资源描述
Chapter29Quanto,Timing,andConvexityAdjustments,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,1,ForwardYieldsandForwardPrices,WedefinetheforwardyieldonabondastheyieldcalculatedfromtheforwardbondpriceThereisanon-linearrelationbetweenbondyieldsandbondpricesItfollowsthatwhentheforwardbondpriceequalstheexpectedfuturebondprice,theforwardyielddoesnotnecessarilyequaltheexpectedfutureyield,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,2,RelationshipBetweenBondYieldsandPrices(Figure29.1,page669),Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,3,ConvexityAdjustmentforBondYields(Eqn29.1,p.670),SupposeaderivativeprovidesapayoffattimeTdependentonabondyield,yTobservedattimeT.Define:G(yT):priceofthebondasafunctionofitsyieldy0:forwardbondyieldattimezerosy:forwardyieldvolatilityTheexpectedbondpriceinaworldthatisFRNwrtP(0,T)istheforwardbondpriceTheexpectedbondyieldinaworldthatisFRNwrtP(0,T)is,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,4,ConvexityAdjustmentforSwapRate,TheexpectedvalueoftheswapratefortheperiodTtoT+tinaworldthatisFRNwrtP(0,T)is(approximately)whereG(y)definestherelationshipbetweenpriceandyieldforabondlastingbetweenTandT+tthatpaysacouponequaltotheforwardswaprate,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,5,Example29.1(page671),Aninstrumentprovidesapayoffin3yearsequaltothe1-yearzero-couponratemultipliedby$1000Volatilityis20%Yieldcurveisflatat10%(withannualcompounding)Theconvexityadjustmentis10.9bpssothatthevalueoftheinstrumentis101.09/1.13=75.95,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,6,Example29.2(Page671-672),Aninstrumentprovidesapayoffin3years=tothe3-yearswapratemultipliedby$100PaymentsaremadeannuallyontheswapVolatilityis22%Yieldcurveisflatat12%(withannualcompounding)Theconvexityadjustmentis36bpssothatthevalueoftheinstrumentis12.36/1.123=8.80,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,7,TimingAdjustments(Equation29.4,page673),Theexpectedvalueofavariable,V,inaworldthatisFRNwrtP(0,T*)istheexpectedvalueofthevariableinaworldthatisFRNwrtP(0,T)multipliedbywhereRistheforwardinterestratebetweenTandT*expressedwithacompoundingfrequencyofm,sRisthevolatilityofR,R0isthevalueofRtoday,sVisthevolatilityofF,andristhecorrelationbetweenRandV,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,8,Example29.3(page673),Aderivativeprovidesapayoff6yearsequaltothevalueofastockindexin5years.Theinterestrateis8%withannualcompounding1200isthe5-yearforwardvalueofthestockindexThisistheexpectedvalueinaworldthatisFRNwrtP(0,5)TogetthevalueinaworldthatisFRNwrtP(0,6)wemultiplyby1.00535Thevalueofthederivativeis12001.00535/(1.086)or760.26,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,9,Quantos(Section29.3,page674),QuantosarederivativeswherethepayoffisdefinedusingvariablesmeasuredinonecurrencyandpaidinanothercurrencyExample:contractprovidingapayoffofSTKdollars($)whereSistheNikkeistockindex(ayennumber),Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,10,DiffSwap,DiffswapsareatypeofquantoAfloatingrateisobservedinonecurrencyandappliedtoaprincipalinanothercurrency,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,11,QuantoAdjustment(page675),Theexpectedvalueofavariable,V,inaworldthatisFRNwrtPX(0,T)isitsexpectedvalueinaworldthatisFRNwrtPY(0,T)multipliedbyexp(rVWsVsWT)Wistheforwardexchangerate(unitsofYperunitofX)andrVWisthecorrelationbetweenVandW.,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,12,Example29.4(page675),CurrentvalueofNikkeiindexis15,000Thisgivesone-yearforwardas15,150.75SupposethevolatilityoftheNikkeiis20%,thevolatilityofthedollar-yenexchangerateis12%andthecorrelationbetweenthetwois0.3Theone-yearforwardvalueoftheNikkeiforacontractsettledindollarsis15,150.75e0.30.20.121or15,260.23,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,13,Quantoscontinued,WhenwemovefromthetraditionalriskneutralworldincurrencyYtothetradionalriskneutralworldincurrencyX,thegrowthrateofavariableVincreasesbyrsVsSwheresVisthevolatilityofV,sSisthevolatilityoftheexchangerate(unitsofYperunitofX)andristhecorrelationbetweenthetworsVsS,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,14,SiegelsParadox,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,15,WhenisaConvexity,Timing,orQuantoAdjustmentNecessary,AconvexityortimingadjustmentisnecessarywheninterestratesareusedinanonstandardwayforthepurposesofdefiningapayoffNoadjustmentisnecessaryforavanillaswap,acap,oraswapoption,Options,Futures,andOtherDerivatives,8thEdition,CopyrightJohnC.Hull2012,16,
展开阅读全文