期权复习题11.docx

期权复习题11.docx

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期权复习题11

期权复习题

1.Explanation

speculatorswishtotakeapositioninthemarket.Eithertheyarebettingthatapricewillgouportheyarebettingthatitwillgodown.Theyusederivativestogetextraleverage

Hedgersareinterestedinreducingariskthattheyalreadyface.

Arbitrageinvolveslockinginarisk-lessprofitbyenteringsimultaneouslyintotransactionsintwoormoremarkets.

Acalloptiongivestheholdertherighttobuyanassetbyacertaindateforacertainprice.

Putoption:

Aputoptiongivestheholdertherighttosellanassetbyacertaindateforacertainprice.

Futurescontract:

Itisanagreementtobuyorsellanassetforacertainpriceatacertaintimeinthefuture.

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

Since

Where

=

25.CompaniesAandBhavebeenofferedthefollowingratesperannumon$20millionfive-yearloan:

CompanyAwantstoborrowfloating-rateloan,CompanyBwantsto

borrowfixed-rateloan.Designaswapthatwillnetbank,actingas

intermediary,0.04%perannumintotalandappeartobeequallyattractivetobothcompanies

Solution:

Differencebetweentheinterestratesinfl–ratemarketsb=0.7%

Differencebetweentheinterestratesinfixed–ratemarketsa=1.2%

TheswaparrangementappearstoimprovethepositionofbothAandB0.23%perannum