JohnHull《期货、期权和衍生证券》13章习题解答

精品文档精品文档--------------------------------------------------------------------------------------------------------精品文档----------------------------------------------------CHAPTER13WienerProcessesandItô’sLemmaPracticeQuestionsProblem13.1.WhatwoulditmeantoassertthatthetemperatureatacertainplacefollowsaMarkovprocess?Doyouthinkthattemperaturesdo,infact,followaMarkovprocess?Imaginethatyouhavetoforecastthefuturetemperaturefroma)thecurrenttemperature,b)thehistoryofthetemperatureinthelastweek,andc)aknowledgeofseasonalaveragesandseasonaltrends.IftemperaturefollowedaMarkovprocess,thehistoryofthetemperatureinthelastweekwouldbeirrelevant.ToanswerthesecondpartofthequestionyoumightliketoconsiderthefollowingscenarioforthefirstweekinMay:(i)MondaytoThursdayarewarmdays;today,Friday,isaverycoldday.(ii)MondaytoFridayareallverycolddays.Whatisyourforecastfortheweekend?Ifyouaremorepessimisticinthecaseofthesecondscenario,temperaturesdonotfollowaMarkovprocess.Problem13.2.Canatradingrulebasedonthepasthistoryofastock’spriceeverproducereturnsthatareconsistentlyaboveaverage?Discuss.Thefirstpointtomakeisthatanytradingstrategycan,justbecauseofgoodluck,produceaboveaveragereturns.Thekeyquestioniswhetheratradingstrategyconsistentlyoutperformsthemarketwhenadjustmentsaremadeforrisk.Itiscertainlypossiblethatatradingstrategycoulddothis.However,whenenoughinvestorsknowaboutthestrategyandtradeonthebasisofthestrategy,theprofitwilldisappear.Asanillustrationofthis,consideraphenomenonknownasthesmallfirmeffect.Portfoliosofstocksinsmallfirmsappeartohaveoutperformedportfoliosofstocksinlargefirmswhenappropriateadjustmentsaremadeforrisk.Researchwaspublishedaboutthisintheearly1980sandmutualfundsweresetuptotakeadvantageofthephenomenon.Thereissomeevidencethatthishasresultedinthephenomenondisappearing.Problem13.3.Acompany’scashposition,measuredinmillionsofdollars,followsageneralizedWienerprocesswithadriftrateof0.5perquarterandavariancerateof4.0perquarter.Howhighdoesthecompany’sinitialcashpositionhavetobeforthecompanytohavealessthan5%chanceofanegativecashpositionbytheendofoneyear?Supposethatthecompany’sinitialcashpositionis.Theprobabilitydistributionofthecashpositionattheendofoneyeariswhereisanormalprobabilitydistributionwithmeanandvariance.Theprobabilityofanegativecashpositionattheendofoneyeariswhereisthecumulativeprobabilitythatastandardizednormalvariable(withmeanzeroandstandarddeviation1.0)islessthan.Fromnormaldistributiontableswhen:i.e.,when.Theinitialcashpositionmustthereforebe$4.58million.Problem13.4.VariablesandfollowgeneralizedWienerprocesseswithdriftratesandandvariancesand.Whatprocessdoesfollowif:Thechangesinandinanyshortintervaloftimeareuncorrelated?Thereisacorrelationbetweenthechangesinandinanyshortintervaloftime?SupposethatX1andX2equala1anda2initially.Afteratimeperiodoflength,X1hastheprobabilitydistributionandhasaprobabilitydistributionFromthepropertyofsumsofindependentnormallydistributedvariables,hastheprobabilitydistributioni.e.,ThisshowsthatfollowsageneralizedWienerprocesswithdriftrateandvariancerate.Inthiscasethechangeinthevalueofinashortintervaloftimehastheprobabilitydistribution:If,,,andareallconstant,argumentssimilartothoseinSection13.2showthatthechangeinalongerperiodoftimeisThevariable,,thereforefollowsageneralizedWienerprocesswithdriftrateandvariancerate.Problem13.5.Consideravariable,,thatfollowstheprocessForthefirstthreeyears,and;forthenextthreeyears,and.Iftheinitialvalueofthevariableis5,whatistheprobabilitydistributionofthevalueofthevariableattheendofyearsix?ThechangeinduringthefirstthreeyearshastheprobabilitydistributionThechangeduringthenextthreeyearshastheprobabilitydistributionThechangeduringthesixyearsisthesumofavariablewithprobabilitydistributionandavariablewithprobabilitydistribution.TheprobabilitydistributionofthechangeisthereforeSincetheinitialvalueofthevariableis5,theprobabilitydistributionofthevalueofthevariableattheendofyearsixisProblem13.6.Supposethatisafunctionofastockprice,andtime.Supposethatandarethevolatilitiesofand.Showthatwhentheexpectedreturnofincreasesby,thegrowthrateofincreasesby,whereisaconstant.FromItô’slemmaAlsothedriftofiswhereistheexpectedreturnonthestock.Whenincreasesby,thedriftofGincreasesbyorThegrowthrateof,therefore,increasesby.Problem13.7.StockAandstockBbothfollowgeometricBrownianmotion.Changesinanyshortintervaloftimeareuncorrelatedwitheachother.DoesthevalueofaportfolioconsistingofoneofstockAandoneofstockBfollowgeometricBrownianmotion?Explainyouranswer.Define,andasthestockprice,expectedreturnandvolatilityforstockA.Define,andasthestockprice,expectedreturnandvolatilityforstockB.Defineandasthechangeinandintime.SinceeachofthetwostocksfollowsgeometricBrownianmotion,whereandareindependentrandomsamplesfromanormaldistribution.Thiscannotbewrittenasforanyconstantsand.(Neitherthedrifttermnorthestochastictermcorrespond.)HencethevalueoftheportfoliodoesnotfollowgeometricBrownianmotion.Problem13.8.Theprocessforthestockpriceinequation(13.8)iswhereandareconstant.Explaincarefullythedifferencebetweenthismodelandeachofthefollowing:Whyisthemodelinequation(13.8)amoreappropriatemodelofstockpricebehaviorthananyofthesethreealternatives?In:theexpectedincreaseinthestockpriceandthevariabilityofthestockpriceareconstantwhenbothareexpressedasaproportion(orasapercentage)ofthestockpriceIn:theexpectedincreaseinthestockpriceandthevariabilityofthestockpriceareconstantinabsoluteterms.Forexample,iftheexpectedgrowthrateis$5perannumwhenthestockpriceis$25,itisalso$5perannumwhenitis$100.Ifthestandarddeviationofweeklystockpricemovementsis$1whenthepriceis$25,itisalso$1whenthepriceis$100.In:theexpectedincreaseinthestockpriceisaconstantproportionofthestockpricewhilethevariabilityisconstantinabsoluteterms.In:theexpectedincreaseinthestockpriceisconstantinabsolutetermswhilethevariabilityoftheproportionalstockpricechangeisconstant.Themodel:isthemostappropriateonesinceitismostrealistictoassumethattheexpectedpercentagereturnandthevariabilityofthepercentagereturninashortintervalareconstant.Problem13.9.Ithasbeensuggestedthattheshort-terminterestrate,,followsthestochasticprocesswhere,,andarepositiveconstantsandisaWienerprocess.Describethenatureofthisprocess.Thedriftrateis.Thus,whentheinterestrateisabovethedriftrateisnegativeand,whentheinterestrateisbelow,thedriftrateispositive.Theinterestrateisthereforecontinuallypulledtowardsthelevel.Therateatwhichitispulledtowardthislevelis.Avolatilityequaltoissuperimposeduponthe“pull”orthedrift.Suppose,andandthecurrentinterestrateis20%perannum.Theinterestrateispulledtowardsthelevelof10%perannum.Thiscanberegardedasalongrunaverage.Thecurrentdriftis%perannumsothattheexpectedrateattheendofoneyearisabout16%perannum.(Infactitisslightlygreaterthanthis,becauseastheinterestratedecreases,the“pull”decreases.)Superimposeduponthedriftisavolatilityof15%perannum.Problem13.10.Supposethatastockprice,,followsgeometricBrownianmotionwithexpectedreturnandvolatility:Whatistheprocessfollowedbythevariable?ShowthatalsofollowsgeometricBrownianmotion.Ifthen,,and.UsingItô’slemma:ThisshowsthatfollowsgeometricBrownianmotionwheretheexpectedreturnisandthevolatilityis.Thestockpricehasanexpectedreturnofandtheexpectedvalueofis.TheexpectedvalueofisProblem13.11.Supposethatistheyieldtomaturitywithcontinuouscompoundingonazero-couponbondthatpaysoff$1attime.Assumethatfollowstheprocesswhere,,andarepositiveconstantsandisaWienerprocess.Whatistheprocessfollowedbythebondprice?Theprocessfollowedby,thebondprice,isfromItô’slemma:Since:therequiredpartialderivativesareHence:Problem13.12(ExcelSpreadsheet)Astockwhosepriceis$30hasanexpectedreturnof9%andavolatilityof20%.InExcelsimulatethestockpricepathover5yearsusingmonthlytimestepsandrandomsamplesfromanormaldistribution.Chartthesimulatedstockpricepath.ByhittingF9observehowthepathchangesastherandomsamplechange.TheprocessisWheretisthelengthofthetimestep(=1/12)andisarandomsamplefromastandardnormaldistribution.FurtherQuestionsProblem13.13.Supposethatastockpricehasanexpectedreturnof16%perannumandavolatilityof30%perannum.Whenthestockpriceattheendofacertaindayis$50,calculatethefollowing:Theexpectedstockpriceattheendofthenextday.Thestandarddeviationofthestockpriceattheendofthenextday.The95%confidencelimitsforthestockpriceattheendofthenextday.WiththenotationinthetextInthiscase,,and.Henceandthatis,Theexpectedstockpriceattheendofthenextdayistherefore50.022Thestandarddeviationofthestockpriceattheendofthenextdayis95%confidencelimitsforthestockpriceattheendofthenextdayarei.e.,Notethatsomestudentsmayconsideronetradingdayratherthanonecalendarday.Then.Theanswerto(a)isthen50.032.Theanswerto(b)is0.945.Theanswerstopart(c)are48.18and51.88.Problem13.14.Acompany’scashposition,measuredinmillionsofdollars,followsageneralizedWienerprocesswithadriftrateof0.1permonthandavariancerateof0.16permonth.Theinitialcashpositionis2.0.Whataretheprobabilitydistributionsofthecashpositionafteronemonth,sixmonths,andoneyear?Whataretheprobabilitiesofanegativecashpositionattheendofsixmonthsandoneyear?Atwhattimeinthefutureistheprobabilityofanegativecashpositiongreatest?Theprobabilitydistributionsare:Thechanceofarandomsamplefrombeingnegativeiswhereisthecumulativeprobabilitythatastandardizednormalvariable[i.e.,avariablewithprobabilitydistribution]islessthan.Fromnormaldistributiontables.Hencetheprobabilityofanegativecashpositionattheendofsixmonthsis0.40%.Similarlytheprobabilityofanegativecashpositionattheendofoneyearisor1.07%.IngeneraltheprobabilitydistributionofthecashpositionattheendofmonthsisTheprobabilityofthecashpositionbeingnegativeismaximizedwhen:isminimized.DefineThisiszerowhenanditiseasytoverifythatforthisvalueof.Itthereforegivesaminimumvaluefor.Hencetheprobabilityofanegativecashpositionisgreatestafter20months.Problem13.15.Supposethatistheyieldonaperpetualgovernmentbondthatpaysinterestattherateof$1perannum.Assumethatisexpressedwithcontinuouscompounding,thatinterestispaidcontinuouslyonthebond,andthatfollowstheprocesswhere,,andarepositiveconstantsandisaWienerprocess.Whatistheprocessfollowedbythebondprice?Whatistheexpectedinstantaneousreturn(includinginterestandcapitalgains)totheholderofthebond?Theprocessfollowedby,thebondprice,isfromItô’slemma:Inthiscasesothat:HenceTheexpectedinstantaneousrateatwhichcapitalgainsareearnedfromthebondistherefore:Theexpectedinterestperunittimeis1.Thetotalexpectedinstantaneousreturnistherefore:Whenexpressedasaproportionofthebondpricethisis:Problem13.16.IffollowsthegeometricBrownianmotionprocessinequation(13.6),whatistheprocessfollowedby(a)y=2S,(b)y=S2,(c)y=eS,and(d)y=er(T-t)/S.IneachcaseexpressthecoefficientsofdtanddzintermsofratherthanS.Inthiscase,,andsothatItô’slemmagivesorInthiscase,,andsothatItô’slemmagivesorInthiscase,,andsothatItô’slemmagivesorInthiscase,,andsothatItô’slemmagivesorProblem13.17.Astockpriceiscurrently50.Itsexpectedreturnandvolatilityare12%and30%,respectively.Whatistheprobabilitythatthestockpricewillbegreaterthan80intwoyears?(Hintwhen.)Thevariableisnormallydistributedwithmeanandstandarddeviation.Inthiscase,,,andsothatthemeanandstandarddeviationofareand,respectively.Also,.Theprobabilitythatisthesameastheprobabilitythat.Thisiswhereistheprobabilitythatanormallydistributedvariablewithmeanzeroandstandarddeviation1islessthan.Fromthetablesatthebackofthebooksothattherequiredprobabilityis0.225.Problem13.18(SeeExcelWorksheet)StockA,whosepriceis$30,hasanexpectedreturnof11%andavolatilityof25%.StockB,whosepriceis$40,hasanexpectedreturnof15%andavolatilityof30%.Theprocessesdrivingthereturnsarecorrelatedwithcorrelationparameter.InExcel,simulatethetwostockpricepathsoverthreemonthsusingdailytimestepsandrandomsamplesfromnormaldistributions.CharttheresultsandbyhittingF9observehowthepathschangeastherandomsampleschange.Considervaluesofequalto0.50,0.75,and0.95.TheprocessesareWheretisthelengthofthetimestep(=1/252)andthe’sarecorrelatedsamplesfromstandardnormaldistributions.