiderdecrementsinopeninterest,ΔOItk,whichoccurwhenexercisingorsellingtothe
marketmakerthecalloption.3ThenwesettheAmericancalloptionvaluetoitsexercisevalue,whichistrueinmostcases.Givenour
definitionofabnormaltrade,itisquitelikelythatontheeventdaytheriseinthestockpriceislargeenoughtoreachtheexercise
region.Ifoptionsaresoldratherthanexercised,ourcalculationofrealizedgainsmayunderestimatetheactualgains.Hencereported
gainsshouldbeinterpretedasconservativeestimates.Forbrevity,werefertodecrementinopeninterestasoptionexercise.Also,we
omitthesuperscriptkandwheneverwerefertoaspecificoptionwemeantheonewhichwasselectedbecauseofitslargest
incrementinopeninterestclosetotradingvolume,i.e.,lowestex-anteprobabilityq.
t
LetG denotethecorrespondingcumulativegainsachievedthroughtheexerciseofoptions
t
Gt:¼t¼Xtþ1τt!ðS˜t−KÞþ−Pt" ð−ΔOI˜tÞ1fΔOI˜tb0g ð4Þ
˜
whereτtissuchthattbτt≤T,withTbeingthematurityoftheselectedoption.Ifthecalloptionswereoptimallyexercised(i.e.,assoon
astheunderlyingassetSt touchestheexerciseregion),thepayoffðSt−KÞþcorrespondstothepriceoftheoptionattimet.
Thecumulativegains˜Gtcouldbeeasilycalculatedforeveryτt≤T.˜ThishashoweverthedisadvantagethatGtcouldincludegains
whicharerealizedthroughtheexerciseofoptionswhichwereissuedbeforetimet.4Toavoidthisinconsistency,thetimeτteisdefined
asfollows
l
τ’t :¼ argl∈ftmþ1a;…x;Tg t¼Xtþ1ð−ΔOI˜tÞ1fΔOI˜tb0g≤Xt!τt:¼ min!τ’t;30"
˜
givingtheoptiontradernomorethan30daystocollecthergains.Ingeneralthesumofnegativedecrementsinopeninteresttilltime
τtwillbesmallerthantheobservedincrementinopeninterestXt.Inthatcase,wewilladdtoGtthegainsrealizedthroughthefraction
ofthenextdecrementinopeninterest.HencethesumofallnegativedecrementsinopeninterestwillbeequaltotheincrementX.
t
CalculatingG foreachdaytandeachoptioninourdatasetprovidesinformationonwhetherornotoptiontradeswithalowex-
t
anteprobabilityq generatelargegainsthroughexercise.UsingthemaximalreturnrmaxinEq.(3),wecancalculatethetime-tex-post
t t
jointhistoricalprobabilityp oftheevent{X,Z,rmax}
t t t t
p : ℙ X≥X ;Z≤Z ;rmax≥rmax 1 N 1 : 5
t ¼ t t t ¼Ni 1 fXi≥Xt;Zi≤Zt;rmiax≥rmtaxg ð Þ
# $ X¼
Thehigherthe(1−p)thelargertheoptionreturnandthemoreunusualtheincrementinopeninterestclosetotradingvolume.
t
3.3.Thirdcriterion:hedgingoptionposition
Optiontradesforwhichthefirsttwocriteriashowabnormalbehaviorcannotbeimmediatelyclassifiedasabnormaltrades.It
couldbethecasethatsuchtransactionswerehedgedbytradersusingtheunderlyingasset.Withoutknowingtheexactcomposition
ofeachtrader'sportfolio,itisnotpossibletoassessdirectlywhethereachoptiontradewashedgedornot.
Weattempttoassessindirectlywhetherunusualtradesincalloptionsareactuallydeltahedgedusingtheunderlyingasset.The
ideaistocomparethetheoreticaltotalamountofsharessoldfornon-hedgingpurposesandtheactualtotalvolumeofseller-
initiatedtransactionsintheunderlyingstock.Ifthelatterissignificantlylargerthantheformer,thenitislikelythatsomeofthe
seller-initiatedtradesoccurforhedgingpurposes.Intheoppositecaseweconcludethatthenewoptionpositionsarenothedged.
Onedifficultyisthatthevolumeduetohedgingistypicallyasmallcomponentofthetotalseller-initiatedvolume.Usually,when
hedgingoccurs,newlyissuedoptionsarehedgedonthesamedaywhichisourworkingassumption.Hedginganalysesatthelevelof
singleoptionarenotpossibleusingourOptionMetricsdataset.Wethereforecheckwhetherallthenewlyissuedoptionsarehedged
onaspecificdayt.Givenourdefinitionofabnormaltrades,suchtradescertainlyaccountforalargefractionofthenewlyissued
options.
Foreachdayt,thetotaltradingvolumeoftheunderlyingstockisdividedintoseller-andbuyer-initiatedusingintradayvolumes
andtransactionpricesaccordingtotheLeeandReady(1991)algorithm.5Thentheseller-initiatedvolumeofunderlyingstock,Vsell,is
t
3 Onagivenday,openingnewpositions(whichincreasesopeninterest)andclosingexistingpositions(whichdecreasesopeninterest)canoff-seteachother.Hence
theobserveddecrementinopeninterestisalowerboundforactualexercisedorsoldoptions.
4 Considerforexampleanoptionwhichexhibitsanunusuallyhighincrementinopeninterestattimet,sayOIt−1=1000andOIt=3000resultinginXt:=OIt−OIt
−1=2000.SupposethatinthedaysfollowingthistransactionthelevelofopeninterestdecreasesandafterhdaysreachesthelevelOIt+h=500.Oneshouldonly
considerthegainsrealizedthroughexercisetilltimeτt≤t+h,whereτtissuchthatthesumofnegativedecrementsinopeninterestduring[t+1,τt]equalsXt=2000.
5 Thealgorithmstatesthatatradewithatransactionpriceabove(below)theprevailingquotemidpointisclassifiedasabuyer-(seller-)initiatedtrade.Atradeatthe
quotemidpointisclassifiedasseller-initiatedifthemidpointmoveddownfromtheprevioustrade(down-tick),andbuyer-initiatedifthemidpointmovedup(up-tick).
Iftherewasnomovementfromthepreviousprice,thepreviousruleissuccessivelyappliedtoseverallagstodeterminewhetheratradewasbuyer-orseller-initiated.
M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275 267
dividedintotradingvolumeduetohedgingandtonon-hedgingpurposes,Vtsell,hedgeandVtsell,non−hedge,respectively.LetΔtC,kbethe
deltaofcalloptionkandKtCbethesetofcalloptions(newlyissuedoralreadyexisting)ondayt.SimilarlyforΔtP,kandKtP.Let
α : OIC;k−OIC;k ΔC;k;γ : OIP;k−OIP;k ΔP;k;β : ΔC;k−ΔC;k OIC;k ;δ : ΔP;k−ΔP;k OIP;k :
t ¼ j t t−1jj t j t ¼ j t t−1j t t ¼ j t t−1j t−1 t ¼ j t t−1j t−1
kX∈KCt kX∈KPt kX∈KCt kX∈KPt
Theαtandγtrepresentthetheoreticalnumberofsharestosellforhedgingthenewcalloptionsissuedatdayt,whereasβtandδt
arethetheoreticalnumberofsharestoselltorebalancetheportfolioofexistingoptionsatdayt.Absolutechangesinopeninterests
anddeltasaccountforthefactthateachoptioncontracthasalongandshortsidethatfollowoppositetradingstrategiesifhedging
occurs.Thetheoreticalseller-initiatedvolumeofstockatdaytforhedgingpurposesisVtsell,hedge‐theory:=αt+βt+γt+δt.
Whenthefirsttwocriteriaofourmethoddonotsignalanyabnormaltrade,weapproximateVtsell,hedgebyVtsell,hedge‐theory.Thenthe
amountofstocksoldfornon-hedgingpurposesiscalculatedasVsell,non‐hedge=Vsell−Vsell,hedge‐theory.
t t t
Whenabnormaltradestakeplaceondayi,Vsell,non‐hedgecannotbecomputedasinthelastequationbecauseVsell,hedge‐theory
i i
wouldbedistortedbytheunhedgedoptionabnormaltrades.WecircumventthisissuebyforecastingthevolumeVsell,non‐hedgeon
i
day i using historical data on Vsell,non ‐hedge. The conditional distribution of Vsell,non ‐hedge is estimated using the adjusted
t i
Nadaraya–WatsonestimatorandthebootstrapmethodproposedbyHalletal.(1999)
T
1 w x K X−x
F yx Xt¼1 fYt≤yg tð Þ Hð t Þ 6
ð j Þ¼ T ð Þ
w x K X−x
e tð Þ Hð t Þ
t 1
X¼
withY :=Vsell,non‐hedge,X :=(|r|,Vsell,non‐hedge),K (⋅)beingamultivariatekernelwithbandwidthmatrixH,w(x)theweighting
t t t t t−1 H t
function,andr thestockreturnatdayt;wereferthereadertoFanandYao(2003)fortheimplementationofEq.(6).
t
Wecannowformallytestthenullhypothesis,H ,thathedgingdoesnottakeplaceatdayi.WhenevertheobservedVsellislarge
0 i
enough,sayabovethe95%quantileofthepredicteddistributionofVsell,non‐hedge,itislikelythatafractionofVsellisduetohedging
i i
purposes.HencewerejectH0atdayiwhen VsiellNqV0:si9el5l;non‐hedge,where qVαsiell;non‐hedge ¼F−1ðαjXiÞ istheα-quantile ofthepredicted
distributionofVsell,non‐hedgeestimatedusingEq.(6).Theseparateappendixshowsthatthepowerofthetestdependsonthe
i
conditioningvariablesX butcanbeashighas20%whenVsellis20%largerthanVsell,neon‐hedge.
i i i
WeremarkthatthenullhypothesisH ofnohedging(whenabnormaltradesoccur)concernsonlylongpositionsinnewly
0
issuedcalloptions.Shortpositionsinthesamecalloptionsdonotaffectourhedgingdetectionmethod.Itissobecausethetotal
volumeoftheunderlyingstockisdividedintobuyer-andseller-initiatedandonlythelattermatterswhenhedginglongcall
options.
3.4.Detectingabnormaltradescombiningthethreecriteria
Letk denotetheselectedabnormaltradeatdaytincalloptionk.Thetwomethodstodetectoptionabnormaltradescanbe
t
describedusingthefollowingfoursetsofevents:Ω1:={ktsuchthatqt≤5%};Ω2:={ktsuchthat“H0:non-hedging”isnotrejected
atdayt};Ω3:={ktsuchthatrmt ax≥qr0mt:9ax0g;andΩ4:={ktsuchthatGt≥qG0:t98g.Thefirstmethoddetectsanabnormaltradewhenit
belongstothefirsttwosets,i.e.,kt∈Ω1∩Ω2.Accordingtothesecondmethodanoptiontradeisabnormalwhenitbelongstoall
foursets,i.e.,kt∈Ω1∩Ω2∩Ω3∩Ω4.
4.Data
Tokeeptheempiricalanalysismanageable,wefocusonthreemainsectors,i.e.,banking,insurance,andairline,andwithineach
sectorweconsidersomeofthemaincompanies.Inaddition,wealsoconsideranumberofrandomlyselectedcompaniesfrom
othersectors,suchasCocaColaandPhilipMorris,tobroadenourempiricalanalysis.Weorganizeourdatasetintwoparts.The
firstpartincludesonlyputoptions,whilethesecondpartputandcalloptions.
Thefirstpartofourdatasetincludes14companiesfromairline,bankingandvariousothersectors.Thelistofcompanies
includes:AmericanAirlines(AMR),UnitedAirlines(UAL),DeltaAirLines(DAL),Boeing(BA)andKLMfortheairlinesector;
BankofAmerica(BAC),Citigroup(C),J.P.Morgan(JPM),MerrillLynch(MER)andMorganStanley(MWD)forthebanking
sector;andAT&T(ATT),CocaCola(KO),HewlettPackard(HP),andPhilipMorris(MO)fortheremainingsectors.Option
dataarefromtheChicagoBoardOptionsExchange(CBOE)asprovidedbyOptionMetrics.Thedatasetincludesthedaily
cross-sectionofavailableputoptionsforeachcompanyfromJanuary1996toApril2006andamountstoabout2.1million
options. Option data for DAL and KLM were available for somewhat shorter periods. Stock prices are downloaded from
OptionMetricsaswelltoavoidnon-synchronicityissuesandareadjustedforstocksplitsandspin-offsusinginformation
fromtheCRSPdatabase.IntradaytransactionpricesandvolumesforeachunderlyingstockpricearefromNYSE'sTradeand
Quote(TAQ)database.Thisdatasetconsistsofseveralmillionsofrecordsforeachstockandisnecessarytoclassifytrading
volumesinbuyer-andseller-initiatedtransactionsinordertocompletetheanalysisrelatedtothehedgingcriterion.Discrepancies
268 M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275
amongdatasetshavebeencarefullytakenintoaccountwhenmergingdatabases.6Additionally,weanalyzeputoptionson3
Europeancompanies,SwissRe,MunichReandEADS,usingdailydatafromtheEUREXprovidedbyDeutscheBank.
Thesecondpartofourdatasetincludes19companiesfromthebankingandinsurancesectors.Putandcalloptionsdataarefrom
January1996toSeptember2009,coveringtherecentfinancialcrisis,andamountstoabout7.5millionoptions.ThelistofAmerican
companiesincludes:AmericanInternationalGroup(AIG),BankofAmericaCorporation(BAC),BearStearnsCorporation(BSC),Citigroup
(C),FannieMae(FNM),FreddieMac(FRE),GoldmanSachs(GS),J.P.Morgan(JPM),LehmanBrothers(LEH),MerrillLynch(MER),
MorganStanley(MS),WachoviaBank(WB)andWellsFargoCompany(WFC).Mostofthesecompaniesbelongtothelistofbanks
whichwerebailedoutand,inwhich,theAmericanTreasuryDepartmentinvestedapproximately$200billionthroughitsCapital
PurchasePrograminanefforttobolstercapitalandsupportnewlending.Optionsandstockdataarefromthesamedatabasesasbefore,
namelyCBOE,TAQ,andCRSP.Furthermoreweanalyze6Europeanbanks:UBS,CreditSuisseGroup(CS)andDeutscheBank(DBK)whose
optionsaretradedonEUREX,andSocietéGénérale(GL),HSBC(HSB)andBNPParibas(BN)withoptionslistedonEuronext.Optiondata
aswellasintradaytransactionpricesandvolumesfortheunderlyingstockareobtainedfromEUREXprovidedbyDeutscheBank,and
fromEURONEXTprovidedbyNYSEEuronextdatabase.AllanalyzedoptionsareinAmericanstyle.
5.Empiricalresults
Thetwoproposedmethodstodetectoptionabnormaltradesareappliedtothecompanieslistedintheprevioussection.Werecall
thatwhentestingabnormaltradesinputoptions,wearetestingthejointhypothesisdiscussedintheintroduction.
Thefirstmethod,whichreliesonlyonex-anteinformation,aimsatdetectingabnormaltradesassoonastheytakeplace.On
average,lessthan0.1%ofthetotalanalyzedtradesbelongstothesetΩ1∩Ω2definedinSection3.4.AsanexampleforAMRour
firstmethoddetects141abnormaltrades,thetotalnumberofanalyzedoptionsbeingmorethan137,000.Thissuggeststhatalready
theex-antemethodcanbequiteeffectiveinsignalingabnormaltrades.
Thesecondmethod,whichreliesalsoonex-postinformation,selectsasignificantlysmallernumberofabnormaltrades.For
example,only5abnormaltradesaredetectedforAMR.Importantly,theempiricalpatternsofabnormaltradesbasedonthetwo
methodsareroughlythesame.Forexample,bothmethodssuggestthatmostabnormaltradesforAMRoccurbeforeanacquisition
announcementinMay2000andthe9/11terroristattacks.
Duetospaceconstraintsweonlydiscussabnormaltradesselectedbytheex-postmethod.Theseparateappendixreportsa
detailedanalysisofvariousabnormaltrades.
Analyzingthefirstpartofourdataset,37transactionsontheCBOEhavebeenidentifiedasbelongingtothesetΩ ∩Ω ∩Ω ∩Ω
1 2 3 4
definedinSection3.4.Nearlyallthedetectedabnormaltradescanbeassociatedtooneofthefollowingthreeeventcategories:merger
andacquisition(M&A)announcements,sixtransactions;quarterlyfinancial/earningsrelatedstatements,14transactions;andtheter-
roristattacksofSeptember11th,13transactions.Fourtransactionscouldnotbeidentified.Tables1and2summarizeabnormaltrades
fortheairlinesector.Correspondingtablesforthebankingsectorandthelastgroupofcompaniesarereportedintheseparateappendix.
Thesecondpartofourdatasetfocusesonthebankingandinsurancesectorsandconsistsofcallandputoptions.Detailedempirical
resultsarecollectedintheseparateappendix.Althoughthesampleperiodspansalmost15years,fromJanuary1996toSeptember
2009,thevastmajorityofdetectedabnormaltradesoccurduringtheSubprimecrisis2007–2009.Mostabnormaltradesinvolve
putoptionswhichspeakstothejointhypothesisdiscussedintheintroduction.Wealsodetectabnormaltradesincalloptionsfor
everybankandinsurancecompanyanalyzed.Calloptionabnormaltradesaremainlyrelatedtopositivequarterlyannouncements
andnewsaboutcertaincompaniesraisingnewcapitalduringthefinancialcrisis2007–2009.
Toprovidesomeinsightsonoptionabnormaltrading,belowwediscussindetailthecaseofanacquisitionannouncementinthe
U.S.airlinesectorinMay2000.Additionalcasesarediscussedintheseparateappendix.
Theex-postmethoddetectstwoputoptionabnormaltradesonMay10thand11th,2000.TheyinvolvedAMRandUAL.OnMay10th
and11th,thenumberofoptionsissuedwithstrike$35andmaturityinJune2000withunderlyingAMRisverylarge:3374onMay10th
and5720thedayafter(at99.7%and99.9%quantileoftheirtwo-yearempiricaldistributions,respectively).Thesetransactionscorre-
spondtothosewhichexhibitthestrongestincrementsinopeninterestduringaspanoffiveyears;seeFig.1(upperleftgraph)andFig.2.
OnMay10th,theunderlyingstockhadavalueof$35.50andtheselectedputwastradedat$2.25.ForUAL2505putoptioncon-
tracts(at98.7%quantileofitstwo-yearempiricaldistribution)withstrike$65andthesamematurityasthoseofAMRwereissuedon
May11thatthepriceof$5.25whentheunderlyinghadavalueof$61.50.Themarketconditionsunderwhichsuchtransactionstook
placearestable.Forexampletheaveragereturnofthestocktheweekbeforeis,inbothcases,positiveandlessthan0.5%.
ThedaysofthedropintheunderlyingstockpriceareMay24thandMay25th,2000,withthefirstdaycorrespondingtothepublic
announcementofUnitedAirline'sregardinga$4.3billionacquisitionofUSAirways.AsreportedintheMay25th,2000editionofthe
NewYorkTimes,“sharesofUALandthoseofitsmainrivalscrashed.”7ThestockpriceofAMRdroppedto$27.13(−23.59%ofvalue
losseswhencomparedtothestockpriceonMay11th)increasingthevalueoftheputoptionsto$7.88(resultinginareturnof250%in
6 ForexampledataforJ.P.MorganfromOptionMetricsandTAQdonotmatch.WhereasthestockvolumereportedinOptionMetricsfortheyears1996–2000isgiven
bythesumofthevolumeofChaseManhattanCorporationandJ.P.Morgan&Co.(ChaseManhattanCorporationacquiredJ.P.Morgan&Co.in2000);TAQonlyreports
thevolumeofJ.P.Morgan&Co.SameissuewasfoundforBankofAmericaCorporationandNationsBankCorporation,whosemergertookplacein1998underthenew
nameofBankofAmericaCorporation.
7 TheNewYorkTimesarticlereportsthatAMRwasconsideredthecompanymostthreatenedbythemerger,explainingthereforethe17%dropinitssharepricein
thedaysafterthepublicannouncement.AccordingtoJamesGoodwin,chairmanandchiefexecutiveofUAL,twomajorhurdleswouldchallengeUAL:“thefirstistoget
USAirwaysshareholderstoapprovethistransaction.[Thesecond]istheregulatorywork,whichrevolvesaroundtheDepartmentofTransportation,theDepartmentof
JusticeandtheEuropeanUnion.”TheskepticismonWallStreetwasimmediatelyreflectedonUALsharepricewhichdeclined$7.19to$53.19ontheannouncementday.
M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275 269
Table1
Abnormaltradesintheairlinesector.Thetableshowsthedayonwhichthetransactiontookplace,Date;identificationnumberoftheputoption,Id;moneyness,
i.e.,stockpricedividedbystrikeprice,S/K;time-to-maturity,τ;levelofopeninterestthedaybeforetheabnormaltrade,OIt−1;incrementinopeninterestfrom
dayt−1todayt,∆OIt;itsquantilewithrespecttoitsempiricaldistributioncomputedoverthelasttwoyears,qtΔOI;totalincrementinopeninterest,i.e.,when
consideringalltheavailableoptionsatdaytandnotonlytheoneswhichhadthehighestincrement,ΔOIttot;correspondingvolume,Volt;maximumreturnrealized
bytheselectedoptionduringthetwo-weekperiodfollowingthetransactionday,rtmax;numberofdaysbetweentransactiondaytandwhenthismaximumreturn
occurs,τ2;gainsrealizedthroughtheexerciseoftheoptionissuedattimetasin(4),Gt;minimumbetweenthenumberofdays(startingfromthetransactionday)
neededfortheexerciseofΔOItand30days,τ3;percentageofΔOItexercisedwithinthefirst30daysafterthetransaction,%ex.;ex-anteprobabilityin
Eq.(2),qt;p-valueofthehypothesisthatdeltahedgingdoesnottakeplaceattimet,p-value;andex-postprobabilityofabnormaltradinginEq.(5),
1−pt.*meansthatthehypothesisofnon-hedgingcanberejectedata5%level.
SummaryofairlinesectorJan1996–Apr2006
Date Id S/K τ OIt−1 ∆OIt qtΔOI ΔOIttot Volt rtmax τ2 Gt τ3 %ex. qt p-Value 1−pt
AmericanAirlines(AMR)Jan1996–Apr2006
10May00 10821216 1.01 38 20 3374 99.7% 3378 3290 106% 9 906,763 11 100% 0.002 0.286 0.998
11May00 10821216 1.02 37 3394 5720 99.9% 5442 5320 98% 10 1,647,844 11 100% 0.002 0.349 0.998
31Aug01 20399554 0.91 22 96 473 95.7% 571 500 455% 7 662,200 11 100% 0.016 0.645 0.984
10Sep01 20428354 0.99 40 258 1312 98.5% 1701 1535 453% 2 1,179,171 26 100% 0.012 0.096 0.998
24Aug05 27240699 0.97 24 1338 4378 93.5% 8395 5319 163% 8 575,105 17 100% 0.048 0.123 0.952
UnitedAirlines(UAL)Jan1996–Jan2003
11May00 11332850 0.95 37 35 2505 98.7% 2534 2505 132% 10 1,156,313 26 100% 0.002 0.373 0.998
6Sep01 20444473 1.06 44 21 1494 96.3% 1189 2000 1322% 7 1,980,387 28 100% 0.030 0.165 0.998
DeltaAirLines(DAL)Jan1996–May2005
*1Oct98 10904865 1.01 16 140 974 97.7% 483 924 261% 6 537,594 12 100% 0.016 0.000 0.996
29Aug01 20402792 0.98 24 1061 202 89.7% 224 215 1033% 9 328,200 13 100% 0.044 0.528 0.998
19Sep02 20718332 0.99 30 275 1728 98.7% 550 1867 132% 7 331,676 22 100% 0.004 0.190 0.998
9Jan03 21350972 1.10 44 274 3933 99.7% 4347 4512 112% 9 1,054,217 30 100% 0.002 0.065 0.998
Boeing(BA)Jan1996–Apr2006
24Nov98 10948064 0.99 53 3758 1047 93.5% 1285 1535 467% 7 883,413 24 100% 0.040 0.481 0.996
29Aug01 20400312 0.92 24 1019 2828 96.7% 3523 3805 382% 10 1,972,534 8 100% 0.028 0.252 0.998
5Sep01 20429078 1.01 45 472 1499 92.1% 2538 1861 890% 8 1,805,929 22 100% 0.048 0.085 0.998
6Sep01 11839316 0.75 135 13228 7105 99.3% 13817 7108 118% 7 2,704,701 3 100% 0.006 0.150 0.998
*7Sep01 20400311 0.90 15 7995 4179 98.5% 4887 5675 306% 6 5,775,710 7 100% 0.016 0.000 0.998
*17Sep01 20400309 0.90 5 116 5026 98.9% 2704 5412 124% 4 2,663,780 5 100% 0.010 0.000 0.998
KLMJan1996–Nov2001
5Sep01 20296159 0.91 17 3 100 99.3% 34 100 467% 9 53,976 9 100% 0.006 0.368 0.998
twotradingweeks).ThesameimpactcanbefoundforUAL:thestockpriceafterthepublicannouncementdroppedto$52.50
(−14.63%whencomparedtothevalueonMay11th)raisingtheput'svalueto$12.63(correspondingtoareturnof140%intwotrad-
ingweeks).InthecaseofAMR,thedeclineintheunderlyingstockcanbeseeninFig.2,wheretheoptionreturnlargelyincreased.
Onthedayofthepublicannouncement4735putoptionsofAMRwereexercised;seeFig.2.Afterthislargedecrementinopen
interest,1494and1376additionalputoptionswereexercisedinthefollowingtwodaysrespectively(reflectedinadditionaldrops
inopeninterestsinFig.2).TheunusualincrementsinopeninterestobservedonMay10thandMay11tharethereforeoffsetby
theexerciseofoptionswhentheunderlyingcrashed.ThecorrespondinggainsG fromthisstrategyaremorethan$1.6millionwithin
t
twotradingweeks.ThesearegraphicallyshowninthelowergraphinFig.1,fromwhichwecanseehowfastthesegainswererealized.
InthecaseofUALsimilarconclusionscanbereached;seeTables1and2.Basedonthesetrades,atotalgainofalmost$3millionwas
realizedwithinafewtradingweeksusingoptionswithunderlyingAMRandUAL.Thenon-hedginghypothesiscannotberejected
suggestingthatsuchtradesareunhedgedoptionpositions.ComparableabnormaltradeshavebeenfoundforAmericanAirlines,
UnitedAirlinesandBoeing(andtoalesserextentforDeltaAirLinesandKLM)beforetheterroristattacksof9/11,andarediscussed
intheseparateappendix.
6.Controllingfalsediscoveriesinabnormaltrades
Anystatisticalmethodcangeneratefalsediscoveriesinabnormaltrades,i.e.,theprobabilitythatanoptiontradecansatisfyvarious
criteriasimplybychanceisnotzero.Controllingforfalsediscoveryisthenanimportanttask,whichallowsabnormaltradeswithhigh
gainstobetrulyseparatedfromliquiditytradesthatluckilyachievedalsohighgains.Toseparatethetwogroupsoftradesweusea
multiplehypothesistestingtechnique.Barrasetal.(2010)adoptedasimilarapproachtodiscriminatebetweenskilledandlucky
mutualfundmanagersbasedonfundperformance.
Forthesakeofpresentation,wephrasethediscussionintermsofinformedversusuninformedtraders.Wesaythatabnormal
tradeswithhighgainsaregeneratedbyinformedtraders(andluckyuninformedtraders).Inpractice,traderswithprivateinformation
and/orwhoarehedgingtheirhumancapitalareprobablyoriginatingputoptionabnormaltrades,whichisthejointhypothesis
discussedintheintroduction.Inthepresentationofthemultiplehypothesistestweomitsuchadistinction.
SupposeweobserveoptionreturnsgeneratedbyMtraderscharacterizedbydifferentdegreesofinformation,rangingfromhighly
accurateprivateinformationtonoinformation(orpossiblyevenmisleadinginformation).Letπ0denotethefractionofuninformed
270 M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275
Table2
Abnormaltradesintheairlinesector:Descriptionofevents.Thetableshowsthedayonwhichthetransactiontookplace,Date;averagereturnofthestockduringthe
lasttwotradingweeks,Return;minimumreturnofthestockduringthetwo-weekperiodfollowingthetransactionday,Min;daywhenthestockdrops,Drop;andwhy
thestockdrops,Event'sdescription.*meansthatthehypothesisofnon-hedgingcanberejectedata5%level.
SummaryofairlinesectorJan1996–Apr2006
Date Return Min Drop Event'sdescription
AmericanAirlines(AMR)Jan1996–Apr2006
10May00 0.4% −17.6% 24/25May00 UAL'sacquisitionofUSAirways
11May00 0.0% −17.6% 24/25May00 UAL'sacquisitionofUSAirways
31Aug01 −0.4% −39.4% 17Sep01 9/11terroristattacks
10Sep01 −1.4% −39.4% 17Sep01 9/11terroristattacks
24Aug05 0.4% −5.3% 30Aug05 HurricaneKatrina
UnitedAirlines(UAL)Jan1996–Jan2003
11May00 0.3% −12% 24May00 UAL'sacquisitionofUSAirways
6Sep01 −1.0% −43.2% 17Sep01 9/11terroristattacks
DeltaAirLines(DAL)Jan1996–May2005
*1Oct98 −1.7% −11.4% 07/08Oct98 Notidentified
29Aug01 0.0% −44.6% 17Sep01 9/11terroristattacks
19Sep02 −5.2% −24.4% 27Sep02 Expectedquarterloss
9Jan03 2.1% −15.7% 21/22Jan03 Restrictionsonalliance
Boeing(BA)Jan1996–Apr2006
24Nov98 −0.2% −22.0% 02/03Dec98 Productionscaleback
29Aug01 −0.4% −25.0% 17/18Sep01 9/11terroristattacks
5Sep01 −0.8% −25.0% 17/18Sep01 9/11terroristattacks
6Sep01 −0.9% −25.0% 17/18Sep01 9/11terroristattacks
*7Sep01 −1.9% −25.0% 17/18Sep01 9/11terroristattacks
*17Sep01 −5.6% −25.0% 17/18Sep01 9/11terroristattacks
KLMJan1996–Nov2001
5Sep01 −1.9% −31.6% 17/18Sep01 9/11terroristattacks
tradersandδm,m=1,…,M,theexpectedreturngeneratedbytraderm.Underthenullhypothesisalloptiontradersareuninformed.
Formally,thismultiplehypothesisreadsH0,m:δm=0,m=1,…,M.Eachhypothesisistestedatsignificancelevelγ,e.g.,γ=10%,
usingatwo-sidet-statistic,i.e.,H isrejectedwhenthecorrespondingt-statisticiseitherbelowthe5thorabovethe95thpercentiles
0,m
ofitsdistributionunderH .Whenthenullhypothesisistrue,allp-valuesbasedont-statisticsareuniformlydistributedbetween0
0,m
and1.Whenthenullhypothesisisnottrue,largeoptionreturnsandcorrespondinglowp-valuesaregeneratedbybothinformedand
luckytraders.Undersuchalternativehypothesis,E[S+]istheexpectedfractionofp-valuesbelowγ/2correspondingtopositiveand
γ
significantt-statistics.ThekeystepistoadjustE[S+]forthepresenceofluckytraders.Theexpectedfractionoftrulyinformedtraders
γ
isE[Tγ+]=E[Sγ+]−π0γ/2.8Thelaststepistheestimationofπ0.Intuitively,largep-valuescorrespondtoestimatedδmnotstatistically
awayfromzeroandhencegeneratedbyuninformedtraders.Thefractionofp-valuesaboveacertainthresholdλisextrapolatedover
theinterval[0,1].Multiplyingthisfractionofp-valuesby1/(1−λ)providesanestimateofπ0.Thisestimationapproachhasbeende-
velopedbyStorey(2002);see,e.g.,Romanoetal.(2008)forareview.Wechooseλusingthedata-drivenapproachinBarrasetal.
(2010).Theobservedfractionofpositiveandsignificantt-statisticsprovidesanunbiasedestimateofE[S+].
γ
Obviously,wedonotobserveoptionreturnsachievedbytraderswithvariousdegreesofprivateinformation.Consistentlywith
ourdetectionmethod,weusethehistoricalprobabilityq ofobservingunusualincrementsinopeninterestandvolume,aswellas
t
highgains,asaproxyforprivateinformation.Theworkingassumptionisthatthesmallersuchprobabilityis,thehigherthedegree
ofprivateinformationoftheoptiontrader.
Foreveryunderlyingasset,foreverydayt,andforeveryoptiontradek=1,…,K inoursample,wecomputethehistoricalprob-
t
abilityqtkasinEq.(2)ofobservinganincrementΔOItkinopeninterestanddistanceZtk:=(Vtk−ΔOItk)betweentradingvolumeand
incrementinopeninterest,andcorrespondingmaximalreturnasinEq.(5).Bydefinition,theprobabilityqtkliesintheinterval
[0,1].WesortinascendingorderallqkanddividesuchunitintervalintoM=1000subintervalsI ,…,I suchthatineverysubinterval
t 1 M
thesamenumberofqkisavailable.ThenwegroupalloptiontradesqkandcorrespondingreturnsrkaccordingtowhichsubintervalI
t t t m
theybelong.ThisprocedureallowsustoconstructMhypotheticaloptiontraders,eachoneofthemcharacterizedbyadifferentdegree
ofprivateinformationandoptionreturns.InsubintervalsI ,m=1,…,M,thelowerthevalueofm,themoreinformedthetraderis,
m
andtherefore,themorelikelyitisthatshewillgeneratelargepositivereturnrk.WithineachsubintervalI ,weregressunadjusted
t m
annualizedoptionreturnsrtkonasubinterval-specificconstantδm,estimatingtheexpectedreturnoftraderm.9
8 Notethatunderthenullhypothesisalltradersareuninformed,i.e.,π0=1,andbydefinitionhalfthesizeofthetestγ/2=E[Sγ+].Thereforetheexpectedfractionof
trulyinformedtradersisE[Tγ+]=0.
9 Intheregression,wedonotadjustoptionreturnsformarketreturnoranyothervariablebecausethefocusisontheabilityoftheoptiontradertogeneratelarge
returns,includingthosereturnsbasedonpredictingfuturemarketorothervariablemovements.Inordertomakeleastsquaresestimationmorerobustweexclude
negativereturnsbelowthe5%empiricalquantile.Theimpactofwinsorizingonthefalsediscoveryrateisvirtuallynegligible.
M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275 271
Fig.1.Detectingabnormaltrades:AmericanAirlines'example.Uppergraphsshowonthex-axismaximaldailyincrementinopeninterestacrossallputoptionswith
underlyingAmericanAirlines(AMR),andonthey-axisthecorrespondingtradingvolume.Upper-leftgraphcoverstheperiodJanuary1997–December2001,and
upper-rightgraphstheperiodJanuary1997–January2006.LowergraphshowscumulativegainsGtinUSDasinEq.(4)fordetectedoptionabnormaltradeonAMR.
Gainscorrespondtothoserealizedbydailyexercising/sellingtheoptions.
AsanexampleFig.3showsestimatedδmforAmericanAirlines.Thelowerthevalueofm,thehighertheestimatedδm,andthe
relationisnearlymonotonic.Moreover,forsmallm,theestimatedδmarepositiveandsignificant,whereasforincreasingm,δm
becomesstatisticallyindistinguishablefromzero.
WebrieflydiscussnowtheestimatesoffalsediscoveryratesforAmericanAirlinesandCitigroup.ForAMR,thetotalnumberof
analyzedoptiontradesamountsat137,000,implyingthateachregressioncoefficientδmhasbeencomputedbyrelayingon137option
returnsrk.TheexpectedfractionoftrulyabnormaltradeshasbeenestimatedtobeE[T+]=9.8%(withstandarderror1.15%,optimal
t
λ=0.65,andγ=0.11),correspondingto98trades.Astheex-anteproceduredetects141abnormaltradesforAMR,thetestresult
suggeststhatsomeofthesetradesmaybeactuallyliquidityuninformedtrades.Incontrast,theex-postprocedureismoreconserva-
tiveanddetectsonly5abnormaltrades,whichimpliesthatthesetradesaremostlikelyabnormaltrades.ForthecaseofCitigroup,
optiontradesamountat246,000andtheestimatedfractionoftrulyabnormaltradesE[T+]=10.6%(withstandarderror1.09%,
optimalλ=0.612,andγ=0.07),correspondingto106trades.Theex-postmethoddetectsonly2abnormaltrades.Thusalsoin
thiscasethedetectionprocedureisconservativeanddetectedtradesaremostlikelyabnormal.Fortheremainingcompanieswe
foundsimilarresults.Becauseofspaceconstraints,figuresandtablesarenotreportedbutavailableuponrequestfromtheauthors.
Finally,toassesstheabilityoftheFDRtestatcontrollingforfalsediscoveries,werunthefollowingexperiment.10Weidentifythe
majornaturaldisasters,suchasfloods,hurricanes,volcaniceruptions,oilspills,andearthquakesfrom2000to2011.Astheexact
timingoftheeventisinprincipleunpredictable,thisshouldruleoutanyabnormaltradethatgenerateslargereturnsuponthe
occurrenceofthenaturaldisaster.Then,weconsideralltheoptiontradesoverthetwoweekspriortotherelevanteventinthe
companiesthatwereaffectedex-postbytheevent.Giventhesetup,nooptiontradeshouldbeclassifiedasabnormal.
10 Wethankananonymousrefereeforsuggestingthisexperiment.
272 M.Chesneyetal./JournalofEmpiricalFinance33(2015)263–275
10000 250
Open Interest
Volume
9000 10−May−2000
24−May−2000
8000 200
Return on the option investment
10−May−2000
7000 24−May−2000 nt
e
m
Open Interest/Volume 456000000000 110500n on the option invest
ur
et
3000 R
2000 50
1000
0 0
Apr00 May00 Jun00 Jul00
Fig.2.Selectedabnormaltrade:AmericanAirlines'example.SelectedputoptionforabnormaltradingwithunderlyingstockAmericanAirlines(AMR)beforetheUnited
Airlines(UAL)announcementof$4.3billionacquisitionofUSAirwaysinMay2000.Thesolidlineshowsthedailydynamicofopeninterest,thebarsshowthecorre-
spondingtradingvolume(lefty-axis)andthedash–dotlinetheoptionreturn(righty-axis).Theemptycircleisthedayofthetransaction,thefilledcircleisthedayof
theannouncement(partiallycoveredbythehighestbar).Thisputoptionhadastrikeof$35andmaturedattheendofJune2000.
Table3providesthelistofnaturaldisastersandaffectedcompanies,aswellastherationaleforincludingthesecompanies.Forex-
ampleweconsiderBritishPetroleumbeforetheoilspillintheU.S.GulfCoastin2010.Thelistofcompaniesisconstrainedbyoption
dataavailability,i.e.,openinterestandvolumeforindividualoptions.Computingq asinEq.(2)andconsideringq b5%,averysmall
t t
numberofsuspiciousoptiontradesisdetectedoverthetwoweekspriortoanaturaldisaster.WhenapplyingtheFDRtestallsuch
tradesareattributedtoluck,confirmingthevalidityofourprocedure.Detailedtestresultsareavailablefromtheauthorsupon
request.
7.Robustnesschecks
Theinputparametersinourdetectionprocedureare:thelengthNoftheestimationwindow,chosentobeN=500tradingdays,
usedforthecomputationoftheex-anteprobabilityq,theconditionaldistributionofVsell,non‐hedge,andthequantilesqrmtaxandqGt;the
timeperiodafterthetransactiondayusedforthecomt putationofrtmax,chosentobe10t tradingdays;thetimehorizonατtusedfαo0rthe
calculationofthegainsGt,chosentobe30tradingdays;thequantilelevelsαandα′inqrαmtaxandqGαt0usedforthecomputationofthesets
Ω3andΩ4,chosentobeα=90%andα′=98%;andtheprobabilitylevelusedtoselecttradesbelongingtothesetΩ1,chosentobe
5%.Inwhatfollowswesettheinputparameterstodifferentvaluesandwerepeatallpreviousanalysisforallcompanies.Tosavespace
wereportonlysomeoftheresultsandforafewcompaniesgivingasenseoftherobustnessofourresults.Additionalresultsare
availablefromtheauthorsuponrequest.
WhenvaryingthelengthoftheestimationwindowNbetween200and1000(allotherparametersbeingunchanged),thenumber
ofselectedtransactionsdoesnotchangesignificantly.ForexampleinthecaseofAMR,weselected5abnormaltradeswhenconsid-
eringthelasttwotradingyears(N=500days);forN∈[200,1000]thenumberofdetectedabnormaltradesrangesbetween4and6;
forUAL,wedetected2abnormaltradeswhenconsideringthelasttwotradingyears(N=500days);thisnumberremainsunchanged
withrespecttotheoriginalchoiceforNN450anddecreasesbyonewhenN∈[200,450].InthecaseofBACandAT&T,thedeviation
fromtheoriginalnumberofselectedtradesislessthan2.Withrespecttothechoiceofthetimeperiodusedforthecomputationof
rtmaxandτt,ourresultsarealsorobust.Weletthelengthofthefirstperiodvaryintherange[1,30]daysandthesecondonein[1,40]
days.InthecaseofAMR,thenumberoftransactionsrangesfrom2to8,beingthereforecenteredaroundtheoriginalnumberandwith
asmalldeviationfromit.ForUAL,thecorrespondingrangeisfrom1to4,forBACfrom2to8andforAT&Tfrom1to6.Thenumberof
detectedtradesisobviouslyadecreasingfunctionofαandα′(allotherparametersbeingunchanged).InthecaseofAMR,when
{α,α′}∈[0.85,0.95]×[0.96,1],thenumberoftransactionsselecteddoesnotexceed15.ForUAL,thenumberofselectedtradesvaries
between1and10,forBACbetween5and25,andforAT&Tbetween1and18.Finally,withrespecttotheprobabilitylevelusedto
determinethesetΩ1,ourfindingsareveryrobustaswell.Whenincreasingthelevelfrom1%to10%,thenumberoftradesselected
forAMRvariesbetween1and6;forUALitrangesbetween2to4,andforBACandAT&Tfrom1to7.Wesimultaneouslychanged
severalparametersandfoundthatthenumberofdetectedtransactionsdoesnotchangesignificantlyandinalmostallcasesin