期权复习题11Word格式文档下载.docx
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In-the-moneyoption:
itwouldleadtoapositivecashflowtotheholderifitwereexercisedimmediately.
risk-neutralvaluation:
Firstly,assumethattheexpectedreturnfromthestockpriceistherisk-freerater,thencalculatetheexpectedpayofffromtheoption,atlast,discountingtheexpectedpayoffattherisk-freerate
Butterflyspreads:
Abutterflyspreadinvolvespositionsinoptionswiththreedifferentstrikeprices:
buyingtwocalloptionswithstrikepricesX1andX3,andsellingtwocalloptionswithastrikepriceX2,X1<X2<X3
Factorsaffectingstockoptionpricing:
stockprice,strikeprice,risk-freeinterestrate,volatility,timetomaturity,anddividends.
Bullspreads:
Abullspreadcanbecreatedusingtwocalloptionswiththesamematurityanddifferentstrikeprices.Theinvestorbuysthecalloptionwiththelowerstrikepriceandshortsthecalloptionwiththehigherstrikeprice.Bullspreadscanalsobecreatedbybuyingaputwithalowstrikepriceandsellingaputwithahighstrikeprice.
Bearspreads:
Abearspreadcanbecreatedbysellingacallwithonelowerstrikepriceandbuyingacallwithanotherhigherstrikeprice
2.Whatisthedifferencebetweenalongforwardpositionandashortforwardposition?
Solution:
Whenatraderentersintoalongforwardcontract,sheisagreeingtobuytheunderlyingassetforacertainpriceatacertaintimeinthefuture.Whenatraderentersintoashortforwardcontract,sheisagreeingtoselltheunderlyingassetforacertainpriceatacertaintimeinthefuture.
3.“whenafuturescontractistradedontheflooroftheexchange,itmaybethecasethattheopeninterestincreasesbyone,staysthesame,ordecreasesbyone.”Explainthisstatement.
Ifbothsidesofthetransactionareenteringintoanewcontract,theopeninterestincreasesbyone.Ifbothsidesofthetransactionareclosingoutexistingposition,theopeninterestdecreasesbyone.Ifonepartyisenteringintoanewcontractwhiletheotherpartyisclosingoutanexistingposition,theopenintereststaysthesame.
4.Thepriceofgoldiscurrently$500perounce.Theforwardpricefordeliveryinoneyearis$700.Anarbitrageurcanborrowmoneyat10%perannum.Whatshouldthearbitrageurdo?
Assumethatthecostofstoringgoldiszero.
5.GivetworeasonswhytheearlyexerciseofAmericanoptionsonanon-dividend-payingstockisnotoptimal.
6Threecalloptionsonastockhavethesameexpirationdateandstrikepricesof$45,$50,and$55.Themarketpricesare$7,$5,and$2,respectively.Explainhowabutterflyspreadcanbecreated.Constructatableshowingtheprofitfromthestrategy.
7.WhatisthepriceofaEuropeanputoptiononanon-dividend-payingstockwhenthestockpriceis$69,thestrikepriceis$70,therisk-freeinterestrateis5%perannum,thevolatilityis35%perannum,andthetimetomaturityissixmonths?
IfthisoptionisaEuropeancalloption,whatistheprice?
8.WhatisthepriceofaEuropeanputoptionontheS&
P500thatissixmonthsfrommaturity,thecurrentvalueoftheindexis500,theexercisepriceis500,therisk-freeinterestrateis10%perannum,thevolatilityoftheindexis30%perannum.,Continuousdividendyieldsis2%perannum?
9.ConsideroneyearAmericanputoptiononanon-dividend-payingstockwhenthestockpriceis$300,thestrikepriceis$300,therisk-freeinterestrateis10%perannum,andthevolatilityis40%perannum.Dividetheyearintothree4-monthtimeintervalsandusethetreeapproachtoestimatethevalueoftheoption.
10.Considerthepriceofastock,S,whichfollowingtheprocess
where
isastandardBrownianmotion.Forthefirstthreeyears,
;
forthenextthreeyears,
.Iftheinitialvalueofstockpriceis$10,whatistheexpectvalueofthestockpriceattheendofyear6?
11.Whatisthedifferencebetweenforwardcontractwhentheforwardpriceis$40andacalloptionwithastrikepriceof$40?
(2)SupposethataU.S.companyknowsthatitisduetoreceive£
10,000fromoneofitsBritishexporterin30days.Itisfacedwithasignificantforeignexchangerisk.Ifyouaremanagement,howtohedgingtheforeignexchangeriskbyusingforwardcontractorcalloptionrespectively?
theforeignexchangeratesof30-dayforwardonpoundis1.60
thestrikeexchangerateofa30-daycalloptiononpoundis1.60.
12Explaincarefullythemeaningofthetermsconvenienceyieldandcostofcarry.WhatistherelationshipbetweenthefutuerspriceF,thespotpriceS,theconvenienceyield,y,andthecostofcarry,c?
Theconvenienceyield,y,(便利收益)foracommodityisameasureofthebenefitsrealizedfromownership(所有权)ofthephysicalcommodity(具体商品、实物商品)thatarenotrealizedbytheholdersofafuturescontract.
Thecostofcarry,c,isthestoragecostplustheinterestthatispaidtofinancetheassetlesstheincomeearnedontheasset.
relationshipbetweenandthefutuersprice,thespotprice,theconvenienceyieldandthecostofcarryis
13.Aone-year-longforwardcontractonanon-dividend-payingstockisenteredintowhenthestockpriceis$40andtherisk-freerateofinterestis10%perannumwithcontinuouscompounding.
(a)Whataretheforwardpriceandtheinitialvalueoftheforwardcontract?
(b)Sixmonthslater,thepriceofthestockis$45andtherisk-freeinterestrateisstill10%.Whataretheforwardpriceandthevalueoftheforwardcontract?
14.Explainthedifferencesbetweenforwardcontractandfuturescontract?
15.Explainthedifferencesbetweenexchanged–tradedandOver-the-counter?
•ExchangeTraded
–standardproducts
–tradingfloororcomputertrading
–virtuallynocreditrisk
•Over-the-Counter
–non-standardproducts
–telephonemarket
–somecreditrisk
16.Calculatethevalueofasix-monthat-the-moneyEuropeancalloptiononastockindexwhentheindexisat500,therisk-freeinterestrateis10%perannum,thevolatilityoftheindexis20%perannum,andthedividendyieldontheindexis3%perannum.
17Ifastockprice,S,followsgeometricBrownianmotion
1)Whatistheprocessfollowedbythevariable
?
Showthat
alsofollowsgeometricBrownianmotion.
2)TheexpectedvalueofSTis
.Whatistheexpectedvalueof
3)ThevarainceofSTis
.
Whatisthevarianceof
18ShowthattheprobabilitythataEuropeancalloptionwillbeexercisedinarisk-neutralworldis,
.Usingrisk-neutralvaluationtovaluethecomplicateddigtialoptionwhosepayoffatmaturityis
19SupposethataportfolioisdeltaneutralandhasagammaofΓ=-3,000,ThedeltaandgammaofaparticulartradedcalloptionareΔT=0.62and
=1.50,respectively.Whatpositioninthetradedcalloptionandintheunderlyingassetwouldmaketheportfoliobothgammaneutralanddeltaneutral?
=>
Theportfoliocanbemadegammaneutralbyincludingalongpositionof
wT=-Γ/ΓT=3000/1.5=2,000tradedcalloptionsintheportfolio.
However,thedeltaoftheportfoliowillthenchangefromzeroto
Δ=2,000ΔT=2,000×
0.62=1,240
Aquantity,1,240,oftheunderlyingassetmustbesoldfromtheportfoliotokeepitdeltaneutral.
20.Thestockpriceprocessassumedsatisfies
SupposethatfisthepriceofacalloptionorotherderivativecontingentonS.UsingnoarbitrageopportunitytoderivetheBlack-ScholesDifferentialEquation
21.Givethedefinitionsofdelta,gamma,vega,theta,andrhoofthederivative。
Givetheproof:
22.Astockpriceiscurrently$40,Itisknownthatattheendofsixmonthsitwillbeeither$36or$44.Therisk-freeinterestrateis10%.Supposethat
isthestockpriceattheendofsixmonths.
Whatisthevalueofaderivativethatpaysoff
atthistime?
23.Supposestockpricesfollowtheprocesses
ShowthattheprobabilitythataEuropeancalloptionwillbeexercisedin
arisk-neutralworldis,
.
Since
24.Supposetwostockpricesfollowtheprocesses
Weassumethatpricesofthetwostocksareindependentandthatnodividendsarepaid.Consideraderivativethatpaysoff$QattimeTifthepriceofstockAisbelow
andthepriceofstockBisbelow
Usingrisk-neutralvaluationtovaluethederivative.
Thefinalpayooff:
Inrisk-neturalworld,
Where
=
25.CompaniesAandBhavebeenofferedthefollowingratesperannumon$20millionfive-yearloan:
CompanyAwantstoborrowfloating-rateloan,CompanyBwantsto
borrowfixed-rateloan.Designaswapthatwillnetbank,actingas
intermediary,0.04%perannumintotalandappeartobeequallyattractivetobothcompanies
Differencebetweentheinterestratesinfl–ratemarketsb=0.7%
Differencebetweentheinterestratesinfixed–ratemarketsa=1.2%
TheswaparrangementappearstoimprovethepositionofbothAandB0.23%perannum