defects in the black–scholes model

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1、CHAPTER46defectsintheBlack–ScholesmodelInthisChapter...•whytheBlack–Scholesassumptionsarewrong•howtoimprovetheBlack–Scholesmodel46.1INTRODUCTIONBeforepointingoutsomeoftheflawsintheassumptionsoftheBlack–Scholesworld,Imustemphasizehowwellthemodelhasdoneinpractice,howwidespread

2、isitsuseandhowmuchimpactithashadonfinancialmarkets,nottomentionaNobelPrizefortwoofitsthreecreators.Themodelisusedbyeveryoneworkinginderivatives,whethertheyaresalesmen,tradersorquants.Itisusedconfidentlyinsituationsforwhichitwasnotdesigned,usuallysuccessfully.Thevalueofvanilla

3、optionsareoftennotquotedinmonetaryterms,butinvolatilitytermswiththeunderstandingthatthepriceofacontractisitsBlack–Scholesvalueusingthequotedvolatility.YouuseamodelotherthanBlack–Scholesonlywithextremecaution,andyouwillhavetobeprettyconvincingtopersuadeyourcol-leaguesthatyou

4、knowwhatyouaredoing.Theideasofdeltahedgingandrisk-neutralpricinghavetakenaformidablegriponthemindsofacademicsandpractitionersalike.Inmanyways,especiallywithregardstocommercialsuccess,theBlack–Scholesmodelisremarkablyrobust.Nevertheless,thereisroomforimprovement.Certainly,we

5、canfindmodelsthatbetterdescribetheunderlying;whatisnotcleariswhetherthesemodelsmakeusmoremoney.InthenextfewsectionsIintroducetheideasandmodelsthatwillbeexpandeduponintherestofthebook.Someoftheseideasareclassical,inthesensethattheyareacademicallyrespectable,requirenogreatleap

6、sofimaginationandarethoroughlyharmless.Otherideasaretoonewtosaywhatthefuturehasinstoreforthem.Yetthereisatleastagrainoftruthinallthatfollows.Eachsectioncorrespondstoachapter.FirstIgivetherelevantBlack–Scholesassumption,andthenIsaywhyitiswrongandhow,later,Itrytorelaxtheassum

7、ption.Allthedetailswillbefoundintherelevantchapters.756PartFiveadvancedtopics46.2DISCRETEHEDGINGBlack–Scholesassumes•DeltahedgingiscontinuousConsiderfirst,thecontinuous-timeworldofstochasticcalcu-lus.WhenwederivedtheBlack–Scholesequationweusedthecontinuous-timeIto’slemma.The

8、deltahedgingthatwasnec-ˆessaryforriskeliminationalsohadtotakeplacecontinuously.Ift