HullOFOD10eSolutionsCh05.docx
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HullOFOD10eSolutionsCh05.docx
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HullOFOD10eSolutionsCh05
CHAPTER5
DeterminationofForwardandFuturesPrices
PracticeQuestions
Problem5.1.
Explainwhathappenswhenaninvestorshortsacertainshare.
Theinvestor’sbrokerborrowsthesharesfromanotherclient’saccountandsellsthemintheusualway.Tocloseouttheposition,theinvestormustpurchasetheshares.Thebrokerthenreplacesthemintheaccountoftheclientfromwhomtheywereborrowed.Thepartywiththeshortpositionmustremittothebrokerdividendsandotherincomepaidontheshares.Thebrokertransfersthesefundstotheaccountoftheclientfromwhomtheshareswereborrowed.Occasionallythebrokerrunsoutofplacesfromwhichtoborrowtheshares.Theinvestoristhenshortsqueezedandhastocloseoutthepositionimmediately.Afeemaybechargedforborrowingshares.
Problem5.2.
Whatisthedifferencebetweentheforwardpriceandthevalueofaforwardcontract?
Theforwardpriceofanassettodayisthepriceatwhichyouwouldagreetobuyorselltheassetatafuturetime.Thevalueofaforwardcontractiszerowhenyoufirstenterintoit.Astimepassestheunderlyingassetpricechangesandthevalueofthecontractmaybecomepositiveornegative.
Problem5.3.
Supposethatyouenterintoasix-monthforwardcontractonanon-dividend-payingstockwhenthestockpriceis$30andtherisk-freeinterestrate(withcontinuouscompounding)is5%perannum.Whatistheforwardprice?
Theforwardpriceis30e0.05×0.5=30.76
Problem5.4.
Astockindexcurrentlystandsat350.Therisk-freeinterestrateis4%perannum(withcontinuouscompounding)andthedividendyieldontheindexis3%perannum.Whatshouldthefuturespriceforafour-monthcontractbe?
Thefuturespriceis350e(0.04-0.03)×0.3333=351.17
Problem5.5.
Explaincarefullywhythefuturespriceofgoldcanbecalculatedfromitsspotpriceandotherobservablevariableswhereasthefuturespriceofcoppercannot.
Goldisaninvestmentasset.Ifthefuturespriceistoohigh,investorswillfinditprofitabletoincreasetheirholdingsofgoldandshortfuturescontracts.Ifthefuturespriceistoolow,theywillfinditprofitabletodecreasetheirholdingsofgoldandgolonginthefuturesmarket.Copperisaconsumptionasset.Ifthefuturespriceistoohigh,astrategyofbuycopperandshortfuturesworks.However,becauseinvestorsdonotingeneralholdtheasset,thestrategyofsellcopperandbuyfuturesisnotavailabletothem.Thereisthereforeanupperbound,butnolowerbound,tothefuturesprice.
Problem5.6.
Explaincarefullythemeaningofthetermsconvenienceyieldandcostofcarry.Whatistherelationshipbetweenfuturesprice,spotprice,convenienceyield,andcostofcarry?
Convenienceyieldmeasurestheextenttowhichtherearebenefitsobtainedfromownershipofthephysicalassetthatarenotobtainedbyownersoflongfuturescontracts.Thecostofcarryistheinterestcostplusstoragecostlesstheincomeearned.Thefuturesprice,
andspotprice,
arerelatedby
where
isthecostofcarry,
istheconvenienceyield,and
isthetimetomaturityofthefuturescontract.
Problem5.7.
Explainwhyaforeigncurrencycanbetreatedasanassetprovidingaknownyield.
Aforeigncurrencyprovidesaknowninterestrate,buttheinterestisreceivedintheforeigncurrency.Thevalueinthedomesticcurrencyoftheincomeprovidedbytheforeigncurrencyisthereforeknownasapercentageofthevalueoftheforeigncurrency.Thismeansthattheincomehasthepropertiesofaknownyield.
Problem5.8.
Isthefuturespriceofastockindexgreaterthanorlessthantheexpectedfuturevalueoftheindex?
Explainyouranswer.
Thefuturespriceofastockindexisalwayslessthantheexpectedfuturevalueoftheindex.ThisfollowsfromSection5.14andthefactthattheindexhaspositivesystematicrisk.Foranalternativeargument,let
betheexpectedreturnrequiredbyinvestorsontheindexsothat
.Because
and
itfollowsthat
.
Problem5.9.
Aone-yearlongforwardcontractonanon-dividend-payingstockisenteredintowhenthestockpriceis$40andtherisk-freerateofinterestis5%perannumwithcontinuouscompounding.
a)Whataretheforwardpriceandtheinitialvalueoftheforwardcontract?
b)Sixmonthslater,thepriceofthestockis$45andtherisk-freeinterestrateisstill5%.Whataretheforwardpriceandthevalueoftheforwardcontract?
a)Theforwardprice,
isgivenbyequation(5.1)as:
F0=40e0.05×1=42.05
or$42.05.Theinitialvalueoftheforwardcontractiszero.
b)Thedeliveryprice
inthecontractis$42.05.Thevalueofthecontract,
aftersixmonthsisgivenbyequation(5.5)as:
f=45−42.05e−0.05×0.5=3.99
i.e.,itis$3.99.Theforwardpriceis:
45e0.05×0.5=46.14
or$46.14.
Problem5.10.
Therisk-freerateofinterestis7%perannumwithcontinuouscompounding,andthedividendyieldonastockindexis3.2%perannum.Thecurrentvalueoftheindexis150.Whatisthesix-monthfuturesprice?
Usingequation(5.3)thesixmonthfuturespriceis
or$152.88.
Problem5.11.
Assumethattherisk-freeinterestrateis4%perannumwithcontinuouscompoundingandthatthedividendyieldonastockindexvariesthroughouttheyear.InFebruary,May,August,andNovember,dividendsarepaidatarateof5%perannum.Inothermonths,dividendsarepaidatarateof2%perannum.SupposethatthevalueoftheindexonJuly31is1,300.WhatisthefuturespriceforacontractdeliverableonDecember31ofthesameyear?
Thefuturescontractlastsforfivemonths.Thedividendyieldis2%forthreeofthemonthsand5%fortwoofthemonths.Theaveragedividendyieldistherefore
Thefuturespriceistherefore
1300e(0.04-0.032)×0.4167=1304.34
or$1304.34.
Problem5.12.
Supposethattherisk-freeinterestrateis6%perannumwithcontinuouscompoundingandthatthedividendyieldonastockindexis4%perannum.Theindexisstandingat400,andthefuturespriceforacontractdeliverableinfourmonthsis405.Whatarbitrageopportunitiesdoesthiscreate?
Thetheoreticalfuturespriceis
400e(0.06-0.04)×4/12=402.68
Theactualfuturespriceis405.Thisshowsthattheindexfuturespriceistoohighrelativetotheindex.Thecorrectarbitragestrategyis
1.Sellfuturescontracts
2.Buythesharesunderlyingtheindex.
Problem5.13.
Estimatethedifferencebetweenshort-terminterestratesinJapanandtheUnitedStatesonMay3,2016fromtheinformationinTable5.4.
Thesettlementpricesforthefuturescontractsareto
Jun:
0.94015
Sept:
0.94320
TheSeptemberpriceis0.32%abovetheJuneprice.Thissuggeststhattheshort-terminterestrateinJapanwaslessthantheshort-terminterestrateintheUnitedStatesbyabout0.32%perthreemonthsorabout1.29%peryear.
Problem5.14.
Thetwo-monthinterestratesinSwitzerlandandtheUnitedStatesare,respectively,1%and2%perannumwithcontinuouscompounding.ThespotpriceoftheSwissfrancis$1.0500.Thefuturespriceforacontractdeliverableintwomonthsisalso$1.0500.Whatarbitrageopportunitiesdoesthiscreate?
Thetheoreticalfuturespriceis
1.0500e(0.02-0.01)×2/12=1.0518
Theactualfuturespriceistoolow.ThissuggeststhataSwissarbitrageurshouldsellSwissfrancsforUSdollarsandbuySwissfrancsbackinthefuturesmarket.
Problem5.15.
Thespotpriceofsilveris$25perounce.Thestoragecostsare$0.24perounceperyearpayablequarterlyinadvance.Assumingthatinterestratesare5%perannumforallmaturities,calculatethefuturespriceofsilverfordeliveryinninemonths.
Thepresentvalueofthestoragecostsforninemonthsare
0.06+0.06e-0.05×0.25+0.06e-0.05×0.5=0.178
or$0.178.Thefuturespriceisfromequation(5.11)givenby
where
i.e.,itis$26.14perounce.
Problem5.16.
Supposethat
and
aretwofuturescontractsonthesamecommoditywithtimestomaturity,
and
where
.Provethat
where
istheinterestrate(assumedconstant)andtherearenostoragecosts.Forthepurposesofthisproblem,assumethatafuturescontractisthesameasaforwardcontract.
If
aninvestorcouldmakearisklessprofitby
3.Takingalongpositioninafuturescontractwhichmaturesattime
4.Takingashortpositioninafuturescontractwhichmaturesattime
Whenthefirstfuturescontractmatures,theassetispurchasedfor
usingfundsborrowedatrater.Itisthenhelduntiltime
atwhichpointitisexchangedfor
underthesecondcontract.Thecostsofthefundsborrowedandaccumulatedinterestattime
is
.Apositiveprofitof
isthenrealizedattime
.Thistypeofarbitrageopportunitycannotexistforlong.Hence:
Problem5.17.
Whenaknownfuturecashoutflowinaforeigncurrencyishedgedbyacompanyusingaforwardcontract,thereisnoforeignexchangerisk.Whenitishedgedusingfuturescontracts,thedailysettlementprocessdoesleavethecompanyexposedtosomerisk.Explainthenatureofthisrisk.Inparticular,considerwhetherthecompanyisbetteroffusingafuturescontractoraforwardcontractwhen
a)Thevalueoftheforeigncurrencyfallsrapidlyduringthelifeofthecontract
b)Thevalueoftheforeigncurrencyrisesrapidlyduringthelifeofthecontract
c)Thevalueoftheforeigncurrencyfirstrisesandthenfallsbacktoitsinitialvalue
d)Thevalueoftheforeigncurrencyfirstfallsandthenrisesbacktoitsinitialvalue
Assumethattheforwardpriceequalsthefuturesprice.
Intotalthegainorlossunderafuturescontractisequaltothegainorlossunderthecorrespondingforwardcontract.Howeverthetimi
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