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1、CHAPTER47discretehedgingInthisChapter...•theeffectofhedgingatdiscretetimes•hedgingerror•therealdistributionofprofitandloss47.1INTRODUCTIONInthischapterweconcentrateononeoftheerroneousassumptionsintheBlack–Scholesmodel,thatofcontinuoushedging.TheBlack–Scholesanalysisrequiresthecontin
2、uousrebal-ancingofahedgedportfolioaccordingtoadelta-neutralstrategy.Thisstrategyis,inpractice,impossible.Thestructureofthischapterisasfollows.Webeginbyexaminingtheconceptofdeltahedginginadiscrete-timeframework.WewillseethattakingexpectationsleadstotheBlack–Scholesequationwithoutany
3、needforstochasticcalculus.Ithenshowhowthiscanbeextendedtoahigher-orderapproximation,validwhenthehedgingperiodisnotinfinitesimal.Wethendiscussthenatureofthehedgingerror,theerrorbetweentheexpectedchangeinportfolioandtheactual.Thisquantityiscommonlyignored(perhapsbecauseitaveragesoutto
4、zero)butisimportant,especiallywhenoneexaminestherealdistributionofreturns.47.2MOTIVATINGEXAMPLE:THETRINOMIALMODELWe’veseenhowperfectlyweareabletoeliminateriskinthebinomialandBlack–Scholesworlds.Now,whatisthesimplestgeneralizationtothesemodelsthatwillshowushowwelldeltahedgingworksin
5、othersituations?Thesimplestsuchmodelisthetrinomial,asshowninFigure47.1.Inthisexamplethestockstartsat100andcanriseto101,staythesameat100,orfallto99.Supposethatacalloptionwithastrikeof100simultaneouslyexpires.Thepayoffswillthenbeeither1,ifthestockrises,andzerointheothertwocases.Howca
6、nwehedgethispayoff?Introduceasthequantityofstockwesellinordertohedge.SincethehedgedportfoliovaluecanbewrittenasV−S,764PartFiveadvancedtopics10110010099Figure47.1Thetrinomialrandomwalk.theportfoliovalueatexpirationwillbeoneof1−×101,0−×100or0−×99.Byhedgingwearetryingtomakethepor
7、tfoliohavethesamevaluewhateverstatethestockendsupin.Sowearetryingtofindsuchthat1−×101=0−×100=0−×99.Can’tbedone.Twoequations,oneunknown.Conclusion,eveninthisrathertrivialextensiontothebinomialworldwearenotabletohedge.47.3AMODELFORADISCRETELYHEDGEDPOSITIONNotonlydoesdeltahedgingfa
8、ilinthetrinomialworld,itev
