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HEDGINGVOLATILITYRISKMenachemBrennerSternSchoolofBusinessNewYorkUniversityNewYork,NY10012,U.S.A.Email:mbrenner@stern.nyu.eduErnestY.OuABNAMRO,Inc.Chicago,IL60604,U.S.A.Email:Yi.Ou@abnamro.comJinE.ZhangDepartmentofEconomicsandFinanceCityUniversityofHongKong83TatCheeAvenueKowloon,HongKongEmail:efjzhang@cityu.edu.hkFirstversion:August2000Thisversion:November2000Keywords:Straddle;CompoundOptions,StochasticVolatilityJELclassification:13_________________________________Acknowledgement:ThepracticalideaofusingastraddleastheunderlyingratherthanavolatilityindexwasfirstraisedbyGaryGastineauinadiscussionwithoneoftheauthors(M.Brenner).WewouldliketothankDavidWeinbaumforhishelpfulcomments.2HEDGINGVOLATILITYRISKAbstractVolatilityriskhasplayedamajorroleinseveralfinancialdebacles(forexample,BaringsBank,LongTermCapitalManagement).Thisriskcouldhavebeenmanagedusingoptionsonvolatilitywhichwereproposedinthepastbutwereneverofferedfortradingmainlyduetothelackofatradableunderlyingasset.Theobjectiveofthispaperistointroduceanewvolatilityinstrument,anoptiononastraddle,whichcanbeusedtohedgevolatilityrisk.Thedesignandvaluationofsuchaninstrumentarethebasicingredientsofasuccessfulfinancialproduct.Unliketheproposedvolatilityindexoption,theunderlyingofthisproposedcontractisatradedat-the-money-forwardstraddle,whichshouldbemoreappealingtopotentialparticipants.Inordertovaluetheseoptions,wecombinetheapproachesofcompoundoptionsandstochasticvolatility.Weusethelognormalprocessfortheunderlyingasset,theOrenstein-Uhlenbeckprocessforvolatility,andassumethatthetwoBrownianmotionsareindependent.Ournumericalresultsshowthatthestraddleoptionpriceisverysensitivetothechangesinvolatilitywhichmeansthattheproposedcontractisindeedaverypowerfulinstrumenttohedgevolatilityrisk.3I.INTRODUCTIONRiskmanagementisconcernedwithvariousaspectsofrisk,inparticular,priceriskandvolatilityrisk.Whiletherearevariousinstruments(andstrategies)todealwithpricerisk,exhibitedbythevolatilityofassetprices,therearepracticallynoinstrumentstodealwiththeriskthatvolatilityitselfmaychange.Volatilityriskhasplayedamajorroleinseveralfinancialdisastersinthepast15years.Long-Term-Capital-Management(LTCM)isonesuchexample,“Inearly1998,Long-Termbegantoshortlargeamountsofequityvolatility.”(Lowenstein,R.(2000)p.123)1.LTCMwassellingvolatilityontheS&P500indexandotherEuropeanindexes,bysellingoptions(straddles)ontheindex.Theywereexposedtotheriskthatvolatility,asreflectedinoptionspremiums,willincrease.Theydidnothedgethisrisk2.Thoughonecandeviseadynamicstrategyusingoptionstodealwithvolatilityrisksuchastrategymaynotbepracticalformostusers.Therewereseveralattemptstointroduceinstrumentsthatcanbeusedtohedgevolatilityrisk(e.g.,theGermanDTBlaunchedafuturescontractontheDAXvolatilityindex)butthosewerelargelyunsuccessful3.Giventhelargeandfrequentshiftsinvolatilityintherecentpast4especiallyinperiodslikethesummerof’97andthefallof’98,thereisagrowingneedforinstrumentstohedgevolatilityrisk.Pastproposalsofsuchinstrumentsincludedfuturesandoptionsonavolatilityindex.Theideaofdevelopingavolatilityindexwasfirstsuggestedby1ThequoteandtheinformationaretakenfromRogerLowenstein’sbookWhenGeniusFailed(2000),Ch.7.2AnotherknownexampleisthevolatilitytradingdonebyNickLeesonin’93and’94intheJapanesemarket.HisexposuretovolatilityriskwasamajorfactorinthedemiseofBaringsBank(seeGapperandDenton(1996)).3VolatilityswapshavebeentradingforsometimeontheOTCmarketbutwehavenoindicationoftheirsuccess.4Thevolatilityofvolatilitycanbeobservedfromthebehaviorofavolatilityindex,VIX,providedinFigure1.4BrennerandGalai(1989).Inafollow-uppaper,BrennerandGalai(1993)haveintroducedavolatilityindexbasedonimpliedvolatilitiesfromat-the-moneyoptions5.In1993theChicagoBoardOptionsExchange(CBOE)hasintroducedavolatilityindex,namedVIX,whichisbasedonimpliedvolatilitiesfromoptionsontheSP100index.Sofartherehavebeennooptionsofferedonsuchanindex.Themainissuewithsuchderivativesisthelackofatradableunderlyingassetwhichmarketmakerscouldusetohedgetheirpositionsandtopricethem.Sincetheunderlyingisnottradablewecannotreplicatetheoptionpayoffsandwecannotusetheno-arbitrageargument.Thefirsttheoreticalpaper6tovalueoptionsonavolatilityindexisbyGrunbichlerandLongstaff(1996).TheyspecifyameanrevertingsquarerootdiffusionprocessforvolatilitysimilartothatofSteinandStein(1991)andothers.Sincevolatilityisnottradingtheyassumethatthepremiumforvolatilityriskisproportionaltothelevelofvolatility.ThisapproachisinthespiritoftheequilibriumapproachofCox,IngersollandRoss(1985)andLongstaffandSchwartz(1992).AmorerecentpaperbyDetempleandOsakwe(1997)alsousesageneralequilibriumframeworktopriceEuropeanandAmericanstylevolatilityoptions.Theyemphasizethemean-revertinginlogvolatilitymodel.Sincethepayoffsoftheoptionproposedherecanbereplicatedbyaself-financingportfolio,consistingoftheunderlyingstraddleandborrowing,wevaluetheoptionusinganoarbitrageapproach.Theideaproposedanddevelopedinthispaperaddressesbothrelatedissues:hedgingandpricing.Thekeyfeatureofthestraddleoptionisthattheunderlyingassetisanat-the-money-forward(ATMF)straddl