Robust hedging strategy

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Computers&OperationsResearch39(2012)2528–2536

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Computers&OperationsResearch

journalhomepage:http://wendang.chazidian.com/locate/caor

Robusthedgingstrategies

RaquelJ.Fonsecan,Berc-Rustem

DepartmentofComputing,ImperialCollegeofScience,TechnologyandMedicine,180Queen’sGate,LondonSW72AZ,UK

articleinfo

Availableonline8January2012Keywords:

Robustoptimization

InternationalportfoliooptimizationHedging

ForwardcontractsOptions

abstract

Whileinvestinginforeignassetsmaybringadditionalbene?tsintermsofriskdiversi?cation,itmayalsoexposetheportfoliotoafurthersourceofriskderivedfromchangesinthevalueoftheforeigncurrencies.Hedgingstrategiesforinternationalportfolioshaveusuallyfocusedontheuseofforwardcontractstomitigatethecurrencyrisk.Weproposeanalternativeformulationaimedatthereductionoftheoverallportfolioriskbyassumingthereturnsareuncertainandmaximizingtheportfolioreturnfortheworstpossibleoutcomeofthereturns.Thistechniqueknownasrobustoptimizationprovidesa?rstguaranteeontheportfoliovaluethankstothenon-inferiorityproperty.Wefurthercomplementourapproachwithforwardcontractsontheforeignexchangeratesandoptionsontheassets.Becausethetotalreturnonanyassetwillbetheproductofitslocalreturnandcurrencyreturn,themodelsproposedarebilinearandnonconvex.Areformulationofboththeuncertaintysetandtheobjectivefunctionasasemide?niteproblemwillyieldanapproximatetractablemodel.Wecomparethehedgingalternativesproposedwithsimulatedandhistoricalmarketdataandconcludeontheirrelativebene?ts.

&2012ElsevierLtd.Allrightsreserved.

1.Introduction

Inhisgroundbreakingworkinportfoliotheory,Markowitz[22]suggestsportfoliooptimizationmodelswithatradeoffbetweenriskandreturn.Insuchmodels,portfoliorisk,usuallymeasuredbythevariance(Var½RðwÞ??),isminimizedsubjecttoadesiredexpectedreturn(rtarget)minVar½RðwÞ??

w

ð1aÞð1bÞð1cÞð1dÞ

s:t:

E½RðwÞ??Zrtarget

10w¼1wZ0:

As?nancialassetsarelessthanperfectlycorrelated,acertaindegreeofriskmaybeeliminatedbydiversifyingtheportfolio.ThesameriskreductionpotentialwasrecognizedlateronbyGrubel[16],whosuggestsincludingforeignassetsinaportfolio,asthecorrelationbetweentheseanddomesticassetswouldbelowerthanthatamongdomesticassetsonly.

Whilethepotentialforportfoliodiversi?cationincreases,investmentinforeignassetscomestogetherwithanewsourceofriskduetochangesintheforeignexchangerate.Eveniftheassetreturnsintheirlocalcurrencyimprove,gainsmaybe

Correspondingauthor.Presentaddress:DepartmentofStatisticsandOperationalResearch,FacultyofSciences,UniversityofLisbon,Portugal.

E-mailaddress:rjfonseca@fc.ul.pt(R.J.Fonseca).0305-0548/$-seefrontmatter&2012ElsevierLtd.Allrightsreserved.doi:10.1016/j.cor.2011.12.021

n

completelyerodedbyadepreciationinthevalueofthecurrency.Manyauthors,awareofthisadditionalrisk,studytheimpactofhedgingthecurrencyriskwithforwardcontractsandhowhedgedportfoliosperformedcomparedtounhedgedones,seeforexam-pleLevyandSarnat[20],GlenandJorion[15],andEunandResnick[11].They?ndthat,whiletheuseofforwardsasahedginginstrumenttoreducethecurrencyriskcontributestothebetterperformanceofhedgedportfolios,http://wendang.chazidian.comrsenandResnick[19]drawattentiontothefactthatallpreviousstudieswereex-postinnatureandassesshowhedgingstrategiesperformwhentheinputparametersmustbeestimatedfromhistoricaldata.They?ndthattheuncertaintysurroundingparameterestimationistoohighonwhichtobaseanysophisticatedhedgingstrategies.

Robustoptimizationisarecentparadigmintheoptimization?eldthatdealswiththeuncertaintyinherenttoparameterestimationdirectlyinthemodel,byoptimizingtheobjectivefunctionfortheworstcaseoftheuncertainparameters.Theseareassumedtomaterializewithinsomeintervaldesignatedasanuncertaintyset,andwhichmaybebuiltuponprobabilisticguaranteesand/ortheinvestor’sknowledgeandexpectationsregardingtheirfuturevalue.FirststudiedbyBen-TalandNemirovski[5]andEl-GhaouiandLebret[9]inthe?eldofcontroltheory,therearealreadyseveralapplicationsto?nanceandportfoliooptimization,seeFabozzietal.[12]andRustemandHowe[27].

Theideaofcombiningsampleandstatisticalinformationwithexperts’viewsisalsothebasisforBayesian[2]andBlack-Litterman[6]approaches.Whileitissometimesargued

R.J.Fonseca,B.Rustem/Computers&OperationsResearch39(2012)2528–25362529

thatrobustoptimizationisnodifferentfromshrinkageestimation[12],theworkofSchottleetal.[28]showsthatthesetwoalternativeapproachesmayalsoberobusti?ed.TheystudyhowtheBlack-LittermanmodelmaybeseenasaspecialcaseoftheBayesianapproach,butthenarguehowthetwoapproachesyieldquitedifferentresultswhenparameteruncertaintyisrobustlyaccountedfor.

Notethatwhilethepreviousworkshavefocusedontheissueofuncertaintypostparameterestimation,robusttechniquesmayalsobeappliedfromthestartoftheproblem,attheestimationstage.Ozmenetal.[24]applyrobustoptimizationtoaregressiontoolinthecontextofdataminingandestimationtheory.Inthe?eldofmachinelearningandlinearregression,therehavealsobeenapplicationsofrobustoptimizationtosupportvectormachines,see[3].

Theapplicationofrobustoptimizationtechniquestoaninter-nationalportfoliomodelhasbeenintroducedbythisauthorinFonsecaetal.[13].Thisworkextendstheresultsof[13]inseveralways.First,weextendthesemide?niteprogrammingmodeltoallowtheinvestortobuyforwardcontractsinordertohedgeagainstthecurrencyrisk.Forwardsarethetraditional?nancialinstrumentusedtoprotectinvestmentsfromdecreasesintheforeignexchangerates.Becausetheforeignexchangerateiseffectivelylockedatthevalueoftheforward,theperformanceofthehedgedependsontheinvestor’scapacitytopredicttheamountsheisgoingtoreceiveinthefuture.Second,weusesimulateddatatoassesstheaverageperformanceoftherobustmodelbyitself,http://wendang.chazidian.comstly,wecompareallthreemodelsusinghistoricalmarketdatafromworldindexes.Thebacktestingexperimentsallowustoassesshowthealternativesproposedbehaveintermsofriskreductionandwealthmaximization.

Therestofthispaperisorganizedasfollows.Section2describestherobustinternationalportfoliooptimizationmodelandthesemide?niteprogrammingapproximationthatenablesatractableconvexformulationoftheoriginalproblem.InSection3wepresentdifferent?nancialinstrumentsavailabletotheinves-tor,whosegoalistoreducetheportfolioexposuretoanyunfavorablechangesinthevalueoftheassetandintheforeignexchangerates.WepresentaseriesofnumericalexperimentswiththesuggestedmodelsandassesstheirrelativeperformanceinSection4.WeconcludeinSection5.2.Robustoptimization

Assumetherearenavailableassetsinthemarket,denomi-natedinmforeigncurrencies.ThecurrentandthefuturepriceoftheithassetinitslocalcurrencyisdenotedbyP0iandPi,

respectively.Thelocalreturnofassetiisra

0i¼Pi=Pi.Similarly,

thereturnonanycurrencyjisre00

j¼Ej=Ej,whereEjandEjarethefutureandthecurrentspotexchangerateofthejthcurrency,respectively.Bothquantitiesareexpressedintermsofthebasecurrencyperunitoftheforeigncurrencyj.Thetotalreturnonanyassetiistheproductofthelocalreturnsraiandtherespective

currencyreturnsre

j.Weadditionallyde?neanauxiliarymatrixOthatassignstoeachassetexactlyonecurrency.Ifwede?neoijastheijthelementofO,thenwehave

(

o¼1iftheithassetistradedinthejthcurrency;

ij0otherwise:ð2Þ

AspointedoutbyBroadie[8],mean-varianceproblemsarequitesensitivetoerrorsintheestimationoftheparameters,particularlyofthereturns.Heshowsthatportfoliosonthe

estimatedfrontierwouldusuallyoverweightthosesecuritieswiththelargestpositiveestimationerrorsinthereturns.Inviewofthis,robustoptimizationcouldberegardedasacomplementtoexpectedvalueoptimization,asittriestoaddresstheproblemoftheuncertaintyinherenttotheestimationofthefuturereturnsbyconsideringthesetoberandomvariables.Whileinmean-var-ianceproblems,futurereturnsareassumedtobeequaltotheestimatedmeanreturn,inrobustmodelsreturnsareassumeduncertainbutexpectedtomaterializewithinsomeinterval,commonlydesignatedasuncertaintyset.Therobustcounterpartoftheinternationalportfoliooptimizationmodelismax

w

ðramin,reÞAX

½diagðraÞOre??0w

ð3aÞs:t:

10w¼1ð3bÞwZ0,

ð3cÞ

whereXrepresentstheuncertaintysetwherethefuturereturnsareexpectedtomaterialize.Thecompletede?nitionofXwillbeexplainedlaterinthissection.Theconstructionoftheseuncer-taintysetsre?ectstheinvestor’sexpectationsandbeliefsregard-ingthefuturevalueofthereturnsandmaybecharacterizedbysomeprobabilisticmeasures.

Asboththelocalassetandthecurrencyreturnsareuncertainparameters,formulation(3)isbilinearandthereforenotconvex.Foracompletedescriptionofnonlinearmethodsappliedto?nancialproblems,particularlyportfoliooptimization,pleaseseeBiggs[1].Toovercomethenonlinearity,Fonsecaetal.[13]proposeasemide?niteprogrammingapproximation,wherethebilinearsemi-in?niteconstraintisreplacedwithalinearcombi-nationofmatrices.Theyshowthattheoriginalrobustmodel(3)maybeapproximatedbythefollowingproblem:maxw,k,f

f

ð4aÞ

s:t:SÀ

XtllWlk0

ð4bÞ

l¼1

10w¼1ð4cÞw,kZ0,ð4dÞ

where200

3

S¼6Àf61

400

diagðwÞO770

1

00

5:

OdiagðwÞ

Thisreformulationispossiblethankstothefollowingresults,

whoseproofisgiveninBen-Taletal.[4]:

Lemma1(S-lemma).GiventwosymmetricmatricesWandSofthesamesizeandassumingtheinequalityn0WnZ0isstrictly

feasible,thatis,0

W40forsomeARk,thenthefollowingequivalenceholds

½n0WnZ0)n0SnZ0??3(lZ0:SklW:

ð5Þ

Symbolkindicatesthatthematrixispositivesemide?nite.Lemma2(ApproximateS-lemma).ConsidertsymmetricmatricesWlwithl¼1,y,tandthefollowingpropositions:

(i)(kARtwithkZ0andSÀP

tl¼1llWlk0;

(ii)n0SnZ0,8nAX:¼fnARk:e0

1n¼1,n0WlnZ0,l¼1,...,tg.ForanytAN,(i)implies(ii).

2530R.J.Fonseca,B.Rustem/Computers&OperationsResearch39(2012)2528–2536

Theapplicationoftheaboveresultinvolves?rstaggregatingthereturnsðra,reÞinthesinglevectorx

n¼½1rare??0,

andsubsequentlywritingtheuncertaintysetXintheform

X¼fnARk:e01n¼1,n0WlnZ0,l¼1,...,tg,

ð6Þ

wheree1isabasisvectorinRkwhose?rstelementis1andalltheothers0.Thisconstructionguaranteesthatthe?rstcompo-nentofthevectornisequalto1.EachmatrixWlrepresentsadifferentconstraintinthesetabove.

TheuncertaintysetXwillresultfromtheintersectionoftwodifferentsets.Theriskassociatedwiththeassetandthecurrencyreturnsisexpressedbytheellipsoidalregion

"ðd2À½a00??SÀ1½0e0??0Þ½a00

??SÀ1

#W1¼SÀ1½a00

??0ÀSÀ1:Thisre?ectstheideaofajointcon?denceinterval,where

deviationsofthereturnsfromtheirexpectedvaluesaandareweightedbythecovariancematrixS.Additionally,afurtherconstraintontheforeignexchangeratesisincludedtore?ecttheirtriangularrelationship.Foreachpairofcurrencies(j,k),acrossexchangerateisde?ned,whoseuncertainreturnsxjkareassumedtobewithinalower(L)andanupperbound(U):LrxkjrU.Aftersomealgebraicmanipulation,aquadraticinequalitymaythenbeestablishedbetweeneachpairofforeignexchangeratereturns

ÀULðre2re2ee

kÞÀðjÞþðUþLÞrkrjZ0,

ð7Þ

whichwillallowustode?nethesetofconstraintsn0WlnZ0,forl¼2,y,t,where

20003Wl¼64000700O5,ð8Þ

ewith

Oe¼ÀðULÞeke00kÀeje0jþ11

0ðUþLÞejekþðUþLÞekej,

ð9Þ

whereek,ejarethecanonicalbasisvectorsinRm.

WehaveusedLemma2,becauseouruncertaintysetconsistsofseveralintersectingregions.Thismeans,however,thattherefor-mulatedproblem(4)onthedecisionvariableswandkisonlyaconservativeapproximationtotheoriginalproblem(3).Althoughmorecomplexthanthestandardlinearprogram,formulation(4)carriesafewadvantages.Itallowsfortheseparateconsiderationofthelocalassetsandthecurrencyreturns,withoutanyassump-tionontheircorrelationordistributionalproperties.Furthermore,itincludestheuncertaintyderivedfromtheestimationofthereturnsdirectlyinthemodelformulationwithoutjeopardizingitstractabilityandnumericalimplementation.Theresultingsemide-?niteprogramisaconvexoptimizationproblem,asbothitsobjectivefunctionandconstraintsareconvex.

Theapplicationofrobustoptimizationtechniquesmaybeseenasa?rstlevelofinsuranceavailabletotheinvestor.Itprovidesanalternativeviewofuncertainty,thatitmaybeusefulfortheassessmentofworstcaseperformanceandrisk.AslongasanyvariationoftheassetandthecurrencyreturnsstayswithintheboundariesoftheuncertaintysetX,theinvestorwillobtainaportfolioreturnhigherthan(orintheworstcaseequalto)thevalueoftheobjectivefunctiondeterminedin(4)—non-inferiorityproperty.

Notethatourportfoliooptimizationformulationdoesnotexplicitlyincludeanyriskmeasure.AdditionalconstraintsontheportfoliovarianceoranyotherriskmeasuresuchastheSharperatiomaybeeasilyaddedtothemodel.Wewouldliketopoint

out,however,thatthede?nitionoftheuncertaintyset,particu-larlyitssizemeasuredbyd,includesinitselftheinvestor’s

attitudeanddesiredlevelofrisk.Whileparameterdmaybeequivalenttosomecon?denceintervalfromanappropriateprobabilitydistribution,agreatervalueofdwillleadtoalargerset,thusrenderingthemodelmoreconservative.Thiswouldbemoreadequatetoarisk-averseinvestor,whowishestobepreparedforalltheworstoutcomespossible.Onthecontrary,arisk-takerinvestormightpreferasmalleruncertaintyset,withalowerdvalue,andwithworstcasereturnsclosertotheestimated.AdetailedstudyoftherelationshipbetweenrobustoptimizationandriskmeasuresmaybefoundinNatarajanetal.[23].

IntheirworkEl-Ghaouietal.[10]showthecloserelationshipbetweentheworstcaseo-Value-at-Riskandrobustportfoliooptimization,whenthetotalreturnlinearlydependsontheuncertainparameters,thatis,theassetreturns.Thetwoproblemsareequivalentwhentheparameterdsetatd¼pmeasuring????????????????????ð1ÀoÞ=o?thesizeofanellipsoidaluncertaintyset,is.RecentworksontheapplicationofrobustoptimizationtodownsideriskmeasuresincludethecomputationoftheworstcaseConditionalValue-at-Risk,whenonlypartialinformationontheprobabilitydistributionofthereturnsisknown,ZhuandFukushima[35].Zymleretal.[36]haveextendedtheworkin[10]totheconsiderationofnonlinearportfoliosbyaddingderivatives.Theyconcludethatundercertainconditions,optimizinganon-linearportfoliofortheworstcaseofthereturnsisequivalentto?ndingtheworstcaseValue-at-Riskoverallprobabilitydistributionswiththesame?rsttwoordermoments.TheuseoftheworstcaseValue-at-Riskwouldallowamoredirectcontrolovertheportfolioriskbytheinvestor,whowouldobtainavalueoverwhichlosseswouldonlyoccurwithprobabilityo.

Inthenextsection,weintroducetwoformalhedginginstru-ments:forwardcontractsandoptions.Thesewillallowtheinvestorto?xwithcertaintytheforeignexchangerateortheassetpricetoholdinthefuture.Weinvestigatehowtheavail-abilityoftheseinstrumentsandrobustoptimizationinteractprovidingtheinvestorwithasetoftoolstoreduceportfoliorisk.

3.Hedgingstrategies

Althoughtheoptimizationfortheworstcaseofthereturnsprovidestheinvestorwithsomeguarantees,shemayseektohedgeagainstthecurrencyandtheequityriskthroughmoreformalinstruments.Forwardcontractsandquantooptionsaretwodifferenttypesofhedginginstrumentswhichallowtheinvestortolockinaspeci?cforeignexchangerateorassetprice.Contrarytorobustoptimization,thisguaranteewouldalsobevalidevenwhenthereturnsfalloutsidetheuncertaintysetconsidered.

3.1.Forwardcontracts

Additionallytorobustoptimization,theinvestormayseekfurtherguaranteesintheformofaforwardcontract,wheresheagreestodayonacertainamounttobeexchangedinthefutureatade?nedforeignexchangerate.Notethat,althoughthisisabindingagreementwheretheamountandtheexchangeratearealreadyde?ned,weareneverabletofullyhedgetheportfolioreturn,aswedonotknowapriorithevalueoftheforeignassetsholdingsinthefuture.Thisvaluedependsontheparticularevolutionofthelocalassetreturns.Wede?nethereturnontheforwardagreementasrf¼F=E0,whereFisthecontractedforwardrateandE0thecurrentspotrate.Contrarytothelocalassetandtheforeignexchangeratereturns,thereturnontheforwardrateisknownwithcertainty.Theinternationalportfoliooptimization

R.J.Fonseca,B.Rustem/Computers&OperationsResearch39(2012)2528–25362531

modelwithforwardsmaybewrittenasmax

mine0wf

ð10aÞw,wf

ðra,reÞAX

½diagðraÞOr??0wþðrfÀreÞs:t:

10w¼1ð10bÞw,wf

Z0,

ð10cÞ

wheretheuncertaintysetXisasin(6).

Whenagreeingonaforward,theinvestormustspecifytheamountofforeigncurrencythatshewishestosell.Thatamountwillbeexchangedatmaturitydateatthede?nedforwardrateandtherestoftheamount,ifitexists,willbetranslatedbacktothedomesticcurrencyatthecurrentspotrate.If,onthecontrary,theagreedamountisgreaterthantheforeignholdingsdueforexampletoaunexpecteddecreaseinthelocalassetreturns,theinvestormustbuytheremainderamountinthemarketinordertohonorthecontract.

Informulation(10),nolimitswereimposedontheamountofforwardscontracted.Toavoidbeingtooconservativeandtoallowagainfromapotentialincreaseinthecurrencyreturn,wecouldrestricttheweightoftheforwardswfintheportfolioto:1.Theamountofforeignholdingsatcurrenttime:wfrO0w.2.Theexpectedvalueoftheforeignholdings:wfrO0diagðÞw.3.Theworstcasevalueoftheforeignholdings:wfrO0

diagðraÞw,8raAXa.Whilethe?rsttwoalternativesoffernodif?cultiesinimplemen-tation,thethirdconstraintisofasemi-in?nitetype.Withthehelpofrobustoptimizationtechniques,weareabletoreformulatetheconstraintinatractableway.Westartbynotingthatwecanwritetheconstraintaboveforeachoftheweightswfastheminimizationproblemminr

a

O0idiagðwÞraÀwf

i

ð11aÞs:t:

ðraÀraÞ0SÀ1aa

aðrÀrÞrd2

a

ð11bÞraZ0:

ð11cÞ

Thisisasecondorderconeprogram,andinthatcasestrongdualityholds.Thismeansthatwecancomputethedualandreplacethesemi-in?niteconstraintbythevalueoftheobjectivefunctionofthedualproblemasbelowmax

y

ðraÞ0½diagðwÞOiÀy??ÀdavÀwfi

ð12aÞs:t:

JS1=2aðdiagðwÞOiÀyÞJrv

ð12bÞyZ0:

ð12cÞ

TheobjectivefunctionmaybereformulatedasbeforewiththehelpofLemma2

½diagðraÞOre??0wþðrfÀreÞ0wfZf

ð13aÞ3½diagðraÞOre??0wÀðreÞ0wfþðrfÞ0wfÀfZ0ð13bÞ)x0

SxZ0,ð13cÞwhere

2132

ððrfÞ0wfÀfÞ

0À1ðwfÞ

3

x¼64ra7and

S¼660

1diagðwÞO77re

5

4

À1f

1

05:

w

OdiagðwÞ

ð13dÞ

Theuncertaintysethasnotchanged,thereforetheW-matrices

remainthesame.

Withtheabovereformulationsinplace,andtakingyiastheithcolumnofmatrixY,werewriteour?nalproblemas

maxð14aÞ

w,wf,l,y,f

f

s:t:SÀ

XtllWlk0

ð14bÞ

l¼1

ðraÞ0½diagðwÞOiÀyJS1=2

fi??ÀdaaðdiagðwÞOiÀyiÞJÀwiZ0,8i¼1,...,m

ð14cÞ10w¼1ð14dÞw,wf,k,YZ0:

ð14eÞ

Wemayalsoimposeaconstraintonthedesiredexpectedreturntoavoidbeingoverpessimisticinourportfolioallocation

Eð½diagðraÞOre??0wþðreÀrfÞ0wfÞZrtarget,

ð15aÞ3½diagðaÞO??0wþ1ef0ftarget

traceðSOÞþðÀrÞwZr

,ð15bÞ

where

"

#O¼

0diagðwÞO

O0diagðwÞ

:

ThecompletederivationoftheexpectedvalueoftheproductoftworandomvariablesmaybefoundinRustem[26].Assumingthattheforwardreturnwouldusuallyhaveavaluebelowtheexpectedcurrencyreturnbutabovetheworstcase,withoutthisconstraintandsincewearemaximizingthereturnfortheworstpossibleoutcome,theoptimalsolutionwouldbetoinvestasmuchaspossibleinforwardcontracts.

Themaindisadvantageofusingforwardsasahedginginstru-mentistheirlackof?exibility.Sincetheinvestorcannotmoveawayfromtheagreedcontract,ifthespotrateincreasesbeyondtheagreedforwardrate,shewilloverpassontheopportunityforahigherreturn.Hedging,assuch,doesnotguaranteeabetterreturn,andinsomecases,mayevenleadtoaworseoutcomethanwithnohedgingatall.Options,ontheotherhand,offertheinvestorthepossibilityofnotexercisingherrightsifthesituationinthemarketisinherfavor.3.2.Quantooptions

Duetotheasymmetrybetweentheuncertaintyoftheamounttobereceivedinthefutureinforeigncurrencyandtheneedtospecifyanamounttodaytobuyaforwardcontract,Giddy[14]arguesthatthisisnotthemostadequatehedginginstrumentinsuchsituations.Giventhattheinvestorisunsureoftheamountsheisgoingtoreceiveduetoparticularchangesinthevalueofthelocalasset,currencyoptionswouldbeabettersuitedinstru-menttoprotectagainstthecurrencyrisk.Steil[29],ontheotherhand,statesthattheunderlyingassetincurrencyoptionsistheforeignexchangerate,whichisnotthecontingencyoneistryingtoprotectinthe?rstplace.Therefore,inordertohedgeagainstthecurrencyandtheassetriskinaninternationalportfolio,theinvestorwouldhavetobuybothcurrencyandequityoptions.

Quantooptionsareanexotictypeofoptions,whosepayoffisdenominatedinonecurrency,butwhichissubsequentlytrans-latedata?xedrateintoanothercurrency[34].Thisparticulargroupofoptionsmaybequiteanadequatetoolforthehedgeofinternationalportfolios,astheytakeintoaccountthecorrelationbetweentheassetandtherespectiveforeignexchangerate[17].Topaloglouetal.[31,32]studytheperformanceofquantooptionsasahedginginstrument,but?ndthatonly‘‘in-the-money’’putoptionshaveacomparableperformancewithforwards.

2532R.J.Fonseca,B.Rustem/Computers&OperationsResearch39(2012)2528–2536

ThepayoffofaquantoputoptionQisthedifferencebetweenthestrikepriceKandthespotpriceoftheunderlyingassetPatmaturitydate,translatedtothebasecurrencyoftheinvestorataspeci?edexchangerateQ¼maxf0,ðKÀPÞg:

ð16Þ

BoththestrikepriceKandthepriceoftheassetParedenomi-natedinforeigncurrencyandtranslatedatthe?xedforeignexchangerateexpressedinunitsofthebasecurrencyperunitoftheforeigncurrency.Theexchangeratechosenisusuallytheforwardratewiththesamematurityastheoption.Theoptionsconsideredhavethesamematurityastheportfoliorebalancingperiod,1month.Thepricingmodelforquantooptionswasdevelopedin1992byReiner[25].AlthoughbasedonthesameassumptionsastheBlackandScholesmodel[7],Reiner’smodelincludesthecorrelationcoef?cientrbetweentheforeignequityandtherespectiveforeignexchangerate.

AsinFonsecaetal.[13],inordertoincludequantooptionsintherobustoptimizationmodel,wede?neasrqijthereturnonthejthquantooptionontheithforeignasset

(rqKijÀP)

i

Þij¼max0,(pij

rqKijÀP0ira

)

i

Þij¼max0,p:ð17ÞijWewouldliketheportfolioconstraintstobesatis?edforallthepossiblevaluesofthereturnswithinthesupport.Wedenotebywqthevectorofweightsofquantoputoptionsintheportfolioandformulatethehedgingmodelwithoptionsas

wmax,wq,f

f

ð18aÞ

s:t:

½diagðraÞOre??0wþrq0

wqÀfZ0,

8ðra,reÞAX,rq¼fðraÞ

ð18bÞ

10wþ10wq¼1ð18cÞw,wqZ0:

ð18dÞ

Lemma2mayagainbeappliedinordertoderiveanequivalenttractableformulationtotheoptionsproblem(18).Afterrewritingtheadditionalconstraintsregardingthequantooptionsandaugmentingthevectornbytheoptionsreturnsvectorrq,weareabletoformulatetheoptionsmodelas

wmax,wq,k,f

f

ð19aÞ

s:t:SÀ

XtllWlk0

ð19bÞl¼1

10wþ10wq¼1ð19cÞw,wq,kZ0,ð19dÞ

where

2wÞ

36Àf00

1q0ð07

S¼666

00

diagðwÞO

76007401O0

diagðwÞ

77:1q0

5

w

Byholdinginherportfoliobothanassetandaputoptiononthesameunderlying,theinvestoriseffectivelysettingamini-mumvaluefortheassetattheleveloftheoption’sstrikeprice.Becausewearedealingwithforeignassets,thecurrencyriskisalsoeliminatedbytranslatingthepotentialpayoffstothedomes-ticcurrencyoftheinvestorata?xedforeignexchangerate.Inthe

nextsection,weperformaseriesofexperimentswhereweassesstheperformanceofthemodelsproposed.

4.Numericalresults

Thetheoreticalframeworkdevelopedintheprevioussectionswillnowbeusedtocomputeoptimalsolutionstoourinterna-tionalportfoliomodel.Westartbystudyingtheportfoliocompo-sitionandtheimpactofconsideringeitherforwardsoroptionsintheexpectedportfolioreturn.InSection4.2,weconductaseriesofexperimentswithsimulateddatainordertocomparativelyassesstheperformanceofalltheproposedmodels,whileinSection4.3weusehistoricalmarketdata.ThemodelswereimplementedinYALMIP[21]andsolvedwiththesemide?niteprogrammingsolverSDPT3[30,33].Resultswerebasedonstatis-ticaldataintheperiodfromDecember2001toJanuary2011fortheMSCIIndicesEuro,UK,USA,Japan,BrazilandChina,andtheirrespectivecurrencies,seeTable1.4.1.Portfoliocomposition

Westartbyassessingtherelationshipbetweentheworstcasereturnandthesizeoftheuncertaintysetforallthemodelsproposed,withandwithoutanexpectedtargetreturn,seeFig.1.TheconstructionofthesecurvesfollowsthesamereasoningasintheMarkowitzef?cientfrontiers:wewouldliketomeasuretheimpactofanincreaseinthelevelofriskontheportfolioguaranteedreturn.Thesizeoftheuncertaintysetisconsideredasariskmeasure:largeruncertaintysetsindicateagreaterconcernwiththereturnsuncertaintyandoverallagreaterimportancegiventoriskoverreturns.Ontheotherhand,smalleruncertaintysetsleadtoworstcaseorguaranteedreturnsveryclosetotheexpectedreturns,indicatingtheinvestorislessriskaverseandprivilegeshighreturnsoveralowerrisklevel.

Notethatwhileitwouldbedesirabletobeabovethecurve,asforanygivenlevelofriskmeasuredbydwewouldobtainahigherguaranteedreturn,thisisnotpossiblegiventheexpecteddistributionofthereturns.Similarly,itisnotef?cientnorrationaltobebelowthecurve,asweareabletoobtainahigherguaranteedreturn.Theseportfoliosaredominatedbytheportfo-liosonthecurve.Inthissense,theoverallmeaningofthesecurvesisequivalenttothatofthestandardef?cientfrontiers.Alargervalueofdindicatesalargeruncertaintyset,andtherefore,asexpected,asmallerworstcasereturninallthecasesconsidered.Whenanexpectedreturnconstraintisincluded,asinFig.1(b),theworstcasereturnissmaller,astheweightsneedtobeallocateddifferentlyinordertosatisfytheexpectedreturn.Thisdifferenceismoreaccentuatedforhighervaluesofd.

Table1

AnnualreturnrateandstandarddeviationoftheMSCIIndices(Dec-01toJan-11).MSCIindicesAnnualret.(%)Std.(%)China18.9929.31Euro0.4020.03JapanÀ0.5418.25UK4.1115.18Brazil23.0524.88USA2.5316.39EUR4.448.87GBP1.118.17JPY3.408.19BRL2.6114.12CNY

2.28

1.27