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时间:2018-02-10
《paul wilmott on quantitative finance lookback options》由会员上传分享,免费在线阅读,更多相关内容在工程资料-天天文库。
1、CHAPTER26lookbackoptionsInthisChapter...•thefeaturesthatmakeupalookbackoption•howtoputthelookbackoptionintotheBlack–Scholesframeworkforboththecontinuous-anddiscrete-samplingcases26.1INTRODUCTIONThedreamcontracthastobeonethatpaysthedifferencebetweenthehighestandthelowestassetpricesrealizedbyana
2、ssetoversomeperiod.Anyspeculatoristryingtoachievesuchatrade.Thecontractthatpaysthisisanexampleofalookbackoption,anoptionthatpaysoffsomefunctionoftherealizedmaximumand/orminimumoftheunderlyingassetoversomeprescribedperiod.Sincelookbackoptionshavesuchanextremepayofftheytendtobeexpensive.Wecanpri
3、cethesecontractsintheBlack–Scholesenvironmentquiteeasily,theoretically.Therearetwocasestoconsider:Whetherthemaximum/minimumismeasuredcontinuouslyordiscretely.HereIwillalwaystalkaboutthe‘maximum’oftheasset.Thetreatmentofthe‘minimum’shouldbeobvious.26.2TYPESOFPAYOFFForthebasiclookbackcontracts,t
4、hepayoffcomesintwovar-ieties,liketheAsianoption.Thesearetherateandthestrikeoption,alsocalledthefixedstrikeandthefloatingstrikerespectively.Thesehavepayoffsthatarethesameasvanillaoptionsexceptthatinthestrikeoptionthevanillaexercisepriceisreplacedbythemaximum.Intherateoptionitistheassetvalueinthev
5、anillaoptionthatisreplacedbythemaximum.26.3CONTINUOUSMEASUREMENTOFTHEMAXIMUMInFigure26.1isshownthetermsheetforaforeignexchangelookbackswap.Onecounterpartypaysafloatinginterestrateeverysixmonths,whiletheothercounterpartypaysonmaturity446PartTwoexoticcontractsandpathdependencyFigure26.1Termsheetf
6、orUSD/DEMlookbackswap.alinearfunctionofthemaximumrealizedleveloftheexchangerate.Thefloatinginterestpaymentsofsix-monthLIBORcanberepresentedasamuchsimplercashflow,asexplainedinChapter14.Thenthiscontractbecomesaverystraightforwardlookbackoption.IntroducethenewvariableMastherealizedmaximumoftheasse
7、tfromthestartofthesamplingperiodt=0,say,untilthecurrenttimet:M=maxS(τ).0≤τ≤tInFigure26.2isshownarealizationofanassetpricepathandthecontinuouslysampledmaximum.Anobviouspointaboutthisplot,butonethatisworthmentioning,isthattheassetpriceisa
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