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,andQuantoAdjustmentsPracticeQuestionsProblem30.1.ExplainhowyouwouldvalueaderivativethatpaysoffIooRinfiveyearswhereRistheone-yearinterestrate(annuallycompounded)observedinfouryears.Whatdifferencewoulditmakeifthepayoffwerein(a)4yearsand(b)6years?Thevalueofthederivativeis100/?45P(0,5)whereP(V)isthevalueofat-yearzero-couponbondtodayandR1histheforwardratefortheperiodbetweenr1andt2,expressedwithannualcompounding.Ifthepayoffismadeinfouryearsthevalueis100(5+c)P(0,4)wherecistheconvexityadjustmentgivenbyequation(30.2).Theformulafortheconvexityadjustmentis:C=强生0+¾,5)wherefflyisthevolatilityoftheforwardratebetweentimesZ1andt2.Theexpression100(45+c)istheexpectedpayoffinaworldthatisforwardriskneutralwithrespecttoazero-couponbondmaturingattimefouryears.Ifthepayoffismadeinsixyears,thevalueisfromequation(30.4)givenbyIoO(R4$+C)P(0,6)exp_1+叫,6_wherepisthecorrelationbetweenthe(4,5)and(4,6)forwardrates.AsanapproximationwecanassumethatP=1,45=46,and/?45=/?46.Approximatingtheexponentialfunctionwethengetthevalueofthederivativeas100(R4一C)尸(0,6).Problem30.2.Explainwhetheranyconvexityortimingadjustmentsarenecessarywhen(a) Wewishtovalueaspreadoptionthatpaysoffeveryquartertheexcess(ifany)ofthefive-yearswaprateoverthethree-monthLlBORrateappliedtoaprincipalof$100.Thepayoffoccurs90daysaftertheratesareobserved.(b) Wewishtovalueaderivativethatpaysoffeveryquarterthethree-monthLIBORrateminusthethree-monthTreasuryhillrate.Thepayoffoccurs90daysaftertheratesareobserved.(a) Aconvexityadjustmentisnecessaryfortheswaprate(b) Noconvexityortimingadjustmentsarenecessary.Problem30.3.SupposethatinExample29.3ofSection29.2thepayoffoccursafteroneyear(i.e.,whentheinterestrateisobserved)ratherthanin15months.WhatdifferencedoesthismaketotheinputstoBlack,smodels?Therearetwodifferences.Thediscountingisdoneovera1.0-yearperiodinsteadofovera1.25-yearperiod.Alsoaconvexityadjustmenttotheforwardrateisnecessary.Fromequation(30.2)theconvexityadjustmentis:1 +0.25 × 0.070072×022×0-25×1=0.00005orabouthalfabasispoint.IntheformulaforthecapletWesetFk=0.07005insteadof0.07.Thismeansthat4=-0.5642andd2=-0.7642.Withcontinuouscompoundingthe15-monthrateis6.5%andtheforwardratebetween12and15monthsis6.94%.The12monthrateistherefore6.39%Thecapletpricebecomes0.25×10,000°69394x,00.07005V(-0.5642)-0.08N(-0.7642)=5.29or$5.29.Problem30.4.TheLIBORZswapyieldcurve(whichisusedfordiscounting)isflatat10%perannumwithannualcompounding.Calculatethevalueofaninstrumentwhere,infiveyears,time,thetwo-yearswaprate(Wifhannualcompounding)isreceivedandafixedrateof10%ispaid.Bothareappliedtoanotionalprincipalof$100.Assumethatthevolatilityoftheswaprateis20%perannum.Explainwhythevalueoftheinstrumentisdifferentfromzero.TheconvexityadjustmentdiscussedinSection30.1leadstotheinstrumentbeingworthanamountslightlydifferentfromzero.DefineG(y)asthevalueasseeninfiveyearsofatwo-yearbondwithacouponof10%asafunctionofitsyield.G(y) =0.11.117 + y (i + y)2G'(y) = 0.1(i+y)22.2(i + »Gy) =0.26.6r(l+y)3(l+y)4ItfollowsthatG'(0.1)=-1.7355andG*(0.1)=4.6582andtheconvexityadjustmentthatmustbemadeforthetwo-yearswap-rateis0.5×0.12×0.22×5×4,6582=0.002681.7355Wecanthereforevaluetheinstrumentontheassumptionthattheswapratewillbe10.268%infiveyears.Thevalueoftheinstrumentisor$0.167.0.2681.15= 0.167Problem30.5.WhatdifferencedoesitmakeinProblem30.4iftheswaprateisobservedinfiveyears,buttheexchangeofpaymentstakesplacein(a)sixyears,and(b)sevenyears?Assumethatthevolatilitiesofallforwardratesare20%.Assumealsothattheforwardswapratefortheperiodbetweenyearsfiveandsevenhasacorrelationof0.8withtheforwardinterestratebetweenyearsfiveandsixandacorrelationof0.95withtheforwardinterestratebetweenyearsfiveandseven.exp -0.8× 0.20 ×0.20×0.1×5=0.9856InthiscaseWehavetomakeatimingadjustmentaswellasaconvexityadjustmenttotheforwardswaprate.For(a)equation(30.4)showsthatthetimingadjustmentinvolvesmultiplyingtheswaprateby1+0.1sothatitbecomes10.268×0.9856=10.120.Thevalueoftheinstrumentis0.120C-=0.l)oo1.16or$0,068.For(b)equation(30.4)showsthatthetimingadjustmentinvolvesmultiplyingtheswapratebyexp -0.95 × 0.2 × 0.2 ×0.1×2×5=0.9660sothatitbecomes10.268×0.966=9.919.Thevalueoftheinstrumentisnow一”“or-$0,042.Problem30.6.ThepriceofabondattimeT,measuredintermsofitsyield,isG(y).AssumegeometricBrownianmotionfortheforwardbondyield,y,inaworldthatisforwardriskneutralwithrespecttoabondmaturingattimeT.Supposethatthegrowthrateoftheforwardbondyieldisaanditsvolatilityv.(a) UseIto,slemmatocalculatetheprocessfarthefatardbondpriceintermsofa,v,y,andG(y).(b) Theforwardbondpriceshouldfollowamartingaleintheworldconsidered.Usethisfacttocalculateanexpressionfora.(c) Showthattheexpressionforais,toafirstapproximation,consistentwithequation(30.J).(a) Theprocessforyisdy=aydt-yydzTheforwardbondpriceisG(y).FromIt,slemma,itsprocessisdG(y)=G,(y)ay+gGy)y1dt÷G,(y)yydz(b) SincetheexpectedgrowthrateofG(y)iszeroG(y)y+gG"(y)b3=0ora-1 G(y) 2 ,2 G,(y) a> y(c) Assumingasanapproximationthatyalwaysequalsitsinitialvalueofy0,thisshowsthatthegrowthrateofyis1 G”(为)d，2 GXy0)5为Thevariableystartsaty0andendsasy.Theconvexityadjustmenttoy0whenwearecalculatingtheexpectedvalueofyinaworldthatisforwardriskneutralwithrespecttoazero-couponbondmaturingattimeTisapproximatelyy0timesthisor_1弛Wr2G'(%),%Thisisconsistentwithequation(30.1).Problem30.7.ThevariableSisaninvestmentassetprovidingincomeatrateqmeasuredincurrencyA.ItfollowstheprocessdS=jusSdt+sSdzintherealworld.Definingnewvariablesasnecessary,givetheprocessfallowedbyS,andthecorrespondingmarketpriceofrisk,in(a) Aworldthatisthetraditionalrisk-neutralworldforcurrencyA.(b) Aworldthatisthetraditionalrisk-neutralworldforcurrencyB.(c) Aworldthatisforwardriskneutralwithrespecttoazero-couponcurrencyAbondmaturingattimeT.(d) AworldthatisforwardriskneutralwithrespecttoazerocouponcurrencyBbondmaturingattimeT.(a) Inthetraditionalrisk-neutralworldtheprocessfollowedbySisdS=r-q)Sdt+sSdzwhereristheinstantaneousrisk-freerate.Themarketpriceofdz-riskiszero.(b) Inthetraditionalrisk-neutralworldforcurrencyBtheprocessisdS=(r-q+pQssQ»dt+sSdzwhereQistheexchangerate(unitsofAperunitofB),isthevolatilityofQandpQSisthecoefficientofcorrelationbetweenQandS.Themarketpriceofdz-riskisQQSbQ(c) Inaworldthatisforwardriskneutralwithrespecttoazero-couponbondincurrencyAmaturingattimeTdS=(rq+sp)Sdt+sSdzwherepisthebondpricevolatility.Themarketpriceofdz-riskisp(d) Inaworldthatisforwardriskneutralwithrespecttoazero-couponbondincurrencyBmaturingattimeTdS=(r-+sp+prssf)Sdt+sSdzwhereFistheforwardexchangerate,aFisthevolatilityofF(unitsofAperunitofB,andPFSisthecorrelationbetweenFandS.Themarketpriceofdz-riskisp+pFSaF.Problem30.8.AcalloptionprovidesapayoffattimeTofmax(Sr-C,0)yen,whereSisthedollarpriceofgoldattimeTandKisthestrikeprice.Assumingthatthestoragecostsofgoldarezeroanddefiningothervariablesasnecessary,calculatethevalueofthecontract.Define尸(f,7):Priceinyenattimerofabondpaying1yenattimeTE<.):ExpectationinworldthatisforwardriskneutralwithrespecttoP(t9T)F:DollarforwardpriceofgoldforacontractmaturingattimeTFo:ValueofFattimezeror:VolatilityofFG:Forwardexchangerate(dollarsperyen):VolatilityofGWeassumethatSislognormal.WecanworkinaworldthatisforwardriskneutralwithrespecttoP(t,T)togetthevalueofthecallasP(0,)N(4)-N(4)wherelnEr(S7)K+T2,=JnE(S)K-jT2d2=百Theexpectedgoldpriceinaworldthatisforwardrisk-neutralwithrespecttoazero-coupondollarbondmaturingattimeTisFti.Itfollowsfromequation(30.6)thatEr(Sr)=吊(1+paFaGT)Hencetheoptionprice,measuredinyen,isP(0J)F.(+proT)N(di)-KN(d2)where_ln(l+prcT)/K+T/2_nF0(S+paFaGT)/K-jT22=Problem30.9.ACanadianequityindexis400.TheCanadiandollariscurrentlyworth0.70U.S.dollars.Therisk-freeinterestratesinCanadaandtheU.S.areconstantat6%and4%,respectively.Thedividendyieldontheindexis3%.DefineQasthenumberofCanadiandollarsperU.SdollarandSasthevalueoftheindex.ThevolatilityofSis20%,thevolatilityofQis6%,andthecorrelationbetweenSandQis0.4.UseDerivaGemtodeterminethevalueofatwoyearAmericanstylecalloptionontheindexif(a) ItpaysoffinCanadiandollarstheamountbywhichtheindexexceeds400.(b) ItpaysOffinU.S.dollarstheamountbywhichtheindexexceeds400.(a) ThevalueoftheoptioncanbecalculatedbysettingSO=400,K=400,r0.06,q=0.03,=0.2,and7=2.With100timestepsthevalue(inCanadiandollars)is52.92.(b) ThegrowthrateoftheindexusingtheCDNnumeraireis0.06-0.03or3%.WhenweswitchtotheUSDnumeraireweincreasethegrowthrateoftheindexby0.4×0.2×0.06or0.48%peryearto3.48%.TheoptioncanthereforebecalculatedusingDerivaGemwith5o=4OO,K=400,/=0.04,q=0.040.0348=0.0052,=0.2,andT=2.With100timestepsDerivaGemgivesthevalueas57.51.FurtherQuestionsProblem30.10.ConsideraninstrumentthatwillpayoffSdollarsintwoyearswhereSisthevalueoftheNikkeiindex.Theindexiscurrently20,000.Theyen/dollarexchangerateis100(yenperdollar).Thecorrelationbetweentheexchangerateandtheindexis0.3andthedividendyieldontheindexis1%perannum.ThevolatilityoftheNikkeiindexis20%andthevolatilityoftheyen-dollarexchangerateis12%.Theinterestrates(assumedconstant)intheU.S.andJapanare4%and2%,respectively.(a) Whatisthevalueoftheinstrument?(b) SupposethattheexchangerateatsomepointduringthelifeoftheinstrumentisQandtheleveloftheindexisS.ShowthataU.S.investorcancreateaportfoliothatchangesinvaluebyapproximatelySdollarwhentheindexchangesinvaluebySyenbyinvestingSdollarsintheNikkeiandshortingSQyen.(c) Confirmthatthisiscorrectbysupposingthattheindexchangesfrom20,000to20f050andtheexchangeratechangesfrom100to99.7.(d) Howwouldyoudeltahedgetheinstrumentunderconsideration?(a) WerequiretheexpectedvalueoftheNikkeiindexinadollarrisk-neutralworld.Inayenrisk-neutralworldtheexpectedvalueoftheindexis2O,OOOo2-oo0x2=20,404.03.Inadollarrisk-neutralworldtheanalysisinSection30.3showsthatthisbecomes2O,4O4.O3e3x2x,2x2=20,699.97Thevalueoftheinstrumentistherefore20,699.97e-0042=19,108.48(b) AnamountSQyenisinvestedintheNikkei.ItsvalueinyenchangestoS4号Indollarsthisisworth+whereistheincreaseinQ.Whentermsofordertwoandhigherareignored,thedollarvaluebecomesS(l+SS-)ThegainontheNikkeipositionistherefore5-SAQ/QWhenSQyenareshortedthegainindollarsis(1AQQ+aqJThisequalsSQ/Qwhentermsofordertwoandhigherareignored.Thegainonthewholepositionistherefore5asrequired.(c) Inthiscasetheinvestorinvests$20,000intheNikkei.Theinvestorconvertsthefundstoyenandbuys100timestheindex.Theindexrisesto20,050sothattheinvestmentbecomesworth2,005,000yenor2,005,00099.7dollars.Theinvestorthereforegains$110.33.Theinvestoralsoshorts2,000,000yen.Thevalueoftheyenchangesfrom$0.0100to$0.01003.Theinvestorthereforeloses().(XXX)3×2,(XX),(XX)=60dollarsontheshortposition.Thenetgainis50.33dollars.Thisisclosetotherequiredgainof$50.(d) SupposethatthevalueoftheinstrumentisV.Whentheindexchangesby5yenthevalueoftheinstrumentchangesbyVaoSSdollars.WecancalculateVS.Part(b)ofthisquestionshowshowtomanufactureaninstrumentthatchangesbySdollars.Thisenablesustodelta-hedgeourexposuretotheindex.Problem30.11.SupposethattheLlBORyieldcurveisflatat8%(withcontinuouscompounding).Thepayofffromaderivativeoccursinfouryears.Itisequaltothefive-yearrateminusthetwo-yearrateatthistime,appliedtoaprincipalof$100withbothratesbeingcontinuouslycompounded.(Thepayoffcanbepositiveornegative.)Calculatethevalueofthederivative.Assumethatthevolatilityforallratesis25%.Whatdifferencedoesitmakeifthepayoffoccursinfiveyearsinsteadoffouryears?Assumeallratesareperfectlycorrelated.UseLIBORdiscountingTocalculatetheconvexityadjustmentforthefive-yearratedefinethepriceofafiveyearbond,asafunctionofitsyieldasG(y)=e5yG,(y)=-5e-5yG(y)=25e-5yTheconvexityadjustmentis().5×().082×0.252×4×5=0.(X)4Similarlyforthetwoyearratetheconvexityadjustmentis0.5×0.082×0.252×4×2=0.16Wecanthereforevaluethederivativebyassumingthatthefiveyearrateis8.4%andthetwo-yearrateis8.16%.ThevalueofthederivativeisO24/侬4=oi74Ifthepayoffoccursinfiveyearsratherthanfouryearsitisnecessarytomakeatimingadjustment.Fromequation(30.4)thisinvolvesmultiplyingtheforwardrateby-l×0.25×0.25×0.08×4×lCCCwexp=0.981651.1.08JThevalueofthederivativeis0.24×0.98165e°8x5=().158Problem30.12.Supposethatthepayofffromaderivativewilloccurintenyearsandwillequalthethree-yearU.S.dollarswaprateforasemiannual-payswapobservedatthattimeappliedtoacertainprincipal.AssumethattheLlBORZsxvapyieldcurve,whichisusedfordiscounting,isflatat8%(semiannuallycompounded)perannumindollarsand3%(semiannuallycompounded)inyen.Theforwardswapratevolatilityis18%,thevolatilityofthetenyeartiyenperdollar,forwardexchangerateis12%,andthecorrelationbetweenthisexchangerateandU.S.dollarinterestratesis0.25.(a) Whatisthevalueofthederivative矿theswaprateisappliedtoaprincipalof$100millionsothatthepayoffisindollars?(b) Whatisitsvalueofthederivativeiftheswaprateisappliedtoaprincipalof100millionyensothatthepayoffisinyen?(a)InthiscaseWemustmakeaconvexityadjustmenttotheforwardswaprate.Define64上oG(y)=zr+7T(i+y2y(I÷y2)6sothati÷2i,300S(l+y2),+,(l+y2)7G(y)=*3)U5。.W(l÷y22(l÷y2)8GW)8)=-262.11andG*(0.08)=853.29sothattheconvexityadjustmentis1QCQ29-×0.082×0.182×10x=0.003382262.11Theadjustedforwardswaprateis0.08+0.00338=0.08338andthevalueofthederivativeinmillionsofdollarsis0.08338×100z,。CU=3.o05l.O420(b)Whentheswaprateisappliedtoayenprincipalwemustmakeaquantoadjustmentinadditiontotheconvexityadjustment.FromSection30.3thisinvolvesmultiplyingtheforwardswaprateby¢-°25×0,2×0,8×,°=0.9474.(Notethatthecorrelationisthecorrelationbetweenthedollarperyenexchangerateandtheswaprate.Itistherefore-0.25ratherthan+0.25.)Thevalueofthederivativein millions of yen is= 5.8650.08338×0.9474×100iP5Problem30.13.Thepayofffromaderivativewilloccurin8years.Itwillequaltheaverageoftheone-yearrisk-freeinterestratesobservedattimes5,6t7,and8yearsappliedtoaprincipalof$1,000.Therisk-freeyieldcurveisflatat6%Wifhannualcompoundingandthevolatilitiesofallratesare16%.Assumeperfectcorrelationbetweenallrates.Whatisthevalueofthederivative?l×0.16×0.16× 0.06 ×5×2exp -= 0.9856Noadjustmentisnecessaryfortheforwardrateapplyingtotheperiodbetweenyearssevenandeight.Usingthis,wecandeducefromequation(30.4)thattheforwardrateapplyingtotheperiodbetweenyearsfiveandsixmustbemultipliedby1.06Similarlytheforwardrateapplyingtotheperiodbetweenyearsixandyearsevenmustbemultipliedbyl×0.16×0.16×0.06×6×llexp=0.99131.1.06JSimilarlytheforwardrateapplyingtotheperiodbetweenyeareightandninemustbemultipliedbylx().16x0.16x().()6x8xl11Cexp=1.01171.1.06JTheadjustedforwardaverageinterestrateistherefore0.25×(0.06×0.9856÷0.06×0.9913+0.06+0.06×1.0117=0.05983Thevalueofthederivativeis0.05983×1000×1.068=37.54