HullFund8eCh12ProblemSolutions讲解.docx

HullFund8eCh12ProblemSolutions讲解.docx

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HullFund8eCh12ProblemSolutions讲解

CHAPTER12

IntroductiontoBinomialTrees

PracticeQuestions

Problem12.8.

ConsiderthesituationinwhichstockpricemovementsduringthelifeofaEuropeanoptionaregovernedbyatwo-stepbinomialtree.Explainwhyitisnotpossibletosetupapositioninthestockandtheoptionthatremainsrisklessforthewholeofthelifeoftheoption.

Therisklessportfolioconsistsofashortpositionintheoptionandalongpositioninshares.Becausechangesduringthelifeoftheoption,thisrisklessportfoliomustalsochange.

Problem12.9.

Astockpriceiscurrently$50.Itisknownthatattheendoftwomonthsitwillbeeither$53or$48.Therisk-freeinterestrateis10%perannumwithcontinuouscompounding.Whatisthevalueofatwo-monthEuropeancalloptionwithastrikepriceof$49?

Useno-arbitragearguments.

Attheendoftwomonthsthevalueoftheoptionwillbeeither$4（ifthestockpriceis$53）or$0（ifthestockpriceis$48）.Consideraportfolioconsistingof:

Thevalueoftheportfolioiseitherorintwomonths.If

i.e.,

thevalueoftheportfolioiscertaintobe38.4.Forthisvalueoftheportfolioisthereforeriskless.Thecurrentvalueoftheportfoliois:

whereisthevalueoftheoption.Sincetheportfoliomustearntherisk-freerateofinterest

i.e.,

Thevalueoftheoptionistherefore$2.23.

Thiscanalsobecalculateddirectlyfromequations（12.2）and（12.3）.,sothat

and

Problem12.10.

Astockpriceiscurrently$80.Itisknownthatattheendoffourmonthsitwillbeeither$75or$85.Therisk-freeinterestrateis5%perannumwithcontinuouscompounding.Whatisthevalueofafour-monthEuropeanputoptionwithastrikepriceof$80?

Useno-arbitragearguments.

Attheendoffourmonthsthevalueoftheoptionwillbeeither$5（ifthestockpriceis$75）or$0（ifthestockpriceis$85）.Consideraportfolioconsistingof:

（Note:

Thedelta,ofaputoptionisnegative.Wehaveconstructedtheportfoliosothatitis+1optionandsharesratherthanoptionandsharessothattheinitialinvestmentispositive.）

Thevalueoftheportfolioiseitherorinfourmonths.If

i.e.,

thevalueoftheportfolioiscertaintobe42.5.Forthisvalueoftheportfolioisthereforeriskless.Thecurrentvalueoftheportfoliois:

whereisthevalueoftheoption.Sincetheportfolioisriskless

i.e.,

Thevalueoftheoptionistherefore$1.80.

Thiscanalsobecalculateddirectlyfromequations（12.2）and（12.3）.,sothat

and

Problem12.11.

Astockpriceiscurrently$40.Itisknownthatattheendofthreemonthsitwillbeeither$45or$35.Therisk-freerateofinterestwithquarterlycompoundingis8%perannum.Calculatethevalueofathree-monthEuropeanputoptiononthestockwithanexercisepriceof$40.Verifythatno-arbitrageargumentsandrisk-neutralvaluationargumentsgivethesameanswers.

Attheendofthreemonthsthevalueoftheoptioniseither$5（ifthestockpriceis$35）or$0（ifthestockpriceis$45）.

Consideraportfolioconsistingof:

（Note:

Thedelta,,ofaputoptionisnegative.Wehaveconstructedtheportfoliosothatitis+1optionandsharesratherthanoptionandsharessothattheinitialinvestmentispositive.）

Thevalueoftheportfolioiseitheror.If:

i.e.,

thevalueoftheportfolioiscertaintobe22.5.Forthisvalueoftheportfolioisthereforeriskless.Thecurrentvalueoftheportfoliois

wherefisthevalueoftheoption.Sincetheportfoliomustearntherisk-freerateofinterest

Hence

i.e.,thevalueoftheoptionis$2.06.

Thiscanalsobecalculatedusingrisk-neutralvaluation.Supposethatistheprobabilityofanupwardstockpricemovementinarisk-neutralworld.Wemusthave

i.e.,

or:

Theexpectedvalueoftheoptioninarisk-neutralworldis:

Thishasapresentvalueof

Thisisconsistentwiththeno-arbitrageanswer.

Problem12.12.

Astockpriceiscurrently$50.Overeachofthenexttwothree-monthperiodsitisexpectedtogoupby6%ordownby5%.Therisk-freeinterestrateis5%perannumwithcontinuouscompounding.Whatisthevalueofasix-monthEuropeancalloptionwithastrikepriceof$51?

AtreedescribingthebehaviorofthestockpriceisshowninFigureS12.1.Therisk-neutralprobabilityofanupmove,p,isgivenby

Thereisapayofffromtheoptionofforthehighestfinalnode（whichcorrespondstotwoupmoves）zeroinallothercases.Thevalueoftheoptionistherefore

ThiscanalsobecalculatedbyworkingbackthroughthetreeasindicatedinFigureS12.1.Thevalueofthecalloptionisthelowernumberateachnodeinthefigure.

FigureS12.1TreeforProblem12.12

Problem12.13.

ForthesituationconsideredinProble