Black-ScholesModel(金融工程-华东师范大学汤银才

11.1Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheBlack-ScholesModelChapter1111.2Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheStockPriceAssumptionConsiderastockwhosepriceisSInashortperiodoftimeoflengthDtthechangeinthenstockpriceSisassumedtobenormalwithmeanmSdtandstandarddeviation,thatis,SfollowsgeometricBrownianmotionds=mSdt+Sdz.ThenmisexpectedreturnandisvolatilityStD11.3Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheLognormalPropertyItfollowsfromthisassumptionthatSincethelogarithmofSTisnormal,STislognormallydistributed2020lnln,2orlnln,2TTSSTTSSTTmm11.4Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityModelingStockPricesinFinanceInfinance,frequentlywemodeltheevolutionofstockpricesasageneralizedWienerProcessAlso,assumepricesaredistributedlognormalandreturnsaredistributednormaldzSdtSdSm11.5Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheLognormalDistributionESSeSSeeTTTTT()()()002221varmm11.6Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityContinuouslyCompoundedRateofReturn,h(Equation(11.7))SSeTSSTTTT0012or=or2hhhmln,11.7Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheExpectedReturnTheexpectedvalueofthestockpriceisE(ST)=S0emTTheexpectedcontinuouslycompoundedreturnonthestockisE(h)=m–2/2(thegeometricaverage)misthethearithmeticaverageofthereturnsNotethatE[ln(ST)]isnotequaltoln[E(ST)]ln[E(ST)]=lnS0+mT,E[ln(ST)]=lnS0+(m-2/2)T11.8Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheExpectedReturnExampleTakethefollowing5annualreturns:10%,12%,8%,9%,and11%ThearithmeticaverageisHowever,thegeometricaverageisThus,thearithmeticaverageoverstatesthegeometricaverage.Thegeometricistheactualreturnthatonewouldhaveearned.TheapproximationforthegeometricreturnisThisdiffersfromgasthereturnsarenotnormallydistributed.10.050.0*)11.009.008.012.010.0(511511_niinxx(09991.0111.1*09.1*08.1*12.1*10.11)1(5111nniixg(09988.02/015811.010.02/22m11.9Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityIsNormalityRealistic?Ifreturnsarenormalandthuspricesarelognormalandassumingthatvolatilityisat20%(aboutthehistoricalaverage)–On10/19/87,the2monthS&P500Futuresdropped29%•Thiswasa-27sigmaeventwithaprobabilityofoccurringofonceinevery10160days–On10/13/89,theS&P500indexlostabout6%•Thiswasa-5sigmaeventwithaprobabilityof0.00000027oronceevery14,756years11.10Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheConceptsUnderlyingBlack-ScholesTheoptionprice&thestockpricedependonthesameunderlyingsourceofuncertaintyWecanformaportfolioconsistingofthestockandtheoptionwhicheliminatesthissourceofuncertaintyTheportfolioisinstantaneouslyrisklessandmustinstantaneouslyearntherisk-freerateThisleadstotheBlack-Scholesdifferentialequation11.11Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheAssumptionsUnderlyingBlack-Scholes1.ThestockfollowsaBrownianmotionwithconstantmand2.Shortsellingofsecuritieswithfulluseofproceedsispermitted3.Notransactioncostortaxes4.Securitiesareperfectlydivisible5.Nodividendspaidduringthelifeoftheoption6.Therearenoarbitrageopportunities7.Securitytradingiscontinuous8.Therisk-freerateofinterest,r,isconstantandisthesameforallmaturities11.12Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity1of3:TheDerivationoftheBlack-ScholesDifferentialEquationDDDDDDSStSzSStSStSSzSmmƒƒƒ½ƒƒWesetupaportfolioconsistingof:derivative+ƒ:shares2222111.13Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityThevalueoftheportfolioisgivenbyƒƒThechangeinitsvalueintimeisgivenbyƒƒDDDDSStSS2of3:TheDerivationoftheBlack-ScholesDifferentialEquation11.14Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversity3of3:TheDerivationoftheBlack-ScholesDifferentialEquation222Thereturnontheportfoliomustbetherisk-freerate.HenceWesubstituteforandintheseequationstogettheBlack-Scholesdifferentialequation:12ffSffrSStSrtSDDDD2rf11.15Options,Futures,andOtherDerivatives,4thedition©2000byJohnC.HullTangYincai,©2003,ShanghaiNormalUniversityTheAlgebraoftheDifferentialEquationI(=D)•noticethatallofthem’scancelouttSSffrtSSftftSSffrtSSftftSSffrtSSftSSfSSftftSSffrwStSSfwSSftSSfSSftftSSffrSSfftrSffmmmm