衍生工具与风险管理第5章课件

D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:1Chapter5:OptionPricingModels:TheBlack-ScholesModelWhenIfirstsawtheformulaIknewenoughaboutittoknowthatthisistheanswer.Thissolvedtheancientproblemofriskandreturninthestockmarket.Itwasrecognizedbytheprofessionforwhatitwasasarealtourdeforce.MertonMillerTrillionDollarBet,PBS,February,2000D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:2ImportantConceptsinChapter5TheBlack-ScholesoptionpricingmodelTherelationshipofthemodel’sinputstotheoptionpriceHowtoadjustthemodeltoaccommodatedividendsandputoptionsTheconceptsofhistoricalandimpliedvolatilityHedginganoptionpositionD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:3OriginsoftheBlack-ScholesFormulaBrownianmotionandtheworksofEinstein,Bachelier,Wiener,ItôBlack,Scholes,Mertonandthe1997NobelPrizeD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:4TheBlack-ScholesModelastheLimitoftheBinomialModelRecallthebinomialmodelandthenotionofadynamicrisk-freehedgeinwhichnoarbitrageopportunitiesareavailable.ConsidertheAOLJune125calloption.Figure5.1,p.131showsthemodelpriceforanincreasingnumberoftimesteps.Thebinomialmodelisindiscretetime.Asyoudecreasethelengthofeachtimestep,itconvergestocontinuoustime.D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:5TheAssumptionsoftheModelStockPricesBehaveRandomlyandEvolveAccordingtoaLognormalDistribution.SeeFigure5.2a,p.134,5.2b,p.135and5.3,p.136foralookatthenotionofrandomness.Alognormaldistributionmeansthatthelog(continuouslycompounded)returnisnormallydistributed.SeeFigure5.4,p.137.TheRisk-FreeRateandVolatilityoftheLogReturnontheStockareConstantThroughouttheOption’sLifeThereAreNoTaxesorTransactionCostsTheStockPaysNoDividendsTheOptionsareEuropeanD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:6ANobelFormulaTheBlack-ScholesmodelgivesthecorrectformulaforaEuropeancallundertheseassumptions.Themodelisderivedwithcomplexmathematicsbutiseasilyunderstandable.TheformulaisTσddTσ/2)Tσ(r/X)ln(Sdwhere)N(dXe)N(dSC122c012Tr10cD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:7ANobelFormula(continued)whereN(d1),N(d2)=cumulativenormalprobabilitys=annualizedstandarddeviation(volatility)ofthecontinuouslycompoundedreturnonthestockrc=continuouslycompoundedrisk-freerateD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:8ANobelFormula(continued)ADigressiononUsingtheNormalDistributionThefamiliarnormal,bell-shapedcurve(Figure5.5,p.139)SeeTable5.1,p.140fordeterminingthenormalprobabilityford1andd2.ThisgivesyouN(d1)andN(d2).D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:9ANobelFormula(continued)ANumericalExamplePricetheAOLJune125callS0=125.9375,X=125,rc=ln(1.0456)=.0446,T=.0959,s=.83.SeeTable5.2,p.141forcalculations.C=$13.21.FamiliarizeyourselfwiththeaccompanyingsoftwareExcel:bsbin3.xls.SeeSoftwareDemonstration5.1.NotetheuseofExcel’s=normsdist()function.Windows:bsbwin2.2.exe.SeeAppendix5.B.D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:10ANobelFormula(continued)CharacteristicsoftheBlack-ScholesFormulaInterpretationoftheFormulaTheconceptofriskneutrality,riskneutralprobability,anditsroleinpricingoptionsTheoptionpriceisthediscountedexpectedpayoff,Max(0,ST-X).WeneedtheexpectedvalueofST-XforthosecaseswhereSTX.D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:11ANobelFormula(continued)CharacteristicsoftheBlack-ScholesFormula(continued)InterpretationoftheFormula(continued)Thefirsttermoftheformulaistheexpectedvalueofthestockpricegiventhatitexceedstheexercisepricetimestheprobabilityofthestockpriceexceedingtheexerciseprice,discountedtothepresent.Thesecondtermistheexpectedvalueofthepaymentoftheexercisepriceatexpiration.D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:12ANobelFormula(continued)CharacteristicsoftheBlack-ScholesFormula(continued)TheBlack-ScholesFormulaandtheLowerBoundofaEuropeanCallRecallfromChapter3thatthelowerboundwouldbeTheBlack-ScholesformulaalwaysexceedsthisvalueasseenbylettingS0beveryhighandthenletitapproachzero.)XeSMax(0,Tr0cD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:13ANobelFormula(continued)CharacteristicsoftheBlack-ScholesFormula(continued)TheFormulaWhenT=0Atexpiration,theformulamustconvergetotheintrinsicvalue.Itdoesbutrequirestakinglimitssinceotherwiseitwouldbedivisionbyzero.MustconsidertheseparatecasesofSTXandSTX.D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:14ANobelFormula(continued)CharacteristicsoftheBlack-ScholesFormula(continued)TheFormulaWhenS0=0Herethecompanyisbankruptsotheformulamustconvergetozero.Itrequirestakingthelogofzero,butbytakinglimitsweobtainthecorrectresult.D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.5:15ANobelFormula(continued)CharacteristicsoftheBlack-ScholesFormula(continued)TheFormulaWhens=0Again,thisrequiresdividingbyzero,butwecantakelimitsandobtaintherightanswe