期权期货与其他衍生产品第九版课后习题与答案Chapter-(15)

CHAPTER15TheBlack-Scholes-MertonModelPracticeQuestionsProblem15.1.WhatdoestheBlack–Scholes–Mertonstockoptionpricingmodelassumeabouttheprobabilitydistributionofthestockpriceinoneyear?Whatdoesitassumeabouttheprobabilitydistributionofthecontinuouslycompoundedrateofreturnonthestockduringtheyear?TheBlack–Scholes–Mertonoptionpricingmodelassumesthattheprobabilitydistributionofthestockpricein1year(oratanyotherfuturetime)islognormal.Itassumesthatthecontinuouslycompoundedrateofreturnonthestockduringtheyearisnormallydistributed.Problem15.2.Thevolatilityofastockpriceis30%perannum.Whatisthestandarddeviationofthepercentagepricechangeinonetradingday?Thestandarddeviationofthepercentagepricechangeintimetistwhereisthevolatility.Inthisproblem03and,assuming252tradingdaysinoneyear,12520004tsothat0300040019tor1.9%.Problem15.3.Explaintheprincipleofrisk-neutralvaluation.Thepriceofanoptionorotherderivativewhenexpressedintermsofthepriceoftheunderlyingstockisindependentofriskpreferences.Optionsthereforehavethesamevalueinarisk-neutralworldastheydointherealworld.Wemaythereforeassumethattheworldisriskneutralforthepurposesofvaluingoptions.Thissimplifiestheanalysis.Inarisk-neutralworldallsecuritieshaveanexpectedreturnequaltorisk-freeinterestrate.Also,inarisk-neutralworld,theappropriatediscountratetouseforexpectedfuturecashflowsistherisk-freeinterestrate.Problem15.4.Calculatethepriceofathree-monthEuropeanputoptiononanon-dividend-payingstockwithastrikepriceof$50whenthecurrentstockpriceis$50,therisk-freeinterestrateis10%perannum,andthevolatilityis30%perannum.Inthiscase050S,50K,01r,03,025T,and121ln(5050)(010092)02502417030250302500917dddTheEuropeanputpriceis0102550(00917)50(02417)NeN0102550046345004045237eor$2.37.Problem15.5.WhatdifferencedoesitmaketoyourcalculationsinProblem15.4ifadividendof$1.50isexpectedintwomonths?InthiscasewemustsubtractthepresentvalueofthedividendfromthestockpricebeforeusingBlack–Scholes-Merton.Hencetheappropriatevalueof0Sis01667010501504852SeAsbefore50K,01r,03,and025T.Inthiscase121ln(485250)(010092)02500414030250302501086dddTheEuropeanputpriceis0102550(01086)4852(00414)NeN010255005432485204835303eor$3.03.Problem15.6.Whatisimpliedvolatility?Howcanitbecalculated?TheimpliedvolatilityisthevolatilitythatmakestheBlack–Scholes-Mertonpriceofanoptionequaltoitsmarketprice.Theimpliedvolatilityiscalculatedusinganiterativeprocedure.Asimpleapproachisthefollowing.Supposewehavetwovolatilitiesonetoohigh(i.e.,givinganoptionpricegreaterthanthemarketprice)andtheothertoolow(i.e.,givinganoptionpricelowerthanthemarketprice).Bytestingthevolatilitythatishalfwaybetweenthetwo,wegetanewtoo-highvolatilityoranewtoo-lowvolatility.Ifwesearchinitiallyfortwovolatilities,onetoohighandtheothertoolowwecanusethisprocedurerepeatedlytobisecttherangeandconvergeonthecorrectimpliedvolatility.Othermoresophisticatedapproaches(e.g.,involvingtheNewton-Raphsonprocedure)areusedinpractice.Problem15.7.Astockpriceiscurrently$40.Assumethattheexpectedreturnfromthestockis15%anditsvolatilityis25%.Whatistheprobabilitydistributionfortherateofreturn(withcontinuouscompounding)earnedoveratwo-yearperiod?Inthiscase015and025.Fromequation(15.7)theprobabilitydistributionfortherateofreturnoveratwo-yearperiodwithcontinuouscompoundingis:225.0,225.015.022i.e.,)03125.0,11875.0(Theexpectedvalueofthereturnis11.875%perannumandthestandarddeviationis17.7%perannum.Problem15.8.AstockpricefollowsgeometricBrownianmotionwithanexpectedreturnof16%andavolatilityof35%.Thecurrentpriceis$38.a)WhatistheprobabilitythataEuropeancalloptiononthestockwithanexercisepriceof$40andamaturitydateinsixmonthswillbeexercised?b)WhatistheprobabilitythataEuropeanputoptiononthestockwiththesameexercisepriceandmaturitywillbeexercised?a)Therequiredprobabilityistheprobabilityofthestockpricebeingabove$40insixmonthstime.SupposethatthestockpriceinsixmonthsisTS5.035.0,5.0235.016.038ln~ln22TSi.e.,2247.0,687.3~lnTSSinceln403689,werequiretheprobabilityofln(ST)3.689.Thisis3689368711(0008)0247NNSinceN(0.008)=0.5032,therequiredprobabilityis0.4968.b)Inthiscasetherequiredprobabilityistheprobabilityofthestockpricebeinglessthan$40insixmonthstime.Itis10496805032Problem15.9.Usingthenotationinthechapter,provethata95%confidenceintervalforTSisbetween22(2)196(2)19600andTTTTSeSeFromequation(15.3):TTSST220,2ln~ln95%confidenceintervalsforlnTSaretherefore20ln()1962STTand20ln()1962STT95%confidenceintervalsforTSaretherefore2200ln(2)196ln(2)196andSTTSTTeei.e.22(2)196(2)19600andTTTTSeSeProblem15.10.Aportfoliomanagerannouncesthattheaverageofthereturnsrealizedineachofthelast10yearsis20%perannum.Inwhatrespectisthisstatementmisleading?ThisproblemrelatestothematerialinSection15.3.Thestatementismisleadinginthatacertainsumofmoney,say$1000,wheninvestedfor10yearsinthefundwouldhaverealizedareturn(withannua