toBinomialTrees(金融工程-华东师范大学汤银才)

Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.1IntroductiontoBinomialTreesChapter9Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.2ASimpleBinomialModelofStockPriceMovements•Inabinomialmodel,thestockpriceattheBEGINNINGofaperiodcanleadtoonly2stockpricesattheENDofthatperiodOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.3OptionPricingBasedontheAssumptionofNoArbitrageOpportunities•Procedures:EstablishaportfolioofstockandoptionValuethePortfolionoarbitrageopportunitiesnouncertaintyatmaturitynoriskwiththeportfoliorisk-freeinterestearnedValuetheoptionRisk-freeinterest=valueofportfoliotodayOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.4ASimpleBinomialModel:Example•Astockpriceiscurrently$20•Inthreemonthsitwillbeeither$22or$18StockPrice=$22StockPrice=$18Stockprice=$20Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.5StockPrice=$22OptionPrice=$1StockPrice=$18OptionPrice=$0Stockprice=$20OptionPrice=?ACallOption•A3-monthcalloptiononthestockhasastrikepriceof$21.•Figure9.1(P.202)Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.6•ConsiderthePortfolio:LONGDsharesSHORT1calloption•Figure9.1becomes•Portfolioisrisklesswhen22D–1=18DorD=0.2522D–118DSettingUpaRisklessPortfolioS0=20Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.7ValuingthePortfolio(withRisk-FreeRate12%)•Therisklessportfoliois:LONG0.25sharesSHORT1calloption•Thevalueoftheportfolioin3monthsis22*0.25-1=4.50=18*0.25•Thevalueoftheportfoliotodayis4.50e-0.12*0.25=4.3670Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.8ValuingtheOption•Theportfoliothatis:LONG0.25sharesSHORT1calloptionisworth4.367•Thevalueofthesharesis5.000=0.25*20•Thevalueoftheoptionistherefore0.633=5.000-4.367Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.9Generalization•ConsideraderivativethatlastsfortimeTandthatisdependentonastock•Figure9.2(P.203)S0uƒuS0dƒdS0ƒOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.10Generalization(continued)•Considertheportfoliothatis:LONGDsharesSHORT1derivative•Figure9.2becomes•TheportfolioisrisklesswhenS0uD–ƒu=S0dD–ƒdorwhendSuSffdu00DS0uD–ƒuS0dD–ƒdDS0-fOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.11Generalization(continued)•ValueoftheportfolioattimeTisS0uD–ƒu•Valueoftheportfoliotodayis(S0uD–ƒu)e–rT•AnotherexpressionfortheportfoliovaluetodayisS0D–f•Hence,ƒ=S0D–(S0uD–ƒu)e–rTOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.12Generalization(continued)•SubstitutingforDweobtainƒ=[pƒu+(1–p)ƒd]e–rTwherepedudrTOptions,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.13Generalization(continued):ProofwithanExample•ThisisknownastheNoArbitragemethodology•Inourearlierexamplef=0.633andD=0.25•Iff0.633,e.g.f=0.60==DS0-f=0.25*20-0.6=4.44.367t=0ST=18ST=22Buycall-0.60001SellDShares5.000-18*0.25=-4.50-22*0.25=-5.50Lend4.367atr-4.3674.504.50NetFlows0.03300•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.14Generalization(continued):ProofwithanExample•Iff0.633,e.g.f=0.65==DS0-f=0.25*20-0.65=4.354.367t=0ST=18ST=22BuyDShares-5.00018*0.25=4.5022*0.25=5.50Borrow4.367atr4.367-4.50-4.50Sellcall0.6500-1NetFlows0.01700•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.15IrrelevanceofaStock’sExpectedReturn•Whenwearevaluinganoptionintermsoftheunderlyingstocktheexpectedreturnonthestockisirrelevant•Thisisbecauseinourformulaf=S0D-(S0uD-fu)e-rTfdoesnotinvolvetheprobabilityofthestockmovingupordown•Itdoesnotmatterifwesaytheprobabilityofanincreaseis50%or80%wegetthesameresult•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.16IrrelevanceofaStock’sE(R)Proof:(continued)•Let’scallputheprobabilityofanincreaseinthestockpriceandpd=1-putheprobabilityofastockdecreaseS0D-f=[pu(S0uD-fu)+pd(S0dD-fd)]e-kTwherekistheappropriateratefortheriskinvolved•However,DischosensuchthatS0uD-fu=S0dD-fdandweknowthatpd=1-pu•Substituting,S0D-f=[pu(S0uD-fu)+(1-pu)(S0uD-fu)]e-kT=(S0uD-fu)e-rTassincethisisrisk-free,k=r•Nopu’sorpd’sleft,thusprobabilityofstockincreaseisirrelevant•Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,ShanghaiNormalUniversity9.17IrrelevanceofaStock’sE(R)(continued)•Theprobabilityofanincreaseinthestockpriceisirrelevantbecauseoptionsareredundantsecurities•Inourtwo-stepmodels,weformarisk-lessportfoliowithstockandtheoption•Thus,thereturn/pay-offfromtheoptionisoffsetbythereturn