paul wilmott on quantitative finance jump diffusion

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1、CHAPTER57jumpdiffusionInthisChapter...•thePoissonprocessformodelingjumps•hedginginthepresenceofjumps•howtopricederivativeswhenthepathoftheunderlyingcanbediscontinuous•jumpvolatility57.1INTRODUCTIONThereisplentyofevidencethatfinancialquantities,betheyequities,currenciesor

2、interestrates,forexample,donotfollowthelognormalrandomwalkthathasbeenthefoundationofalmosteverythinginthisbook,andalmosteverythinginthefinancialliterature.Welookatsomeofthisevidenceinamoment.Oneofthestrikingfeaturesofrealfinancialmarketsisthateverynowandthenthereisasudden

3、unexpectedfallorcrash.ThesesuddenmovementsoccurfarmorefrequentlythanwouldbeexpectedfromaNormallydistributedreturnwithareasonablevolatility.Onallbuttheshortesttimescalesthemovelooksdiscontinuous;thepricesofassetshavejumped.Thisisimportantforthetheoryandpracticeofderivati

4、vesbecauseitisusuallynotpossibletohedgethroughthecrash.Onecertainlycannotdeltahedgeasthestockmarkettumblesaroundone’sankles,andtooffloadallone’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’sjustthattheywouldbetoosmalltoseeonthisfigure.Theymaybetoosmalltoseehere,