ch21期权定价.ppt

第21章期权定价OptionValuation,期权定价OptionValuation,21.1期权定价简介21.2期权价值的限制21.3二项式期权定价模型21.4布莱克-斯克尔斯期权定价21.5布莱克-斯克尔斯公式的运用21.6经验证明,期权定价OptionValues,内在价值立即执行该期权能获的收入Intrinsicvalue-profitthatcouldbemadeiftheoptionwasimmediatelyexercised看涨期权:股票价格执行价格Call:stockprice-exerciseprice看跌期权:执行价格-股票价格Put:exerciseprice-stockprice时间价值期权价格与内在价值间的差异Timevalue-thedifferencebetweentheoptionpriceandtheintrinsicvalue,期权的时间价值：看涨期权TimeValueofOptions:Call,期权价值Optionvalue,X,股票价格StockPrice,看涨期权价值ValueofCall,内在价值IntrinsicValue,时间价值Timevalue,影响看涨期权价值的因素FactorsInfluencingOptionValues:Calls,因素Factor对价值的作用Effectonvalue股票价格Stockpriceincreases执行价格Exercisepricedecreases股票价格的波动Volatilityofstockpriceincreases到期Timetoexpirationincreases利息率Interestrateincreases股利DividendRatedecreases,二项式期权定价模型：举例BinomialOptionPricing:TextExample,100,200,50,股票价格StockPrice,二项式期权定价模型：举例BinomialOptionPricing:TextExample,另一个组合AlternativePortfolio买1股100元的股票借$46.308%的利率）净支出是$53.70Buy1shareofstockat$100Borrow$46.30(8%Rate)Netoutlay$53.70收入PayoffValueofStock50200Repayloan-50-50NetPayoff0150,53.70,150,0,收入结构正好是看涨期权的2倍PayoffStructureisexactly2timestheCall,二项式期权定价模型：举例BinomialOptionPricing:TextExample,53.70,150,0,2C=$53.70C=$26.85,收入和期权价值的另一种观点AnotherViewofReplicationofPayoffsandOptionValues,另一个组合1股股票和两个售出的看涨期权的组合恰好被套期保值AlternativePortfolio-oneshareofstockand2callswritten(X=125)Portfolioisperfectlyhedged股票价值StockValue50200两个售出的看涨期权的义务CallObligation0-150净收入Netpayoff5050Hence100-2C=46.30orC=26.85,布莱克-斯科尔斯期权定价模型Black-ScholesOptionValuation,Co=Soe-dTN(d1)-Xe-rTN(d2)d1=ln(So/X)+(rd+s2/2)T/(sT1/2)d2=d1-(sT1/2)式中whereCo=当前看涨期权的价值Currentcalloptionvalue.So=当前股票价格CurrentstockpriceN(d)=随机的偏离标准正态分布的概率小于dprobabilitythatarandomdrawfromanormaldist.willbelessthand.,布莱克-斯科尔斯期权定价模型Black-ScholesOptionValuation,X=执行价格Exerciseprice.d=标的股票的年股利收益率Annualdividendyieldofunderlyingstocke=2.71828,自然对数函数的底数thebaseofthenat.log.r=无风险利率Risk-freeinterestrate(annualizescontinuouslycompoundedwiththesamematurityastheoption.,布莱克-斯科尔斯期权定价模型Black-ScholesOptionValuation,T=期权到期前的时间（以年为单位）timetomaturityoftheoptioninyears.ln=自然对数Naturallogfunctions=股票连续复利年收益率的标准差Spoundedrateofreturnonthestock,标准正态曲线布莱克-斯科尔斯看涨期权举例CallOptionExample,股票价格So=100执行价格X=95利率r=.10到期时间T=.25(quarter)标准差s=.50股利收益率d=0d1=ln(100/95)+(.10-0+(.52/2)/(.5.251/2)=.43d2=.43-(.5)(.251/2)=.18,正态分布的概率ProbabilitiesfromNormalDist.,N(.43)=.6664Table17.2dN(d).42.6628.43.6664添写Interpolation.44.6700,正态分布的概率ProbabilitiesfromNormalDist.,N(.18)=.5714Table17.2dN(d).16.5636.18.5714.20.5793,看涨期权价值CallOptionValue,Co=Soe-dTN(d1)-Xe-rTN(d2)Co=100X.6664-95e-.10X.25X.5714Co=13.70隐含波动性ImpliedVolatility用布莱克-斯科尔斯公式和实际期权价格来解决波动性UsingBlack-Scholesandtheactualpriceoftheoption,solveforvolatility.隐含波动性是否与股票具有一致性？Istheimpliedvolatilityconsistentwiththestock?,看跌期权价值：布莱克-斯科尔斯PutOptionValue:Black-Scholes,P=Xe-rT1-N(d2)-S0e-dT1-N(d1)用上例的数据UsingthesampledataP=$95e(-.10X.25)(1-.5714)-$100(1-.6664)P=$6.35,看跌期权定价：用看涨-看跌期权平价关系PutOptionValuation:UsingPut-CallParity,P=C+PV(X)-So=C+Xe-rT-SoUsingtheexampledataC=13.70X=95S=100r=.10T=.25P=13.70+95e-.10X.25-100P=6.35,用布莱克-斯科尔斯公式UsingtheBlack-ScholesFormula,套期：套期保值率或d系数Hedging:Hedgeratioordelta股票价格需要套期保值防范所持有期权的价格变化风险Thenumberofstocksrequiredtohedgeagainstthepriceriskofholdingoneoption看涨期权的套期保值率Call=N(d1)看跌期权的套期保值率Put=N(d1)-1,用布莱克-斯科尔斯公式UsingtheBlack-ScholesFormula,期权弹性OptionElasticity期权价值变化1%所对应的股票价值的百分比变化Percentagechangeintheoptionsvaluegivena1%changeinthevalueoftheunderlyingstock,投资组合保险保护股票价值的下跌PortfolioInsurance-ProtectingAgainstDeclinesinStockValue,买看跌期权-价格下跌保护和没有限制的向上的潜在收益蔼BuyingPuts-resultsindownsideprotectionwithunlimitedupsidepotential限制Limitations如果使用指数到看跌期权则会产生循迹误差Trackingerrorsifindexesareusedfortheputs看跌期权的到期时间可能很短Maturityofputsmaybetooshort套期保值率或d系数在股票价值变化时随之变化Hedgeratiosordeltaschangeasstockvalueschange,Summary,期权的价值包括内在价值与时间价值，或“波动性”价值。波动性价值是如果股票价格与预测变动方向相反则选择不执行期权的权利。因此，不论股票价格如何变动，期权拥有者的损失不会超过获得期权的成本。Optionvaluesmaybeviewedasthesumofintrinsicvalueplustimeor“volatility”value.Thevolatilityvalueistherighttochoosenottoexerciseifthestockpricemovesagainsttheholder.Thustheoptionholdercannotlosemorethanthecostoftheoptionregardlessofstockpriceperformance.,Summary,当期权的执行价格较低，股票的价格较高，利率较高，到期时间长，风险大时，看涨期权更有价值。Calloptionsaremorevaluablewhentheexercisepriceislower,whenthestockpriceishigher,whentheinterestrateishigher,whenthetimetomaturityisgreater,whenthestocksvolatilityisgreater,andwhendividendsarelower.,Summary,看涨期权的价值应该至少等于股票价格减去执行价格与到期前支付的红利的现值，这说明不支付红利的股票看涨期权的价格可能比立即执行所获得的收入要高。因为不支付红利的美式看涨期权的提前执行没有价值，所以欧式看涨期权与不支付红利的股票美式看涨期权具有相同的价值。Calloptionsmustsellforatleastthestockpricelessthepresentvalueoftheexercisepriceanddividendstobepaidbeforematurity.Thisimpliesthatacalloptiononanon-dividend-payingstockmaybesoldformorethantheproceedsfromimmediateexercise.ThusEuropeancallsareworthasmuchasAmericancallsonstocksthatpaynodividends,becausetherighttoexercisetheAmericancallearlyhasnovalue.,Summary,可以用两时期、两状态定价模型对期权进行定价。随时期数的增加，期权公式可以更近似地反映股票价格的分布。布莱克-舒尔斯期权定价公式可以看作是当时间间隔持续地分为更小的期间时，在利率与股票的波动性保持不变的情况下，二项式期权定价公式的极限情况。Optionsmaybepricedrelativetotheunderlyingstockpriceusingasimpletwo-period,two-statepricingmodel.Asthenumberofperiodsincreases,themodelcanapproximatemorerealisticstockpricedistributions.TheBlack-Scholesformulamaybeseenasalimitingcaseofthebinomialoptionmodel,astheholdingperiodisdi-videdintoprogressivelysmallersubperiodswhentheinterestrateandstockvolatilityareconstant.,Summary,布莱克-舒尔斯期权定价公式对于不支付红利的股票期权定价是正确的，它对于支付红利的股票欧洲看涨期权的定价也是充分了。但是，对于支付红利的股票美式看涨期权的定价则需要更复杂的公式。TheBlack-Scholesformulaisvalidforoptionsonstocksthatpaynodividends.DividendadjustmentsmaybeadequatetopriceEuropeancallsondividend-payingstocks,butthepropertreatmentofAmericancallsondividend-payingstocksrequiresmorecomplexformulas.,Summary,不管股票是否支付红利，看跌期权都可提前执行。因此，一般来讲，美式看跌期权比欧式看跌期权更有价值。Putoptionsmaybeexercisedearly,whetherthestockpaysdividendsornot.Therefore,AmericanputsgenerallyareworthmorethanEuropeanputs.,Summary,欧式看跌期权的价值可以从与看涨期权的平价关系中得到，但是由于美式看跌期权有提前执行的可能，欧式看跌期权的定价方法不适用于美式看跌期权。Europeanputvaluescanbederivedfromthecallvalueandtheput-callparityrelation-ship.ThistechniquecannotbeappliedtoAmericanputsforwhichearlyexerciseisapossibility.,Summary,套期保值率是在出售期权时，为抵消期权的价格风险所需要的股票的数量，深度虚值看涨期权的套期保值率接近于0，而深度实值的看涨期权的套期保值率接近1。Thehedgeratioisthenumberofsharesofstockrequiredtohedgethepriceriskinvolvedinwritingoneoption