paul wilmott on quantitative finance lookback options

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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