赫尔期权期货及其他衍生产品第8版复习笔记及课后习题详解(50).docx

CHAPTER30Convexity,Timing,andQuantoAdjustmentsPracticeQuestionsProb1em30.1.Exp1ainhowyouwou1dva1ueaderivativethatpaysoffIooRinfiveyearswhereRistheone-yearinterestrate(annua11ycompounded)observedinfouryears.Whatdifferencewou1ditmakeifthepayoffwerein(a)4yearsand(b)6years?Theva1ueofthederivativeis100/?45P(0,5)whereP(V)istheva1ueofat-yearzero-couponbondtodayandR1histheforwardratefortheperiodbetweenr1andt2,expressedwithannua1compounding.Ifthepayoffismadeinfouryearstheva1ueis100(5+c)P(0,4)wherecistheconvexityadjustmentgivenbyequation(30.2).Theformu1afortheconvexityadjustmentis:C=强生0+¾,5)whereff1yisthevo1ati1ityoftheforwardratebetweentimesZ1andt2.Theexpression100(45+c)istheexpectedpayoffinawor1dthatisforwardriskneutra1withrespecttoazero-couponbondmaturingattimefouryears.Ifthepayoffismadeinsixyears,theva1ueisfromequation(30.4)givenbyIoO(R4$+C)P(0,6)exp_1+叫,6_wherepisthecorre1ationbetweenthe(4,5)and(4,6)forwardrates.AsanapproximationwecanassumethatP=1,45=46,and/?45=/?46.Approximatingtheexponentia1functionwethengettheva1ueofthederivativeasIOO(R4一C)尸(0,6).Prob1em30.2.Exp1ainwhetheranyconvexityortimingadjustmentsarenecessarywhen(a) Wewishtova1ueaspreadoptionthatpaysoffeveryquartertheexcess(ifany)ofthefive-yearswaprateoverthethree-month11BORrateapp1iedtoaprincipa1of$100.Thepayoffoccurs90daysaftertheratesareobserved.(b) Wewishtova1ueaderivativethatpaysoffeveryquarterthethree-month1IBORrateminusthethree-monthTreasuryhi11rate.Thepayoffoccurs90daysaftertheratesareobserved.(a) Aconvexityadjustmentisnecessaryfortheswaprate(b) Noconvexityortimingadjustmentsarenecessary.Prob1em30.3.SupposethatinExamp1e29.3ofSection29.2thepayoffoccursafteroneyear(i.e.,whentheinterestrateisobserved)ratherthanin15months.WhatdifferencedoesthismaketotheinputstoB1ack,smode1s?Therearetwodifferences.Thediscountingisdoneovera1.0-yearperiodinsteadofovera1.25-yearperiod.A1soaconvexityadjustmenttotheforwardrateisnecessary.Fromequation(30.2)theconvexityadjustmentis:1+0.25×0.070072×022×0-25×1=0.00005oraboutha1fabasispoint.Intheformu1aforthecap1etWesetFk=0.07005insteadof0.07.Thismeansthat4=-0.5642andd2=-0.7642.Withcontinuouscompoundingthe15-monthrateis6.5%andtheforwardratebetween12and15monthsis6.94%.The12monthrateistherefore6.39%Thecap1etpricebecomes0.25×10,000°69394x,00.07005V(-0.5642)-0.08N(-0.7642)=5.29or$5.29.Prob1em30.4.The1IBORZswapyie1dcurve(whichisusedfordiscounting)isf1atat10%perannumwithannua1compounding.Ca1cu1atetheva1ueofaninstrumentwhere,infiveyears,time,thetwo-yearswaprate(Wifhannua1compounding)isreceivedandafixedrateof10%ispaid.Bothareapp1iedtoanotiona1principa1of$100.Assumethatthevo1ati1ityoftheswaprateis20%perannum.Exp1ainwhytheva1ueoftheinstrumentisdifferentfromzero.TheconvexityadjustmentdiscussedinSection30.11eadstotheinstrumentbeingworthanamounts1ight1ydifferentfromzero.DefineG(y)astheva1ueasseeninfiveyearsofatwo-yearbondwithacouponof10%asafunctionofitsyie1d.G(y)=0.11.117+y(i+y)2G'(y)=0.1(i+y)22.2(i+»Gy)=0.26.6r(1+y)3(1+y)4Itfo11owsthatG'(0.1)=-1.7355andG*(0.1)=4.6582andtheconvexityadjustmentthatmustbemadeforthetwo-yearswap-rateis0.5×0.12×0.22×5×4,6582=0.002681.7355Wecanthereforeva1uetheinstrumentontheassumptionthattheswapratewi11be10.268%infiveyears.Theva1ueoftheinstrumentisor$0.167.Prob1em30.5.WhatdifferencedoesitmakeinProb1em30.4iftheswaprateisobservedinfiveyears,buttheexchangeofpaymentstakesp1acein(a)sixyears,and(b)sevenyears?Assumethatthevo1ati1itiesofa11forwardratesare20%.Assumea1sothattheforwardswapratefortheperiodbetweenyearsfiveandsevenhasacorre1ationof0.8withtheforwardinterestratebetweenyearsfiveandsixandacorre1ationof0.95withtheforwardinterestratebetweenyearsfiveandseven.exp-0.8×0.20×0.20×0.1×5=0.9856InthiscaseWehavetomakeatimingadjustmentaswe11asaconvexityadjustmenttotheforwardswaprate.For(a)equation(30.4)showsthatthetimingadjustmentinvo1vesmu1tip1yingtheswaprateby1+0.1sothatitbecomes10.268×0.9856=10.120.Theva1ueoftheinstrumentis0.120C-=0.1)oo1.16or$0,068.For(b)equation(30.4)showsthatthetimingadjustmentinvo1vesmu1tip1yingtheswapratebyexp-0.95×0.2×0.2×0.1×2×5=0.9660sothatitbecomes10.268×0.966=9.919.Theva1ueoftheinstrumentisnow一”“or-$0,042.Prob1em30.6.ThepriceofabondattimeT,measuredintermsofitsyie1d,isG(y).AssumegeometricBrownianmotionfortheforwardbondyie1d,y,inawor1dthatisforwardriskneutra1withrespecttoabondmaturingattimeT.Supposethatthegrowthrateoftheforwardbondyie1disaanditsvo1ati1ityv.(a) UseIto,s1emmatoca1cu1atetheprocessfarthefatardbondpriceintermsofa,v,y,andG(y).(b) Theforwardbondpriceshou1dfo11owamartinga1einthewor1dconsidered.Usethisfacttoca1cu1ateanexpressionfora.(c) Showthattheexpressionforais,toafirstapproximation,consistentwithequation(30.J).(a) Theprocessforyisdy=aydt-yydzTheforwardbondpriceisG(y).FromIt,s1emma,itsprocessisdG(y)=G,(y)ay+gGy)y1dt÷G,(y)yydz(b) SincetheexpectedgrowthrateofG(y)iszeroG(y)y+gG"(y)b3=0(c