JohnHull《期货、期权和衍生证券》13章习题解答

CHAPTER13WienerProcessesandIt?’sLemmaPracticeQuestionsProblem13.1.WhatwoulditmeantoassertthatthetemperatureatacertainplacefollowsaMarkovprocess?Doyouthinkthattemperaturesdo,infact,followaMarkovImaginethatyouhavetoforecastthefuturetemperaturefroma)thecurrenttemperature,b)thehistoryofthetemperatureinthelastweek,andc)aknowledgeofseasonalaveragesandseasonaltrends.IftemperaturefollowedaMarkovprocess,theofthetemperatureinthelastweekwouldbeirrelevant.ToanswerthesecondpartofthequestionyoumightliketoconsiderthefollowingscenarioforthefirstweekinMay:MondaytoThursdayarewarmdays;today,Friday,isaverycoldday.MondaytoFridayareallverycolddays.Whatisyourforecastfortheweekend?Ifyouaremorepessimisticinthecaseofthescenario,temperaturesdonotfollowaMarkovprocess.Problem13.2.Canatradingrulebasedonthepasthistoryofastock’spriceeverproducereturnsthatareconsistentlyaboveaverage?Discuss.Thefirstpointtomakeisthatanytradingstrategycan,justbecauseofgoodluck,produceaboveaveragereturns.Thekeyquestioniswhetheratradingstrategyconsistentlyoutperformsthemarketwhenadjustmentsaremadeforrisk.Itiscertainlypossiblethatatradingstrategycoulddothis.However,whenenoughinvestorsknowaboutthestrategytradeonthebasisofthestrategy,theprofitwilldisappear.Asanillustrationofthis,consideraphenomenonknownasthesmallfirmeffect.Portfoliosofstocksinsmallfirmsappeartohaveoutperformedportfoliosofstocksinlargefirmswhenappropriateadjustmentsaremadeforrisk.Researchwaspublishedaboutthisintheearly1980sandmutualfundsweresetuptotakeadvantageofthephenomenon.Theresomeevidencethatthishasresultedinthephenomenondisappearing.Problem13.3.Acompany’scashposition,measuredinmillionsofdollars,followsageneralizedWienerprocesswithadriftrateof0.5perquarterandavariancerateof4.0perquarter.Howhighdoesthecompany’sinitialcashpositionhavetobeforthecompanytohavealessthan5%chanceofanegativecashpositionbytheendofoneyear?Supposethatthecompany’sinitialcasisp.obabilitydistributionofthecashpositionattheendofoneyearis(x444)(x2016)where(mv)isanormalprobabilitydistributionwithmeanmandvariancev.TheprobabilityofanegativecashpositionattheendofoneyearisNx20 4 4whereN(x)isthecumulativeprobabilitythatastandardizednormalvariable(withzeroandstandarddeviation1.0)islessthanx.FromnormaldistributiontablesNx20005 4 4when:x204i.e.,whenx45796.Theinitialcashpositionmustthereforebe$4.58million.Problem13.4.VariablesX1

andX2

followgeneralizedWienerprocesseswithdriftrates1

and andvariances2and2.WhatprocessdoesXX

followif:2 1 2 1 2ThechangesinX1

andX2

inanyshortintervaloftimeareuncorrelated?Thereisacorrelation betweenthechangesinX1oftime?

andX2

inanyshortintervalSupposethatX1andX2equala1anda2initially.AfteratimeperiodoflengthT, X1hastheprobabilitydistribution(aT2T)andX2

1 1 1hasaprobabilitydistribution(aT2T)2 2 2Fromthepropertyofsumsofindependentnormallydistributedvariables,XX1 2the probabilitydistribution

hasTaT2T2Ti.e.,

1 1 2 2 1 2ThisshowsthatXX1 2

aa(221 2 1 2 1 2 followsageneralizedWienerprocesswithdriftrate 1 2andvariancerate22.1 2InthiscasethechangeinthevalueofXX1 2probabilitydistribution:

inashortintervaloftimehasthe()t222)t1 2 1 2 12If,1

,,1

and areallconstant,argumentssimilartothoseinSection13.2showthatthechangeinalongerperiodoftimeTis(T222

)T1 2 1 2 12Thevariable,X1

X,thereforefollowsageneralizedWienerprocesswithdriftrate2 andvariancerate222.1 2 1 2 12Problem13.5.Consideravariable,S,thatfollowstheprocessdSdtdzForthefirstthreeyears,2and3;forthenextthreeyears,3and4.Iftheinitialvalueofthevariableis5,whatistheprobabilitydistributionofthevalueofthevariableattheendofyearsix?ThechangeinSduringthefirstthreeyearshastheprobabilitydistribution(2393)(627)Thechangeduringthenextthreeyearshastheprobabilitydistribution(33163)(948)Thechangeduringthesixyearsisthesumofavariablewithprobabilitydistribution(627)andavariablewithprobabilitydistribution(948).Theprobabilitydistributionthechangeistherefore(692748)(1575)Sincetheinitialvalueofthevariableis5,theprobabilitydistributionofthevalueofvariableattheendofyearsixis(2075)Problem13.6.SupposethatGisafunctionofastockprice,Sandtime.SupposethatS

and arethevolatilitiesofSandG.ShowthatwhentheexpectedreturnofSincreasesGbyS

,thegrowthrateofGincreasesby

,where isaconstant.GFromIt?’slemmaAlsothedriftofGis

GGSG S S G G 1 S 2S2S t 2S2whereistheexpectedreturnonthestock.WhenincreasesbyS

,thedriftofGincreasesbyor

GSS SGGThegrowthrateofG,therefore,increasesby .GProblem13.7.StockAandstockBbothfollowgeometricBrownianmotion.Changesinanyshortintervaloftimeareuncorrelatedwitheachother.DoesthevalueofaportfolioconsistingoneofstockAandoneofstockBfollowgeometricBrownianmotion?Explainyouranswer.DefineS,A A

andA

asthestockprice,expectedreturnandvolatilityforstockA.DefineS,B B

andB

asthestockprice,expectedreturnandvolatilityforstockB.DefineSA

andSB

asthechangeinSA

andSB

intime.SinceeachofthetwostocksfollowsgeometricBrownianmotion,A ASS

A

StAAttStB BB BBBwhereA

andB

areindependentrandomsamplesfromanormaldistribution.SSA B

(SA

SBB

)t

SAA

S)tBBtThiscannotbewritten

SS

(SA

S)t(S

S)ttforanyconstants and.(NeitherthedrifttermnorthestochastictermHencethevalueoftheportfoliodoesnotfollowgeometricBrownianmotion.Problem13.8.Theprocessforthestockpriceinequation(13.8)istSStStwhere andareconstant.Explaincarefullythedifferencebetweenthismodeleachofthefollowing:tStttSStttStStWhyisthemodelinequation(13.8)amoreappropriatemodelofstockpricethananyofthesethreealternatives?In:

tSStSttheexpectedincreaseinthestockpriceandthevariabilityofthestockpriceareconstantwhenbothareexpressedasaproportion(orasapercentage)ofthestockpriceIn:tStttheexpectedincreaseinthestockpriceandthevariabilityofthestockpriceareconstantinabsoluteterms.Forexample,iftheexpectedgrowthrateis$5perannumwhenthestockpriceis$25,itisalso$5perannumwhenitis$100.Ifthestandarddeviationofweeklypricemovementsis$1whenthepriceis$25,itisalso$1whenthepriceis$100.In:tSStttheexpectedincreaseinthestockpriceisaconstantproportionofthestockpricewhilevariabilityisconstantinabsoluteterms.In:tStSttheexpectedincreaseinthestockpriceisconstantinabsolutetermswhilethevariabilitytheproportionalstockpricechangeisconstant.Themodel:tSStStisthemostappropriateonesinceitismostrealistictoassumethattheexpectedpercentagereturnandthevariabilityofthepercentagereturninashortintervalareconstant.Problem13.9.Ithasbeensuggestedthattheshort-terminterestrate,r,followsthestochasticprocessdra(br)dtrcdzwherea,b,andcarepositiveconstantsanddzisaWienerprocess.Describenatureofthisprocess.Thedriftrateisa(br).Thus,whentheinterestrateisabovebthedriftrateisnegativeand,whentheinterestrateisbelowb,thedriftrateispositive.Theinterestratethereforecontinuallypulledtowardsthelevelb.Therateatwhichitispulledtowardthislevelisa.Avolatilityequaltocissuperimposeduponthe“pull”orthedrift.Supposea,bandc015andthecurrentinterestrateis20%perannum.Theinterestrateispulledtowardsthelevelof10%perannum.Thiscanberegardedasalongrunaverage.Thecurrentdriftis4%perannumsothattheexpectedrateattheendofoneyearisabout16%perannum.(Infactitisslightlygreaterthanthis,becauseasinterestratedecreases,the“pull”decreases.)Superimposeduponthedriftisavolatilityof15%perannum.Problem13.10.Supposethatastockprice,S,followsgeometricBrownianmotionwithexpectedreturn andvolatility:dSSdtSdzWhatistheprocessfollowedbythevariableSn?ShowthatSnalsofollowsgeometricBrownianmotion.IfG(St)Snthen0,GSnSn1,and2GS2n(n1)Sn2.UsingIt?’slemma:dG[nG1n(n2G]dtnGdz2ThisshowsthatGSnfollowsgeometricBrownianmotionwheretheexpectedreturnisn1n(n1)22andthevolatilityisn.ThestockpriceShasanexpectedreturnof andtheexpectedvalueofS isS.TheexpectedvalueofSnisT 0 T2Sne[n1n(n1)2]T20Problem13.11.Supposethatxistheyieldtomaturitywithcontinuouscompoundingonzero-couponbondthatpaysoff$1attimeT.Assumethatxfollowstheprocessdxa(x0

x)dtsxdzwherea,x0

,andsarepositiveconstantsanddzisaWienerprocess.Whatistheprocessfollowedbythebondprice?TheprocessfollowedbyB,thebondprice,isfromIt?’slemma:B

B 12

dBa(xx)s2x2dtsxdzx 0

t 2

xSince:

Bex(Tt)therequiredpartialderivativesareBtB

xex(Tt)xBx(Tt)ex(Tt)(Tt)B2BHence:

x2

(Tt)2ex(Tt)(Tt)2B 1 dBa(xx)(Tt)x s2x2t)2Bdtsx(Tt)Bdz 0 2 Problem13.12(ExcelSpreadsheet)Astockwhosepriceis$30hasanexpectedreturnof9%andavolatilityof20%.InExcelsimulatethestockpricepathover5yearsusingmonthlytimestepsandrandomsamplesanormaldistribution.Chartthesimulatedstockpricepath.ByhittingF9observehowthepathchangesastherandomsamplechange.TheprocessistS0.09St0.20StWheretisthelengthofthetimestep(=1/12)andisarandomsamplefromastandardnormaldistribution.FurtherQuestionsProblem13.13.Supposethatastockpricehasanexpectedreturnof16%perannumandavolatilityof30%perannum.Whenthestockpriceattheendofacertaindayis$50,calculatethefollowing:Theexpectedstockpriceattheendofthenextday.Thestandarddeviationofthestockpriceattheendofthenextday.The95%confidencelimitsforthestockpriceattheendofthenextday.WiththenotationinthetextS (t2t)SInthiscaseS50,016,030and1365000274.HenceS (016000274009000274)50(0000440000247)andS (500000445020000247)thatis,

S (002206164)Theexpectedstockpriceattheendofthenextdayistherefore50.02206154Thestandarddeviationofthestockpriceattheendofthenextday0615495%confidencelimitsforthestockpriceattheendofthenextdayare500221960785 and 500221960785

0785i.e.,4848 and 5156Notethatsomestudentsmayconsideronetradingdayratherthanonecalendarday.Then1252000397.Theanswerto(a)isthen50.032.Theanswerto(b)is0.945.Theanswerstopart(c)are48.18and51.88.Problem13.14.Acompany’scashposition,measuredinmillionsofdollars,followsageneralizeWienerprocesswithadriftrateof0.1permonthandavariancerateof0.16permonth.initialcashpositionis2.0.Whataretheprobabilitydistributionsofthecashpositionafteronemonth,months,andoneyear?Whataretheprobabilitiesofanegativecashpositionattheendofsixmonthsoneyear?Atwhattimeinthefutureistheprobabilityofanegativecashpositiongreatest?Theprobabilitydistributionsare:(2001016)(21016)(20060166)(26096)(201201612)(32196)Thechanceofarandomsamplefrom(26096)beingnegativeis096N26096whereN(x)isthecumulativeprobabilitythatastandardizednormalvariable[i.e.,variablewithprobabilitydistribution(01)]islessthanx.FromnormaldistributiontablesN(265)00040.Hencetheprobabilityofanegativecashpositionattheendofsixmonthsis0.40%.Similarlytheprobabilityofanegativecashpositionattheendofoneyearis196N32N(2196or1.07%.Ingeneraltheprobabilitydistributionofthecashpositionattheendofxmonthsis(2001x016x)Theprobabilityofthecashpositionbeingnegativeismaximizedwhen:22001x016xisminimized.Define2001x 04xy 5x102504x2 2dy 25x30dy dx 2 22x3(250125x)2Thisiszerowhenx20anditiseasytoverifythatd2ydx20forthisvalueofx.Itthereforegivesaminimumvaluefory.Hencetheprobabilityofanegativecashpositionisgreatestafter20months.Problem13.15.Supposethatxistheyieldonaperpetualgovernmentbondthatpaysinterestatrateof$1perannum.Assumethatxisexpressedwithcontinuouscompounding,thatinterestispaidcontinuouslyonthebond,andthatxfollowstheprocessdxa(x0

x)dtsxdzwherea,x0

,andsarepositiveconstantsanddzisaWienerprocess.Whatistheprocessfollowedbythebondprice?Whatistheexpectedinstantaneousreturn(includinginterestandcapitalgains)totheholderofthebond?TheprocessfollowedbyB,thebondprice,isfromIt?’slemma:B B 12B dB a(xx) s2x2dt sxdzInthiscasesothat:

x 0

t 2x2 xB1xB0

B1

2B2Hence

t x x2

x2 x3 1 1 2 1dBa(xx) s2x2 dt sxdz 0 x2 2 x3 x2 1 s2 sa(xx) dt dz 0 x2 xxTheexpectedinstantaneousrateatwhichcapitalgainsareearnedfromthebondistherefore:a(x0

1s2x2 xTheexpectedinterestperunittimeis1.Thetotalexpectedinstantaneousreturnistherefore:1a(x0

x)1x2

s2x 1a(xx)

s2 1 0 x2

x xxa(x

x)s2x 0Problem13.16.IfSfollowsthegeometricBrownianmotionprocessinequation(13.6),whatisprocessfollowedby(a)y=2S, (b)y=S2,(c)y=eS,and(d)y=er(T-t)/S. Ineachcaseexpressthecoefficientsof dt anddzintermsofyratherthanS.InthiscaseyS2,2yS20,andyt0sothatIt?’slemmagivesdy2Sdt2SdzordyydtydzInthiscaseyS2S,2yS22,andyt0sothatIt?’slemmagivesdy(2S22S2)dt2S2dzordy(22)ydt2ydzInthiscaseySeS,2yS2eS,andyt0sothatIt?’slemmagivesdy(SeS2S2eS2)dtSeSdzordy[ylny2y(lny)22]dtylnydzInthiscaseySer(Tt)S2yS,2yS22er(Tt)S32yS2,andytrer(Tt)SrysothatIt?’