the black–scholes model

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1、CHAPTER5theBlack–ScholesmodelInthisChapter...•thefoundationsofderivativestheory:deltahedgingandnoarbitrage•thederivationoftheBlack–Scholespartialdifferentialequation•theassumptionsthatgointotheBlack–Scholesequation•howtomodifytheequationforcommodityandcurrenc

2、yoptions5.1INTRODUCTIONThisis,withoutdoubt,themostimportantchapterinthebook.InitIdescribeandexplainthebasicbuildingblocksofderivativestheory.Thesebuildingblocksaredeltahedgingandnoarbitrage.Theyformamoderatelysturdyfoundationtothesubjectandhaveperformedwellsi

3、nce1973whentheideasbecamepublic.InthischapterIbeginwiththestochasticdifferentialequationmodelforequitiesandexploitthecorrelationbetweenthisassetandanoptiononthisassettomakeaperfectlyrisk-freeportfolio.Ithenappealtonoarbitragetoequatereturnsonallrisk-freeportf

4、oliostotherisk-freeinterestrate,thesocalled‘nofreelunch’argument.Theargumentsaretriviallymodifiedtoincorporatedividendsontheunderlyingandalsotopricecommodityandcurrencyoptionsandoptionsonfutures.Thischapterisquitetheoretical,yetalloftheideascontainedherearereg

5、ularlyusedinpractice.Eventhoughalloftheassumptionscanbeshowntobewrongtoagreaterorlesserextent,theBlack–Scholesmodelisprofoundlyimportantbothintheoryandinpractice.5.2AVERYSPECIALPORTFOLIOInChapter2Idescribedsomeofthecharacteristicsofoptionsandoptionsmarkets.Ii

6、ntroducedtheideaofcallandputoptions,amongstothers.Thevalueofacalloptionisclearlygoingtobeafunctionofvariousparametersinthecontract,suchasthestrikepriceEandthetimetoexpiryT−t,whereTisthedateofexpiryandtisthecurrenttime.Thevaluewillalso92PartOnemathematicalandfi

7、nancialfoundationsdependonpropertiesoftheassetitself,suchasitsprice,itsdriftanditsvolatility,aswellastherisk-freerateofinterest.1WecanwritetheoptionvalueasV(S,t;σ,µ;E,T;r).Noticethatthesemicolonsseparatedifferenttypesofvariablesandparameters:•Sandtarevariable

8、s;•σandµareparametersassociatedwiththeassetprice;•EandTareparametersassociatedwiththedetailsoftheparticularcontract;•risaparameterassociatedwiththecurrencyinwhichtheassetisquoted.I’mnotgo