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1、ABSTRACTAmericanoptionisthemostfrequentlytradedoptionsinthe¯nancialmar-ket,whilethestockoptioniswellstudiedintheliterature.EmpiricalresearchhasfoundthatthestockpricemovementisnotsimplyfollowtheGeometryBrowianMotion,buttakesonacharacteristicofhighpeakandfattail.So
2、thetraditionalB-S-Mmodelisnolonggiveagoodanswer,thiscanbedonebyintroducingtheJump-Di®usionmodel.TheAmericanoptionpricingundertheJump-Di®usionmodelisatoughproblemintheoptionpricingtheorem.ThisisduotothecharacterofearlyexerciseinAmericanoptionandnonlocalproblemcaus
3、edbyjump.Nowthemostlyusedmethodisthe¯niteelementmethodbasedonthevariationalinequalities.Sincethe¯niteelementmethodissophisticatedinitselfandishardtoprogramonthecomputer,thisthesisistryingtousetheextendedfouriertransformmethodtoprovideanotherperspectiveandideastoo
4、vercomethedi±cultiesmetinpricingAmericanoptionsunderJump-Di®usionprocesses.The¯stchapterofthisthesissimplyreferthesituationofstudyinthevalua-tionofAmericanoptionsundertheJump-Di®usionprocesses.Andwecomparedthemainmethodsinrecentlyusageinthis¯led.Thenweprovidethem
5、ethodadoptedinthispaper,andpresentthemainworkinourthesis.Thesecondchapteristhebasisofthefollowingcontent.Inthebeginning,weintroducedtheimportantconceptsandformulationsin¯nancialmathematicssuchasarbitragefreepricingtheorem,martingale,Ito^lemmaetc.Thenwecarefullyin
6、troducedB-S-MmodelandJump-Di®usionmodel.Basedonthese,wegetthemainpointthisthesisisgoingtosolve
7、thefreeboundaryproblemofAmericanoptions.Afterintroducedallthemathematicalmodelsabove,wethenexplainedthemostimportantmethodusedinthispaper
8、theextendedfouriertransformmet
9、hod,andpresentthemainideasofCauchyresidualtheoremandcontourintegrationincomplexvariablefunctions.Thethirdchapteristheprimarychapterofthisthesis.At¯rst,wecomparedanotherfourierapproachinpricingAmericanoptions,andthenputforwardthekeyassumptionusedinthispaper.Undert
10、heconditionofthekeyassumption,weusetheextendedfouriertransformmethodtopricetheAmericanputoptionsintheJump-Di®usionmodel,andtakethecontourintegrationtoobtainthe
