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1、CHAPTER57jumpdiffusionInthisChapter...?thePoissonprocessformodelingjumps?hedginginthepresenceofjumps?howtopricederivativeswhenthepathoftheunderlyingcanbediscontinuous?jumpvolatility57.1INTRODUCTIONThereisplentyofevidencethat?nancialquantities,betheyequities,currenciesor
2、interestrates,forexample,donotfollowthelognormalrandomwalkthathasbeenthefoundationofalmosteverythinginthisbook,andalmosteverythinginthe?nancialliterature.Welookatsomeofthisevidenceinamoment.Oneofthestrikingfeaturesofreal?nancialmarketsisthateverynowandthenthereisasudden
3、unexpectedfallorcrash.ThesesuddenmovementsoccurfarmorefrequentlythanwouldbeexpectedfromaNormallydistributedreturnwithareasonablevolatility.Onallbuttheshortesttimescalesthemovelooksdiscontinuous;thepricesofassetshavejumped.Thisisimportantforthetheoryandpracticeofderivati
4、vesbecauseitisusuallynotpossibletohedgethroughthecrash.Onecertainlycannotdeltahedgeasthestockmarkettumblesaroundone’sankles,andtoof?oadallone’spositionswillleadtorealinsteadofpaperlosses,andmayevenmakethefallworse.InthischapterIexplainclassicalwaysofpricingandhedgingwhe
5、ntheunderlyingfollowsajump-diffusionprocess.57.2EVIDENCEFORJUMPSLet’slookatsomedatatoseejusthowfarfromNormalthereturnsreallyare.Thereareseveralwaystovisualizethedif-ferencebetweentwodistributions,inourcasethedifferencebetweentheempiricaldistributionandtheNormaldistribu-
6、tion.Onewayistooverlaythetwoprobabilitydistributions.InFigure57.1weseethedistributionofXeroxreturns,from1986until1997,normalizedtounitstandarddeviation.The928PartFiveadvancedtopics0.70.6PDF0.50.40.30.20.10?3?2.6?2.2?1.8?1.4?1?0.6?0.20.20.611.41.82.22.63ScaledreturnsFigu
7、re57.1TheprobabilitydensityfunctionsforXeroxdailyreturns,scaledtozeromeanandunitstandarddeviation,andthestandardizedNormaldistribution.peakoftherealdistributionisclearlyhigherthantheNormaldistribution.Becausebothofthesedistributionshavethesamestandarddeviationthenthehig
8、herpeakmustbebalancedbyfattertails,it’sjustthattheywouldbetoosmalltoseeonthis?gure.Theymaybetoosmalltoseehere,
