期权期货与其他衍生产品第九版课后习题与答案Chapter.docx

期权期货与其他衍生产品第九版课后习题与答案Chapter.docx

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期权期货与其他衍生产品第九版课后习题与答案Chapter

期权期货与其他衍生产品第九版课后习题与答案Chapter

CHAPTER29

InterestRateDerivatives:

TheStandardMarketModels

PracticeQuestions

Problem29.1.

Acompanycapsthree-monthLIBORat10%perannum.Theprincipalamountis$20million.Onaresetdate,three-monthLIBORis12%perannum.Whatpaymentwouldthisleadtounderthecap?

Whenwouldthepaymentbemade?

Anamount

20000000002025100000$$,,?

.?

.=,

wouldbepaidout3monthslater.

Problem29.2.

Explainwhyaswapoptioncanberegardedasatypeofbondoption.

Aswapoption（orswaption）isanoptiontoenterintoaninterestrateswapatacertaintimeinthefuturewithacertainfixedratebeingused.Aninterestrateswapcanberegardedastheexchangeofafixed-ratebondforafloating-ratebond.Aswaptionisthereforetheoptiontoexchangeafixed-ratebondforafloating-ratebond.Thefloating-ratebondwillbeworthitsfacevalueatthebeginningofthelifeoftheswap.Theswaptionisthereforeanoptiononafixed-ratebondwiththestrikepriceequaltothefacevalueofthebond.

Problem29.3.

UsetheBlack’smodeltovalueaone-yearEuropeanputoptionona10-yearbond.Assumethatthecurrentvalueofthebondis$125,thestrikepriceis$110,theone-yearrisk-freeinterestrateis10%perannum,thebond’sforwardpricevolatilityis8%perannum,andthepresentvalueofthecouponstobepaidduringthelifeoftheoptionis$10.

Inthiscase,0110（12510）12709Fe.?

=-=.,110K=,011（0）PTe-.?

=,008Bσ=.,and10T=..2121ln（12709110）（0082）18456008

00817656

ddd./+./==..=-.=.Fromequation（29.2）th

evalueo

ftheputoptionis

011011110（17656）12709（18456）012eNeN-.?

-.?

-.-.-.=.

or$0.12.

Problem29.4.

Explaincarefullyhowyouwoulduse（a）spotvolatilitiesand（b）flatvolatilitiestovalueafive-yearcap.

Whenspotvolatilitiesareusedtovalueacap,adifferentvolatilityisusedtovalueeach

caplet.Whenflatvolatilitiesareused,thesamevolatilityisusedtovalueeachcapletwithinagivencap.Spotvolatilitiesareafunctionofthematurityofthecaplet.Flatvolatilitiesarea

functionofthematurityofthecap.

Problem29.5.

Calculatethepriceofanoptionthatcapsthethree-monthrate,startingin15months’time,at13%（quotedwithquarterlycompounding）onaprincipalamountof$1,000.Theforwardinterestratefortheperiodinquestionis12%perannum（quotedwithquarterly

compounding）,the18-monthrisk-freeinterestrate（continuouslycompounded）is11.5%perannum,andthevolatilityoftheforwardrateis12%perannum.

Inthiscase1000L=,025kδ=.,012kF=.,013KR=.,0115r=.,012kσ=.,125kt=.,1（0）08416kPt+,=..

250kLδ=

212052*********6637

dd==-.=-.-.=-.Th

evalueo

ftheoptionis

25008416[012（05295）013（06637）]NN?

.?

.-.-.-.

059=.or$0.59.

Problem29.6.

AbankusesBlack’smodeltopriceEuropeanbondoptions.Supposethatanimpliedpricevolatilityfora5-yearoptiononabondmaturingin10yearsisusedtopricea9-yearoptiononthebond.Wouldyouexpecttheresultantpricetobetoohighortoolow?

Explain.

Theimpliedvolatilitymeasuresthestandarddeviationofthelogarithmofthebondpriceatthematurityoftheoptiondividedbythesquarerootofthetimetomaturity.Inthecaseofafiveyearoptiononatenyearbond,thebondhasfiveyearsleftatoptionmaturity.Inthecaseofanineyearoptiononatenyearbondithasoneyearleft.Thestandarddeviationofaoneyearbondpriceobservedinnineyearscanbenormallybeexpectedtobeconsiderablylessthanthatofafiveyearbondpriceobservedinfiveyears.（SeeFigure29.1.）Wewouldthereforeexpectthepricetobetoohigh.

Problem29.7.

Calculatethevalueofafour-yearEuropeancalloptiononbondthatwillmaturefiveyearsfromtodayusingBlack’smodel.Thefive-yearcashbondpriceis$105,thecashpriceofafour-yearbondwiththesamecouponis$102,thestrikepriceis$100,thefour-yearrisk-freeinterestrateis10%perannumwithcontinuouscompounding,andthevolatilityforthebondpriceinfouryearsis2%perannum.

Thepresentvalueoftheprincipalinthefouryearbondis40110067032e-?

.=..Thepresentvalueofthecouponsis,therefore,1026703234968-.=..Thismeansthattheforwardpriceofthefive-yearbondis

401（10534968）104475e?

.-.=.TheparametersinBlack’smodelaretherefore104475BF=.,100K=,01r=.,4T=,

and002B=.σ.

212111144010744

ddd==.=-.=.Th

epriceo

ftheEuropeancallis

014[104475（11144）100（10744）]319eNN-.?

..-.=.

or$3.19.

Problem29.8.

Iftheyieldvolatilityforafive-yearputoptiononabondmaturingin10yearstimeis

specifiedas22%,howshouldtheoptionbevalued?

Assumethat,basedontoday’sinterestratesthemodifieddurationofthebondatthematurityoftheoptionwillbe4.2yearsandtheforwardyieldonthebondis7%.

TheoptionshouldbevaluedusingBlack’smodelinequation（29.2）withthebondpricevolatilitybeing

4200702200647.?

.?

.=.or6.47%.

Problem29.9.

Whatotherinstrumentisthesameasafive-yearzero-costcollarwherethestrikepriceofthecapequalsthestrikepriceofthefloor?

Whatdoesthecommonstrikepriceequal?

A5-yearzero-costcollarwherethestrikepriceofthecapequalsthestrikepriceoftheflooristhesameasaninterestrateswapagreementtoreceivefloatingandpayafixedrateequaltothestrikeprice.Thecommonstrikepriceistheswaprate.Notethattheswapisactuallyaforwardswapthatexcludesthefirstexchange.（SeeBusinessSnapshot29.1）

Problem29.10.

Deriveaput–callparityrelationshipforEuropeanbondoptions.

Therearetwowayofexpressingtheput–callparityrelationshipforbondoptions.Thefirstisintermsofbondprices:

0RTcIKepB-++=+

wherecisthepriceofaEuropeancalloption,pisthepriceofthecorrespondingEuropeanputoption,Iisthepresentvalueofthebondcouponpaymentsduringthelifeoftheoption,Kisthestrikeprice,Tisthetimetomaturity,0Bisthebondprice,andR

istherisk-freeinterestrateforamaturityequaltothelifeoftheoptions.Toprovethiswecanconsidertwoportfolios.ThefirstconsistsofaEuropeanputoptionplusthebond;thesecondconsistsoftheEuropeancalloption,andanamountofcashequaltothepresentvalueofthecouponsplusthepresentvalueofthestrikeprice.Bothcanbeseentobeworththesameatthematurityoftheoptions.

Thesecondwayofexpressingtheput–callparityrelationshipis

RTRTBcKepFe--+=+

whereBFistheforwardbondprice.Thiscanalsobeprovedbyconsideringtwoportfolios.ThefirstconsistsofaEuropeanputoptionplusaforwardcontractonthebondplusthepresentvalueoftheforwardprice;thesecondconsistsofaEuropeancalloptionplusthe

presentvalueofthestrikeprice.Bothcanbeseentobeworththesameatthematurityoftheoptions.

Problem29.11.

Deriveaput–callparityrelationshipforEuropeanswapoptions.

Theput–callparityrelationshipforEuropeanswapoptionsis

+=

cVp

wherecisthevalueofacalloptiontopayafixedrateof

sandreceivefloating,pis

K

thevalueofaputoptiontoreceiveafixedrateof

sandpayfloating,andVisthevalue

K

oftheforwardswapunderlyingtheswapoptionwhere

sisreceivedandfloatingispaid.

K

Thiscanbeprovedbyconsideringtwoportfolios.Thefirstconsistsoftheputoption;thesecondconsistsofthecalloptionandtheswap.Supposethattheactualswaprateatthe

s.Thecallwillbeexercisedandtheputwillnotbematurityoftheoptionsisgreaterthan

K

exercised.Bothportfoliosarethenworthzero.Supposenextthattheactualswaprateatthe

s.Theputoptionisexercisedandthecalloptionisnotmaturityoftheoptionsislessthan

K

sisreceivedandfloatingispaid.exercised.Bothportfoliosareequivalenttoaswapwhere

K

InallstatesoftheworldthetwoportfoliosareworththesameattimeT.Theymustthereforebeworththesametoday.Thisprovestheresult.

Problem29.12.

ExplainwhythereisanarbitrageopportunityiftheimpliedBlack（flat）volatilityofacapisdifferentfromthatofafloor.DothebrokerquotesinTable29.1presentanarbitrageopportunity?

Supposethatthecapandfloorhavethesamestrikepriceandthesametimetomaturity.Thefollowingput–callparityrelationshipmusthold:

+=

capswapfloor

wheretheswapisanagreementtoreceivethecaprateandpayfloatingoverthewholelifeofthecap/floor.IftheimpliedBlackvolatilitiesforthecapequalthoseforthefloor,theBlackformulasshowthatthisrelationshipholds.Inothercircumstancesitdoesnotholdandthereisanarbitrageopportunity.ThebrokerquotesinTable29.1donotpresentanarbitrageopportunitybecausethecapofferisalwayshigherthanthefloorbidandthefloorofferisalwayshigherthanthecapbid.

Problem29.13.

Whenabond’spriceislognormalcanthebond’syieldbenegative?

Explainyouranswer.

Yes.Ifazero-couponbondpriceatsomefuturetimeislognormal,thereissomechancethatthepricewillbeabovepar.Thisinturnimpliesthattheyieldtomaturityonthebondisnegative.

Problem29.14.

WhatisthevalueofaEuropeanswapoptionthatgivestheholdertherighttoenterintoa

3-yearannual-payswapinfouryearswhereafixedrateof5%ispaidandLIBORisreceived?

Theswapprincipalis$10million.AssumethattheLIBOR/swapyieldcurveisusedfordiscountingandisflatat5%perannumwithannualcompoundingandthevolatilityoftheswaprateis20%.CompareyouranswertothatgivenbyDerivaGem.Nowsupposethatall

swapratesare5%andallOISratesare4.7%.UseDerivaGemtocalculatetheLIBORzerocurveandtheswapoptionvalue?