金融工程数值方法分析4067431436.ppt

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1、CHAPTER5theBlack–ScholesmodelInthisChapterthefoundationsofderivativestheory:deltahedgingandnoarbitragethederivationoftheBlack–ScholespartialdifferentialequationtheassumptionsthatgointotheBlack–ScholesequationhowtomodifytheequationforcommodityandcurrencyoptionsA

2、VERYSPECIALPORTFOLIO:Usetodenotethevalueofaportfolioofonelongoptionpositionandashortpositioninsomequantity,delta,oftheunderlying:(5.1)WewillassumethattheunderlyingfollowsalognormalrandomwalkItisnaturaltoaskhowthevalueoftheportfoliochangesfromtimettot+dt.Thechan

3、geintheportfoliovalueisduepartlytothechangeintheoptionvalueandpartlytothechangeintheunderlying:FromItˆowehave:Thustheportfoliochangesby:(5.2)ELIMINATIONOFRISK:DELTAHEDGING:Ifwechoose(5.3)thentherandomnessisreducedtozero.Anyreductioninrandomnessisgenerallytermed

4、hedging,whetherthatrandomnessisduetofluctuationsinthestockmarketortheoutcomeofahorserace.Theperfecteliminationofrisk,byexploitingcorrelationbetweentwoinstruments(inthiscaseanoptionanditsunderlying)isgenerallycalleddeltahedging.NOARBITRAGE:Afterchoosingthequanti

5、tydeltaassuggestedabove,weholdaportfoliowhosevaluechangesbytheamount:(5.4)Thischangeiscompletelyriskless:(5.5)Thisisanexampleofthenoarbitrageprinciple.THEBLACK–SCHOLESEQUATION：Substituting(5.1),(5.3)and(5.4)into(5.5)wegetthat(5.6)