金融工程二叉树.ppt

Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 1 BinomialTrees Chapter11 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 2 ASimpleBinomialModel Astockpriceiscurrently 20Inthreemonthsitwillbeeither 22or 18 StockPrice 22 StockPrice 18 Stockprice 20 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 3 StockPrice 22OptionPrice 1 StockPrice 18OptionPrice 0 Stockprice 20OptionPrice ACallOption A3 monthcalloptiononthestockhasastrikepriceof21 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 4 ConsiderthePortfolio longDsharesshort1calloptionPortfolioisrisklesswhen22D 1 18DorD 0 25 SettingUpaRisklessPortfolio Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 5 ValuingthePortfolio Risk FreeRateis12 Therisklessportfoliois long0 25sharesshort1calloptionThevalueoftheportfolioin3monthsis22 0 25 1 4 50Thevalueoftheportfoliotodayis4 5e 0 12 0 25 4 3670 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 6 ValuingtheOption Theportfoliothatislong0 25sharesshort1optionisworth4 367Thevalueofthesharesis5 000 0 25 20 Thevalueoftheoptionistherefore0 633 5 000 4 367 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 7 Generalization AderivativelastsfortimeTandisdependentonastock Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 8 Generalization continued ConsidertheportfoliothatislongDsharesandshort1derivativeTheportfolioisrisklesswhenS0uD u S0dD dor S0uD u S0dD d Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 9 Generalization continued ValueoftheportfolioattimeTisS0uD uValueoftheportfoliotodayis S0uD u e rTAnotherexpressionfortheportfoliovaluetodayisS0D fHence S0D S0uD u e rT Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 10 Generalization continued SubstitutingforDweobtain p u 1 p d e rTwhere注 公式不涉及标的股票上涨或者下跌的概率 我们本来可以很正常的认为股票上涨概率大 看涨期权的价值就应该大 但事实不是这样 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 11 pasaProbability Itisnaturaltointerpretpand1 pasprobabilitiesofupanddownmovements Thevalueofaderivativeisthenitsexpectedpayoffinarisk neutralworlddiscountedattherisk freerate p 1 p Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 12 Risk neutralValuation Whentheprobabilityofanupanddownmovementsarepand1 ptheexpectedstockpriceattimeTisS0erTThisshowsthatthestockpriceearnstherisk freerate Binomialtreesillustratethegeneralresultthattovalueaderivativewecanassumethattheexpectedreturnontheunderlyingassetistherisk freerateanddiscountattherisk freerateThisisknownasusingrisk neutralvaluation Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 13 OriginalExampleRevisited Sincepistheprobabilitythatgivesareturnonthestockequaltotherisk freerate Wecanfinditfrom20e0 12 0 25 22p 18 1 p whichgivesp 0 6523Alternatively wecanusetheformula S0u 22 u 1 S0d 18 d 0 S0 p 1 p Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 14 ValuingtheOptionUsingRisk NeutralValuation Thevalueoftheoptionise 0 12 0 25 0 6523 1 0 3477 0 0 633 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 15 IrrelevanceofStock sExpectedReturn Whenwearevaluinganoptionintermsofthethepriceoftheunderlyingasset theprobabilityofupanddownmovementsintherealworldareirrelevantThisisanexampleofamoregeneralresultstatingthattheexpectedreturnontheunderlyingassetintherealworldisirrelevant Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 16 ATwo StepExampleFigure11 3 page246 Eachtimestepis3monthsK 21 r 12 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 17 ValuingaCallOptionFigure11 4 page247 ValueatnodeB e 0 12 0 25 0 6523 3 2 0 3477 0 2 0257ValueatnodeA e 0 12 0 25 0 6523 2 0257 0 3477 0 1 2823 201 2823 22 18 24 23 2 19 80 0 16 20 0 2 0257 0 0 A B C D E F Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 18 APutOptionExample K 52Figure11 7 page250 K 52 timestep 1yrr 5 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 19 WhatHappensWhenanOptionisAmerican Figure11 8 page251 Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 20 Delta Delta D istheratioofthechangeinthepriceofastockoptiontothechangeinthepriceoftheunderlyingstockThevalueofDvariesfromnodetonode Options Futures andOtherDerivatives6thEdition Copyright JohnC Hull2005 11 21 Choosinguand