衍生产品市场-课后答案-第八章

Chapter8Swaps„Question8.1Wefirstsolveforthepresentvalueofthecostpertwobarrels:2$22$2341.033.1.06(1.065)+=Wethenobtaintheswappriceperbarrelbysolving:241.033(1.065)1.0622.483,xxx+=⇒=whichwastobeshown.„Question8.21.Wefirstsolveforthepresentvalueofthecostperthreebarrels,basedontheforwardprices:23$20$21$2255.3413.1.06(1.065)(1.07)++=Hencewecouldspend$55.3413todaytoreceive1barrelineachofthenextthreeyears.Wethenobtaintheswappriceperbarrelbysolving:2355.34131.06(1.065)(1.07)20.9519xxxx++=⇒=2.Wefirstsolveforthepresentvalueofthecostpertwobarrels(Year2andYear3):23$21$2236.473.(1.065)(1.07)+=Hencewecouldspend$36.473todayandreceive1barrelofoilinYear2andYear3.WeobtaintheswappriceperbarrelbyfindingtwoequalpaymentswewouldmakeinYears2and3thathavethesamepresentvalue:2336.473(1.065)(1.07)21.481xxx+=⇒=94McDonald•FundamentalsofDerivativesMarkets„Question8.3Sincethedealerispayingfixedandreceivingfloating,eachyearshehasacashflow$20.9519.TS−Shecanhedgethisriskbyselling1barrelforward(i.e.,shortoneforward)ineachofthethreeyears.Herpayoffsfromtheswap,theshortforwardcontracts,andthenetaresummarizedinthefollowingtable:YearNetSwapPaymentShortForwardsNetPosition11$20.9519S−1$20S−−0.951922$20.9519S−2$21S−+0.048133$20.9519S−3$22S−+1.0481Weneedtodiscountthenetcashflowstoyearzero.Wehave:230.95190.04811.0481PV(netCF)0.1.06(1.065)(1.07)−=++=Indeed,thepresentvalueofthenetcashflowiszero.„Question8.4Thefairswapratewasdeterminedtobe$20.9519.Therefore,comparedtotheforwardcurvepriceof$20inoneyear,weareoverpaying$0.9519.Inyeartwo,withinterest,thisoverpaymentincreasesto$0.95191.070024$1.01853,×=whereweusedtheappropriateforwardratetocalculatetheinterestpayment.Inyeartwo,weunderpayby$0.0481,sothatourtotalaccumulativeunderpaymentis$1.01856−$0.0481=$0.97042.Inyearthree,usingtheappropriate1-yearforwardrateof8.007%,thisnetoverpaymentincreasesto$0.970461.08007$1.0481.×=However,inyearthree,wereceiveafixedpaymentof20.9519,whichunderpaysrelativetotheforwardcurvepriceof$22by$22$20.9519$1.0481.−=Therefore,ourcumulativebalanceisindeedzero,whichwastobeshown.„Question8.5Question8.3showedthat,attheinitialyieldcurve,theswaphasazeropresentvalue.Tolookathowtheyieldcurveaffectsthevalueofthedealer’sswapposition,werepeatthehedgeandthenlookatthepresentvalueofthehedgedcashflowsunderthenewyieldcurve.AsinQuestion8.3,thedealerispayingfixedandreceivingfloating;henceeachyearshehasacashflow$20.9519.TS−Shecanhedgethisriskbyselling1barrelforward(i.e.,shortoneforward)ineachofthethreeyears.Herpayoffsfromtheswap,theshortforwardcontracts,andthenetaresummarizedinthefollowingtable:YearNetSwapPaymentShortForwardsNetPosition11$20.9519S−1$20S−−0.951922$20.9519S−2$21S−+0.048133$20.9519S−3$22S−+1.0481Chapter8Swaps95Weneedtodiscountthenetpositionstoyearzero,takingintoaccounttheuniformshiftofthetermstructure.Wehave:230.95190.04811.0481PV(netCF)0.0081.1.065(1.07)(1.075)−=++=−Thepresentvalueofthenetcashflowisnegative;thedealerneverrecoversfromtheincreasedinterestratehefacesontheoverpaymentofthefirstswappayment.Whentheinterestratesfall,thevalueoftheswapbecomes:230.95190.04811.0481PV(netCF)0.00831.055(1.06)(1.065)−=++=+Thepresentvalueofthenetcashflowispositive.Thedealermakesmoney,becausehegetsafavorableinterestrateontheloanheneedstotaketofinancethefirstoverpayment.Thedealercouldhavetriedtohedgehisexposurewithaforwardrateagreementoranyotherderivativeprotectingagainstinterestraterisk.„Question8.6Inordertoanswerthisquestion,weuseEquation(8.13)ofthemaintextwiththesubstitution:00,().iitftF=Withaneffectivequarterlyinterestof1.5%,thezero-couponpriceforquarteriis1.015i.Usingthisresult,thezero-couponpricesare:QuarterZero-couponprice10.985220.970730.956340.942250.928360.914570.901080.8877UsingEquation(8.13),the4-quarterswappriceis:.985221.970721.1.956320.8.942220.520.8533.9852.9707.9563.9422R×+×+×+×==+++Asimilarcalculationwillyieldan8-quarterswappriceof$20.4284.Thetotalcostsofprepaid4-and8-quarterswapsarethepresentvaluesofthepaymentobligations.NotethatasimplealgebraexerciseshowsthisisjustthenumeratorinEquation(8.13).Forthe4-quarterswap,wefindthisisequalto:4-quarterprepaidswapprice20.8533(.9852.9707.9563.9422)$80.3768=×+++=Asimilarcalculationwillyieldan8-quarterprepaidswappriceof$152.9256.96McDonald�FundamentalsofDerivativesMarkets„Question8.7Inordertoanswerthisquestion,weuseEquation(8.13)ofthemaintextwiththesubstitution:00,().iitftF=Forexample,the3-quarterswappriceis3.985221.970121.1.954620.820.9677.9852.9701.9546R×+×+×==++Similarcalculationsleadtothefollowingswapprices:QuarterSwapprice121.0000221.0496320.9677420.8536520.7272620.6110720.5145820.4304„Question8.8WeuseFormula(8.4),andreplacetheforwardinterestratewiththeforwardoilprices.Inparticular,wecalculate:600,3603(0,).954620.8.938820.5.923120.2.90752020.3807.9546.9388.9231.9075(0,)iitiiiPtFRPt==×+×+×+×===+++∑∑Therefore,theswappriceofa4-quarteroilswapwiththefirstsettlementoccurringinthethirdquarteris$20.3807.„Question8.9WeuseEquation(8.6)ofthemaintexttoanswerthisquestion:800,1801(0,),where[1,2,1,2,1,2,1,2](0,)iiitititiiQPtFRQQPt====∑∑Afterpluggingintherelevan