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1、DIRECTANALYSISOFIMPLIEDVOLATILITYFOREUROPEANOPTIONSByYanWeiADISSERTATIONSubmittedtoMichiganStateUniversityinpartialful llmentoftherequirementsforthedegreeofDOCTOROFPHILOSOPHYAppliedMathematics2012UMINumber:3505149AllrightsreservedINFORMATIONTOALLUSERSThequalityofthisreproductionisdependen
2、tonthequalityofthecopysubmitted.Intheunlikelyeventthattheauthordidnotsendacompletemanuscriptandtherearemissingpages,thesewillbenoted.Also,ifmaterialhadtoberemoved,anotewillindicatethedeletion.UMI3505149Copyright2012byProQuestLLC.Allrightsreserved.Thiseditionoftheworkisprotectedagainstunau
3、thorizedcopyingunderTitle17,UnitedStatesCode.ProQuestLLC.789EastEisenhowerParkwayP.O.Box1346AnnArbor,MI48106-1346ABSTRACTDIRECTANALYSISOFIMPLIEDVOLATILITYFOREUROPEANOPTIONSByYanWeiWeshowexistenceanduniquenessofastrongsolutiontoalinearnon-uniformlyparabolicequation,whichgivesthefairpriceof
4、anormalizedEuropeancalloption.Wethenprovideadirectlinkbetweenlocalandimpliedvolatilitiesintheformofaquasilineardegenerateparabolicpartialdi erentialequation.Wealsoestablishclosed-formasymptoticformulaefortheimpliedvolatilitynearexpiryaswellasfordeepinandoutofthemoneyoptions,usingagenerali
5、zedcomparisonprincipleonboundeddomains.Tomyparents,fortheirfaithandloveiiiACKNOWLEDGMENTSItwouldnothavebeenpossibletowritethisdoctoraldissertationwithoutthehelpandsupportofthekindpeoplearoundme,onlysomeofwhomisitpossibletogiveparticularmentionhere.Aboveall,Iamheartilythankfultomyadvisor,P
6、eterW.Bates,whogavemeencourage-ment,guidanceandsupportfromthepreliminarytotheconcludinglevelofmydissertation.IoweadeepdebtofgratitudetoProfessorHenriBerestycki,forsuggestingthisinterestingandchallengingproject.ItisapleasuretothankProfessorDon.R.Aronson,forreferencesthathesharedwithme.Spec
7、ialthanksgotomycommittee,fortheirprecioustime.Last,butbynomeansleast,Iwouldliketothankmyfriends,SamanthaL.Dahlberg,JacquelineM.Dresch,TimMiller,andmyformersupervisorPavelSikorskii,fortheirconsistentsupportandfriendship.Foranyerrorsorinadequaciesthatmayremaininthiswo
