期权期货与其他衍生产品第九版课后习题与答案Chapter

1、期权期货与其他衍生产品第九版课后习题与答案ChapterCHAPTER29InterestRateDerivatives:TheStandardMarketModelsPracticeQuestionsProblem29.1.Acompanycapsthree-monthLIBORat10%perannum.Theprincipalamountis$20million.Onaresetdate,three-monthLIBORis12%perannum.Whatpaymentwouldthisleadtounderthecap?Whenwouldthepaymentbemade?Anamoun

2、t20000000002025100000$,?.?.=,wouldbepaidout3monthslater.Problem29.2.Explainwhyaswapoptioncanberegardedasatypeofbondoption.Aswapoption(orswaption)isanoptiontoenterintoaninterestrateswapatacertaintimeinthefuturewithacertainfixedratebeingused.Aninterestrateswapcanberegardedastheexchangeofafixed-ratebon

3、dforafloating-ratebond.Aswaptionisthereforetheoptiontoexchangeafixed-ratebondforafloating-ratebond.Thefloating-ratebondwillbeworthitsfacevalueatthebeginningofthelifeoftheswap.Theswaptionisthereforeanoptiononafixed-ratebondwiththestrikepriceequaltothefacevalueofthebond.Problem29.3.UsetheBlacksmodelto

4、valueaone-yearEuropeanputoptionona10-yearbond.Assumethatthecurrentvalueofthebondis$125,thestrikepriceis$110,theone-yearrisk-freeinterestrateis10%perannum,thebondsforwardpricevolatilityis8%perannum,andthepresentvalueofthecouponstobepaidduringthelifeoftheoptionis$10.Inthiscase,0110(12510)12709Fe.?=-=.

5、,110K=,011(0)PTe-.?,=,008B(T=.,and10T=.21211n(12709110)(0082)1845600800817656ddd./+./=.=-.=.Fromequation(29.2)thevalueoftheputoptionis011011110(17656)12709(18456)012eNeN-.?-.?-.-.-.=.or$0,12.Problem29.4.Explaincarefullyhowyouwoulduse(a)spotvolatilitiesand(b)flatvolatilitiestovalueafive-yearcap.Whens

6、potvolatilitiesareusedtovalueacap,adifferentvolatilityisusedtovalueeachcaplet.Whenflatvolatilitiesareused,thesamevolatilityisusedtovalueeachcapletwithinagivencap.Spotvolatilitiesareafunctionofthematurityofthecaplet.Flatvolatilitiesareafunctionofthematurityofthecap.Problem29.5.Calculatethepriceofanop

7、tionthatcapsthethree-monthrate,startingin15monthstime,at13%(quotedwithquarterlycompounding)onaprincipalamountof$1,000.Theforwardinterestratefortheperiodinquestionis12%perannum(quotedwithquarterlycompounding),the18-monthrisk-freeinterestrate(continuouslycompounded)is11.5%perannum,andthevolatilityofth

8、eforwardrateis12%perannum.Inthiscase1000L=,025k5=.,012kF=.,013KR=.,0115r=.,012k卡.,125kt=.,1(0)08416kPt+,=.250kL游0.12JL252120529505295006637dd=.=-.-.=-.ThevalueoftheoptionisVL2525008416012(05295)013(06637)NN?.?.-.-.-.059=.or$0.59.Problem29.6.AbankusesBlacksmodeltopriceEuropeanbondoptions.Supposethata

9、nimpliedpricevolatilityfora5-yearoptiononabondmaturingin10yearsisusedtopricea9-yearoptiononthebond.Wouldyouexpecttheresultantpricetobetoohighortoolow?Explain.Theimpliedvolatilitymeasuresthestandarddeviationofthelogarithmofthebondpriceatthematurityoftheoptiondividedbythesquarerootofthetimetomaturity.

10、Inthecaseofafiveyearoptiononatenyearbond,thebondhasfiveyearsleftatoptionmaturity.Inthecaseofanineyearoptiononatenyearbondithasoneyearleft.Thestandarddeviationofaoneyearbondpriceobservedinnineyearscanbenormallybeexpectedtobeconsiderablylessthanthatofafiveyearbondpriceobservedinfiveyears.(SeeFigure29.

11、1.)Wewouldthereforeexpectthepricetobetoohigh.Problem29.7.Calculatethevalueofafour-yearEuropeancalloptiononbondthatwillmaturefiveyearsfromtodayusingBlacksmodel.Thefive-yearcashbondpriceis$105,thecashpriceofafour-yearbondwiththesamecouponis$102,thestrikepriceis$100,thefour-yearrisk-freeinterestrateis1

12、0%perannumwithcontinuouscompounding,andthevolatilityforthebondpriceinfouryearsis2%perannum.Thepresentvalueoftheprincipalinthefouryearbondis40110067032e-?.=.Thepresentvalueofthecouponsis,therefore,1026703234968-.=.Thismeansthattheforwardpriceofthefive-yearbondis401(10534968)104475e?.-.=.Theparameters

13、inBlacksmodelaretherefore104475BF=.,100K=,01r=.,4T=,and002B=.u0.02/212111144010744ddd=.=.=.ThepriceoftheEuropeancallis014104475(11144)100(10744)319eNN-.?.-.=.or$3.19.Problem29.8.Iftheyieldvolatilityforafive-yearputoptiononabondmaturingin10yearstimeisspecifiedas22%,howshouldtheoptionbevalued?Assumeth

14、at,basedontodaysinterestratesthemodifieddurationofthebondatthematurityoftheoptionwillbe4.2yearsandtheforwardyieldonthebondis7%.TheoptionshouldbevaluedusingBlacksmodelinequation(29.2)withthebondpricevolatilitybeing4202202202247.?.?.=.or6.47%.Problem29.9.Whatotherinstrumentisthesameasafive-yearzero-co

15、stcollarwherethestrikepriceofthecapequalsthestrikepriceofthefloor?Whatdoesthecommonstrikepriceequal?A5-yearzero-costcollarwherethestrikepriceofthecapequalsthestrikepriceoftheflooristhesameasaninterestrateswapagreementtoreceivefloatingandpayafixedrateequaltothestrikeprice.Thecommonstrikepriceistheswa

16、prate.Notethattheswapisactuallyaforwardswapthatexcludesthefirstexchange.(SeeBusinessSnapshot29.1)Problem29.10.Deriveaput-callparityrelationshipforEuropeanbondoptions.Therearetwowayofexpressingtheput-callparityrelationshipforbondoptions.Thefirstisintermsofbondprices:0RTcIKepB-+=+wherecisthepriceofaEu

17、ropeancalloption,pisthepriceofthecorrespondingEuropeanputoption,Iisthepresentvalueofthebondcouponpaymentsduringthelifeoftheoption,Kisthestrikeprice,Tisthetimetomaturity,0Bisthebondprice,andRistherisk-freeinterestrateforamaturityequaltothelifeoftheoptions.Toprovethiswecanconsidertwoportfolios.Thefirs

18、tconsistsofaEuropeanputoptionplusthebond;thesecondconsistsoftheEuropeancalloption,andanamountofcashequaltothepresentvalueofthecouponsplusthepresentvalueofthestrikeprice.Bothcanbeseentobeworththesameatthematurityoftheoptions.Thesecondwayofexpressingtheput-callparityrelationshipisRTRTBcKepFe-+=+whereB

19、Fistheforwardbondprice.Thiscanalsobeprovedbyconsideringtwoportfolios.ThefirstconsistsofaEuropeanputoptionplusaforwardcontractonthebondplusthepresentvalueoftheforwardprice;thesecondconsistsofaEuropeancalloptionplusthepresentvalueofthestrikeprice.Bothcanbeseentobeworththesameatthematurityoftheoptions.

20、Problem29.11.Deriveaput-callparityrelationshipforEuropeanswapoptions.Theput-callparityrelationshipforEuropeanswapoptionsis+=cVpwherecisthevalueofacalloptiontopayafixedrateofsandreceivefloating,pisKthevalueofaputoptiontoreceiveafixedrateofsandpayfloating,andVisthevalueKoftheforwardswapunderlyingthesw

21、apoptionwheresisreceivedandfloatingispaid.KThiscanbeprovedbyconsideringtwoportfolios.Thefirstconsistsoftheputoption;thesecondconsistsofthecalloptionandtheswap.Supposethattheactualswaprateatthes.ThecallwillbeexercisedandtheputwillnotbematurityoftheoptionsisgreaterthanKexercised.Bothportfoliosarethenw

22、orthzero.Supposenextthattheactualswaprateatthes.TheputoptionisexercisedandthecalloptionisnotmaturityoftheoptionsislessthanKsisreceivedandfloatingispaid.exercised.BothportfoliosareequivalenttoaswapwhereKInallstatesoftheworldthetwoportfoliosareworththesameattimeT.Theymustthereforebeworththesametoday.T

23、hisprovestheresult.Problem29.12.ExplainwhythereisanarbitrageopportunityiftheimpliedBlack(flat)volatilityofacapisdifferentfromthatofafloor.DothebrokerquotesinTable29.1presentanarbitrageopportunity?Supposethatthecapandfloorhavethesamestrikepriceandthesametimetomaturity.Thefollowingput-callparityrelati

24、onshipmusthold:+=capswapfloorwheretheswapisanagreementtoreceivethecaprateandpayfloatingoverthewholelifeofthecap/floor.IftheimpliedBlackvolatilitiesforthecapequalthoseforthefloor,theBlackformulasshowthatthisrelationshipholds.Inothercircumstancesitdoesnotholdandthereisanarbitrageopportunity.Thebrokerq

25、uotesinTable29.1donotpresentanarbitrageopportunitybecausethecapofferisalwayshigherthanthefloorbidandthefloorofferisalwayshigherthanthecapbid.Problem29.13.Whenabondspriceislognormalcanthebondsyieldbenegative?Explainyouranswer.Yes.Ifazero-couponbondpriceatsomefuturetimeislognormal,thereissomechancetha

26、tthepricewillbeabovepar.Thisinturnimpliesthattheyieldtomaturityonthebondisnegative.Problem29.14.WhatisthevalueofaEuropeanswapoptionthatgivestheholdertherighttoenterintoa3-yearannual-payswapinfouryearswhereafixedrateof5%ispaidandLIBORisreceived?Theswapprincipalis$10million.AssumethattheLIBOR/swapyiel

27、dcurveisusedfordiscountingandisflatat5%perannumwithannualcompoundingandthevolatilityoftheswaprateis20%.CompareyouranswertothatgivenbyDerivaGem.Nowsupposethatallswapratesare5%andallOISratesare4.7%.UseDerivaGemtocalculatetheLIBORzerocurveandtheswapoptionvalue?百Inequation(29.10),10000000L=,005Ks=.,0005

28、s=.,10202d=.=.,2.02-=d,and56711122404105105105A=+=.Thevalueoftheswapoption(inmillionsofdollars)is1022404005(02)005(02)0178NN?.-.-.=.ThisisthesameastheanswergivenbyDerivaGem.(ForthepurposesofusingtheDerivaGemsoftware,notethattheinterestrateis4.879%withcontinuouscompoundingforallmaturities.)WhenOISdis

29、countingisusedtheLIBORzerocurveisunaffectedbecauseLIBORswapratesarethesameforallmaturities.(ThiscanbeverifiedwiththeZeroCurveworksheetinDerivaGem).Theonlydifferenceisthat2790.2047.11047.11047.11765=+=Asothatthevalueischangedto0.181.ThisisalsothevaluegivenbyDerivaGem.(NotethattheOISrateis4.593%withan

30、nualcompounding.)Problem29.15.Supposethattheyield,R,onazero-couponbondfollowstheprocessdRdtdz=萨+whereand(TarefunctionsofRandt,anddzisaWienerprocess.UseItoslemmatoshowthatthevolatilityofthezero-couponbondpricedeclinestozeroasitapproachesmaturity.Thepriceofthebondattimetis()RTte-whereTisthetimewhenthe

31、bondmatures.UsingIt?slemmathevolatilityofthebondpriceis()()()RTtRTteTteR(T-?=-?ThistendstozeroastapproachesT.Problem29.16.CarryoutamanualcalculationtoverifytheoptionpricesinExample29.2.Thecashpriceofthebondis005050005100005100051044410012282ee生e-.?.-.?.-.?-.?+=.Asthereisnoaccruedinterestthisisalsoth

32、equotedpriceofthebond.Theinterestpaidduringthelifeoftheoptionhasapresentvalueof00505005100515005244441504eTheforwardpriceofthebondistherefore005225(122821504)12061e.?.-.=.Theyieldsemiannualcompoundingis5.0630%.Thedurationofthebondatoptionmaturity005025005775005775005025005075005775005775025477547751

33、00444100e/eee耳-.?.-.?.-.?.-.?.-.?.-.?.-.?.?+.?+.?+or5.994.modifieddurationis5.994/1.025315=5.846.Thebondpricevolatilityistherefore584600506300202292.?.?.=.WecanthereforevaluethebondoptionusingBlackmodelwith12061BF=.,005225(0225)08936P-.?.,.=.,592B%=.(T,and225T=.Whenthestrikepriceisthecashprice115K=a

34、ndthevalueoftheoptionis1.74. Whenthestrikepriceisthequotedprice117K=andthevalueoftheoptionis2.36.ThisisinagreementwithDerivaGem.Problem29.17.Supposethatthe1-year,2-year,3-year,4-yearswapratesforswapswithwithiseThe,seand5-yearLIBOR-for-fixedsemiannualpaymentsare6%,6.4%,6.7%,6.9%,and7%.Thepriceofa5-ye

35、arsemiannualcapwithaprincipalof$100atacaprateof8%is$3.UseDerivaGem(thezerorateandCap_and_swap_optworksheets)todetermine(a)The5-yearflatvolatilityforcapsandfloorswithLIBORdiscounting(b)Thefloorrateinazero-cost5-yearcollarwhenthecaprateis8%andLIBORdiscountingisused(c)Answer(a)and(b)ifOISdiscountingisu

36、sedandOISswapratesare100basispointsbelowLIBORswaprates.(a)FirstwecalculatetheLIBORzerocurveusingthezerocurveworksheetofDerivaGem.The1-,2-,3-,4-,and5_yearzerorateswithcontinuouscompoundingare5.9118%,6.3140%,6.6213%,6.8297%,and6.9328%,respectively.WethentransferthesetothechoosetheCapsandSwapOptionswor

37、ksheetandchooseCap/FloorastheUnderlyingType.WeenterSemiannualfortheSettlementFrequency,100forthePrincipal,0fortheStart(Years),5fortheEnd(Years),8%fortheCap/FloorRate,and$3forthePrice.WeselectBlack-EuropeanasthePricingModelandchoosetheCapbutton.WechecktheImplyVolatilityboxandCalculate.Theimpliedvolat

38、ilityis25.4%.(b)WethenuncheckImpliedVolatility,selectFloor,checkImplyBreakevenRate.Thefloorratethatiscalculatedis6.71%.Thisisthefloorrateforwhichthefloorisworth$3.Acollarwhenthefloorrateis6.61%andthecaprateis8%haszerocost.(c)ThezerocurveworksheetnowshowsthatLIBORzeroratesfor1-,2-,3-,4-,5-yearmaturit

39、iesare5.9118%,6.3117%,6.6166%,6.8227%,and6.9249%.TheOISzeroratesare4.9385%,5.3404%,5.6468%,5.8539%,and5.9566%,respectively.WhenthesearetransferredtothecapandswaptionworksheetandtheUseOISDiscountingboxischecked,theanswertoa)becomes24.81%andtheanswertob)becomes6.60%.Problem29.18.Showthat12VfV+=where1V

40、isthevalueofaswaptiontopayafixedrateofKsandreceiveLIBORbetweentimes1Tand2T,fisthevalueofaforwardswaptoreceiveafixedrateofKsandpayLIBORbetweentimes1Tand2T,and2VisthevalueofaswapoptiontoreceiveafixedrateofKsbetweentimes1Tand2T.Deducethat12VV=whenKsequalsthecurrentforwardswaprate.Weprovethisresultbycon

41、sideringtwoportfolios.ThefirstconsistsoftheswapoptiontoreceiveKs;thesecondconsistsoftheswapoptiontopayKsandtheforwardswap.SupposethattheactualswaprateatthematurityoftheoptionsisgreaterthanKs.TheswapoptiontopayKswillbeexercisedandtheswapoptiontoreceiveKswillnotbeexercised.Bothportfoliosarethenworthze

42、rosincetheswapoptiontopayKsisneutralizedbytheforwardswap.SupposenextthattheactualswaprateatthematurityoftheoptionsislessthanKs.TheswapoptiontoreceiveKsisexercisedandtheswapoptiontopayKsisnotexercised.BothportfoliosarethenequivalenttoaswapwhereKsisreceivedandfloatingispaid.Inallstatesoftheworldthetwo

43、portfoliosareworththesameattime1T.Theymustthereforebeworththesametoday.Thisprovestheresult.WhenKsequalsthecurrentforwardswaprate0f=and12VV=.Aswapoptiontopayfixedisthereforeworththesameasasimilarswapoptiontoreceivefixedwhenthefixedrateintheswapoptionistheforwardswaprate.Problem29.19.SupposethatLIBORz

44、eroratesareasinProblem29.17.UseDerivaGemtodeterminethevalueofanoptiontopayafixedrateof6%andreceiveLIBORonafive-yearswapstartinginoneyear.Assumethattheprincipalis$100million,paymentsareexchangedsemiannually,andtheswapratevolatilityis21%.UseLIBORdiscounting.WechoosetheCapsandSwapOptionsworksheetofDeri

45、vaGemandchooseSwapOptionastheUnderlyingType.Weenter100asthePrincipal,1astheStart(Years),6astheEnd(Years),6%astheSwapRate,andSemiannualastheSettlementFrequency.WechooseBlack-Europeanasthepricingmodel,enter21%astheVolatilityandcheckthePayFixedbutton.WedonotchecktheImplyBreakevenRateandImplyVolatilityb

46、oxes.Thevalueoftheswapoptionis5.63.Problem29.20.Describehowyouwould(a)calculatecapflatvolatilitiesfromcapspotvolatilitiesand(b)calculatecapspotvolatilitiesfromcapflatvolatilities.(a) Tocalculateflatvolatilitiesfromspotvolatilitieswechooseastrikerateandusethespotvolatilitiestocalculatecapletprices.We

47、thensumthecapletpricestoobtaincappricesandimplyflatvolatilitiesfromBlacksmodel.Theanswerisslightlydependentonthestrikepricechosen.Thisprocedureignoresanyvolatilitysmileincappricing.(b) Tocalculatespotvolatilitiesfromflatvolatilitiesthefirststepisusuallytointerpolatebetweentheflatvolatilitiessothatwe

48、haveaflatvolatilityforeachcapletpaymentdate.Wechooseastrikepriceandusetheflatvolatilitiestocalculatecapprices.Bysubtractingsuccessivecappricesweobtaincapletpricesfromwhichwecanimplyspotvolatilities.Theanswerisslightlydependentonthestrikepricechosen.Thisprocedurealsoignoresanyvolatilitysmileincapletp

49、ricing.FurtherQuestionsProblem29.21.Consideraneight-monthEuropeanputoptiononaTreasurybondthatcurrentlyhas14.25yearstomaturity.Thecurrentcashbondpriceis$910,theexercisepriceis$900,andthevolatilityforthebondpriceis10%perannum.Acouponof$35willbepaidbythebondinthreemonths.Therisk-freeinterestrateis8%for

50、allmaturitiesuptooneyear.UseBlacksmodeltodeterminethepriceoftheoption.Considerboththecasewherethestrikepricecorrespondstothecashpriceofthebondandthecasewhereitcorrespondstothequotedprice.Thepresentvalueofthecouponpaymentis008025353431e-.?.=.Equation(29.2)canthereforebeusedwith008812(9103431)92366BFe

51、.?/=-.=.,008r=.,010B卡.and06667T=.Whenthestrikepriceisacashprice,900K=andOJJO.666712103587002770ddd=.=.=.Theoptionpriceistherefore70.666700806667900(02770)87569(03587)1834eNN-.?.-.-.-.=.or$18,34.Whenthestrikepriceisaquotedprice5monthsofaccruedinterestmustbeaddedto900togetthecashstrikeprice.Thecashstr

52、ikepriceis900350833392917+?.=.Inthiscase(1M.666712100319001136ddd=.=-.=.andtheoptionpriceisVb-66670080666792917(01136)87569(00319)3122eNN-.?.-.=.or$31,22.Problem29.22.Calculatethepriceofacaponthe90-dayLIBORrateinninemonthstimewhentheprincipalamountis$1,000.UseBlacksmodelwithLIBORdiscountingandthefol

53、lowinginformation:(a) Thequotednine-monthEurodollarfuturesprice=92. (Ignoredifferencesbetweenfuturesandforwardrates.)(b) Theinterest-ratevolatilityimpliedbyanine-monthEurodollaroption=15%perannum.(c) Thecurrent12-monthrisk-freeinterestratewithcontinuouscompounding=7.5%perannum.(d) Thecaprate=8%peran

54、num.(Assumeanactual/360daycount.)Thequotedfuturespricecorrespondstoaforwardrateof8%perannumwithquarterlycompoundingandactual/360.TheparametersforBlacksmodelaretherefore:008kF=.,008K=.,0075R=.,015k卡.,075kt=.,and007511(0)09277kPte-.?+,=.0.15/7521220225000650dd=.=-.andthecallprice,c,isgivenby0.15血75-02

55、5100009277008(00650)008(00650)096c=?-=,Problem29.23.SupposethattheLIBORyieldcurveisflatat8%withannualcompounding.Aswaptiongivestheholdertherighttoreceive7.6%inafive-yearswapstartinginfouryears.Paymentsaremadeannually.Thevolatilityoftheforwardswaprateis25%perannumandtheprincipalis$1million.UseBlacksm

56、odeltopricetheswaptionwithLIBORdiscounting.CompareyouranswertothatgivenbyDerivaGem.Thepayofffromtheswaptionisaseriesoffivecashflowsequaltomax00760Ts.-,inmillionsofdollarswhereTsisthefive-yearswaprateinfouryears.Thevalueofanannuitythatprovides$1peryearattheendofyears5,6,7,8,and9is95129348108ii=.EThev

57、alueoftheswaptioninmillionsofdollarsistherefore21293480076()008()NdNd.-.where2103526d=.and“25班2202274d=-.Thevalueoftheswaptionis293480076(01474)008(03526)003955NN.-.-.=.or$39,550.ThisisthesameanswerasthatgivenbyDerivaGem.NotethatforthepurposesofusingDerivaGemthezerorateis7.696%continuouslycompounded

58、forallmaturities.Problem29.24.UsetheDerivaGemsoftwaretovalueafive-yearcollarthatguaranteesthatthemaximumandminimuminterestratesonaLIBOR-basedloan(withquarterlyresets)are7%and5%respectively.TheLIBORandOISzerocurvesarecurrentlyflatat6%and5.8%respectively(withcontinuouscompounding).Useaflatvolatilityof20%.Assumethattheprincipalis$100.UseOISdiscountingWeusetheCapsandSwapOpti