JohnHull《期货、期权和衍生证券》13章习题解答

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CHAPTER13

WienerProcessesandItô’sLemma

PracticeQuestions

Problem13.1.

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.Theprobabilitydistributionofthecashpositionattheendofoneyearis

whereisanormalprobabilitydistributionwithmeanandvariance.Theprobabilityofanegativecashpositionattheendofoneyearis

whereisthecumulativeprobabilitythatastandardizednormalvariable<withmeanzeroandstandarddeviation1.0>islessthan.Fromnormaldistributiontables

when:

i.e.,when.Theinitialcashpositionmustthereforebe$4.58million.

Problem13.4.

VariablesandfollowgeneralizedWienerprocesseswithdriftratesandandvariancesand.Whatprocessdoesfollowif:

Thechangesinandinanyshortintervaloftimeareuncorrelated?

Thereisacorrelationbetweenthechangesinandinanyshortintervaloftime?

SupposethatX1andX2equala1anda2initially.Afteratimeperiodoflength,X1hastheprobabilitydistribution

andhasaprobabilitydistribution

Fromthepropertyofsumsofindependentnormallydistributedvariables,hastheprobabilitydistribution

i.e.,

ThisshowsthatfollowsageneralizedWienerprocesswithdriftrateandvariancerate.

Inthiscasethechangeinthevalueofinashortintervaloftimehastheprobabilitydistribution:

If,,,andareallconstant,argumentssimilartothoseinSection13.2showthatthechangeinalongerperiodoftimeis

Thevariable,,thereforefollowsageneralizedWienerprocesswithdriftrateandvariancerate.

Problem13.5.

Consideravariable,,thatfollowstheprocess

Forthefirstthreeyears,and;forthenextthreeyears,and.Iftheinitialvalueofthevariableis5,whatistheprobabilitydistributionofthevalueofthevariableattheendofyearsix?

Thechangeinduringthefirstthreeyearshastheprobabilitydistribution

Thechangeduringthenextthreeyearshastheprobabilitydistribution

Thechangeduringthesixyearsisthesumofavariablewithprobabilitydistributionandavariablewithprobabilitydistribution.Theprobabilitydistributionofthechangeistherefore

Sincetheinitialvalueofthevariableis5,theprobabilitydistributionofthevalueofthevariableattheendofyearsixis

Problem13.6.

Supposethatisafunctionofastockprice,andtime.Supposethatandarethevolatilitiesofand.Showthatwhentheexpectedreturnofincreasesby,thegrowthrateofincreasesby,whereisaconstant.

FromItô’slemma

Alsothedriftofis

whereistheexpectedreturnonthestock.Whenincreasesby,thedriftofGincreasesby

or

Thegrowthrateof,therefore,increasesby.

Problem13.7.

StockAandstockBbothfollowgeometricBrownianmotion.Changesinanyshortintervaloftimeareuncorrelatedwitheachother.DoesthevalueofaportfolioconsistingofoneofstockAandoneofstockBfollowgeometricBrownianmotion?Explainyouranswer.

Define,andasthestockprice,expectedreturnandvolatilityforstockA.Define,andasthestockprice,expectedreturnandvolatilityforstockB.Defineandasthechangeinandintime.SinceeachofthetwostocksfollowsgeometricBrownianmotion,

whereandareindependentrandomsamplesfromanormaldistribution.

Thiscannotbewrittenas

foranyconstantsand.<Neitherthedrifttermnorthestochastictermcorrespond.>HencethevalueoftheportfoliodoesnotfollowgeometricBrownianmotion.

Problem13.8.

Theprocessforthestockpriceinequation<13.8>is

whereandareconstant.Explaincarefullythedifferencebetweenthismodelandeachofthefollowing:

Whyisthemodelinequation<13.8>amoreappropriatemodelofstockpricebehavior

thananyofthesethreealternatives?In:

theexpectedincreaseinthestockpriceandthevariabilityofthestockpriceareconstantwhenbothareexpressedasaproportion<orasapercentage>ofthestockprice

In:

theexpectedincreaseinthestockpriceandthevariabilityofthestockpriceareconstantinabsoluteterms.Forexample,iftheexpectedgrowthrateis$5perannumwhenthestockpriceis$25,itisalso$5perannumwhenitis$100.Ifthestandarddeviationofweeklystockpricemovementsis$1whenthepriceis$25,itisalso$1whenthepriceis$100.

In:

theexpectedincreaseinthestockpriceisaconstantproportionofthestockpricewhilethevariabilityisconstantinabsoluteterms.

In:

theexpectedincreaseinthestockpriceisconstantinabsolutetermswhilethevariabilityoftheproportionalstockpricechangeisconstant.

Themodel:

isthemostappropriateonesinceitismostrealistictoassumethattheexpectedpercentagereturnandthevariabilityofthepercentagereturninashortintervalareconstant.

Problem13.9.

Ithasbeensuggestedthattheshort-terminterestrate,,followsthestochasticprocess

where,,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:

ThisshowsthatfollowsgeometricBrownianmotionwheretheexpectedreturnis

andthevolatilityis.Thestockpricehasanexpectedreturnofandtheexpectedvalueofis.Theexpectedvalueofis

Problem13.11.

Supposethatistheyieldtomaturitywithcontinuouscompoundingonazero-couponbondthatpaysoff$1attime.Assumethatfollowstheprocess

where,,andarepositiveconstantsandisaWienerprocess.Whatistheprocessfollowedbythebondprice?

Theprocessfollowedby,thebondprice,isfromItô’slemma:

Since:

therequiredpartialderivativesare

Hence:

Problem13.12<ExcelSpreadsheet>

Astockwhosepriceis$30hasanexpectedreturnof9%andavolatilityof20%.InExcelsimulatethestockpricepathover5yearsusingmonthlytimestepsandrandomsamplesfromanormaldistribution.Chartthesimulatedstockpricepath.ByhittingF9observehowthepathchangesastherandomsamplechange.

Theprocessis

Wheretisthelengthofthetimestep<=1/12>andisarandomsamplefromastandardnormaldistribution.

FurtherQuestions

Problem13.13.

Supposethatastockpricehasanexpectedreturnof16%perannumandavolatilityof30%perannum.Whenthestockpriceattheendofacertaindayis$50,calculatethefollowing:

Theexpectedstockpriceattheendofthenextday.

Thestandarddeviationofthestockpriceattheendofthenextday.

The95%confidencelimitsforthestockpriceattheendofthenextday.

Withthenotationinthetext

Inthiscase,,and.Hence

and

thatis,

Theexpectedstockpriceattheendofthenextdayistherefore50.022

Thestandarddeviationofthestockpriceattheendofthenextdayis

95%confidencelimitsforthestockpriceattheendofthenextdayare

i.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:

Thechanceofarandomsamplefrombeingnegativeis

whereisthecumulativeprobabilitythatastandardizednormalvariable[i.e.,avariablewithprobabilitydistribution]islessthan.Fromnormaldistributiontables.Hencetheprobabilityofanegativecashpositionattheendofsixmonthsis0.40%.

Similarlytheprobabilityofanegativecashpositionattheendofoneyearis

or1.07%.

Ingeneraltheprobabilitydistributionofthecashpositionattheendofmonthsis

Theprobabilityofthecashpositionbeingnegativeismaximizedwhen:

isminimized.Define

Thisiszerowhenanditiseasytoverifythatforthisvalueof.Itthereforegivesaminimumvaluefor.Hencetheprobabilityofanegativecashpositionisgreatestafter20months.

Problem13.15.

Supposethatistheyieldonaperpetualgovernmentbondthatpaysinterestattherateof$1perannum.Assumethatisexpressedwithcontinuouscompounding,thatinterestispaidcontinuouslyonthebond,andthatfollowstheprocess

where,,andarepositiveconstantsandisaWienerprocess.Whatistheprocessfollowedbythebondprice?Whatistheexpectedinstantaneousreturn<includinginterestandcapitalgains>totheholderofthebond?

Theprocessfollowedby,thebondprice,isfromItô’slemma:

Inthiscase

sothat:

Hence

Theexpectedinstantaneousrateatwhichcapitalgainsareearnedfromthebondistherefore:

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’slemmagives

or

Inthiscase,,andsothatItô’slemmagives

or

Inthiscase,,andsothatIt