CFA二级复习冲刺-衍生.docx

S-MBle-US >ro>Q SuI-BOIU OJOwluo。OUO-e>o17 SlU IUl-EE0。PBMOL1_OU-2-B>PUB M一 6m TO MraPUeuo=en-e>:SUlEls>>*3e:寸1SSWeulOOi:Sesosvos- SsTOMil扈andValuationofForwardCommitmentsPricingandvaluationofforwardcommitmentsPricingofforwardPricingoffuturesValuationofforwardPricingofswapValuationofswapPricingofForwardPricingofforwardIftheunderlyinggeneratesnoperiodiccashflow:F0(T)=S0×(l+r)IfF(7)>Sy,+r):Cash-and-CarryarbitrageAtinitiation:borrowmoneySgtR,fbuythespotasset,andselltheforwardatF0(T).Atexpiration:settletheshortpositiononforwardcontractbydeliveringtheasset.Profitatexpiration:Fq)-SQ+r)LPricingofForwardPricingofforward(Cent.)IfFg)<Sy.+r)T:ReverseCash-and-CarryarbitrageAtinitiation:borrowandsellthespotasset,investtheproceedS0atRpandbuytheforwardatF0(T).Atexpiration:payFJ)tosettlethelongpositiononforwardcontract,anddeliverthespotassettoclosetheshortpositiononspotasset.Profitatexpiration:S(jl+r)-F(J).PricingofForwardPricingofforward(Cont.)Iftheunderlyinggeneratesperiodiccashflow:F0(T)=(S0-+)(l÷r)Pricingofequityforwardwithdiscretedividend:F0(T)=(S0-PVD0)×(l÷r)or:Fd(T)=So×(l÷r)-FVDPricingofequityindexforward:continuousdividend(c)F0(T)=SoXRj-fic)Pricingoffixedincomeforward:discretecouponF0(T)=(S0-PVC0)X(l+r)or:F0(T)二S0X(l+r)-FVCPricingofForwardPricingofforward(Cont.)PricingofFRA:Theunbiasedestimateofforwardrate,calculatedfromthespotrates(forwardratemodel).Pricingofcurrencyforward:1+Rdc'J良兄(T)=SX-or:F(T)=SXe(RDCRFC)T十00、FCOL。(tw瞬三s悟<大情太遇打m一ETW瞬三selx眯#匿Ow×i三-il-oDX盗虫经二+d父朕持赞。w玄W国一(、<厘匾楹ss-aws。碘既海胆×三-i三s三-sssssS黑胆离US工袈藤屈一peM£0MU-d三nsPractice1JimTrent,CFAhasbeenaskedtopriceathreemonthforwardcontracton10,000sharesofGlobalIndustriesstock.Thestockiscurrentlytradingat$58andwillpayadividendof$2today.Iftheeffectiveannualrisk-freerateis6%,whatpriceshouldtheforwardcontracthave?Assumethestockpricewillchangevalueafterthedividendispaid.A)$55.85.B)$58.85.C)$56.82.Practice1Answer:CF(T)=(58-2)*(l+6%)1/4=$56.82OPricingofFuturesPricingoffixedincomefuturesQuotedfuturesprice=(SGPVC)X(l+r)-AlCF二S0×(l+r)-Al-FVCCFS亍QUotedPriCe+Al0Al=-×PMTT-tPMTPMT+FIIIIOsettlement1ndateValuationofForwardValuationofforwardValueofforward:thedifferencebetweenzzwiththeposition"and"withouttheposition".Generallydefinevalueasthevaluetothelongposition.Atinitiation(t=0):VO(T)=0.Duringitslife(t<T):Vt(T)=(St-t+t)-F0(T)(l÷r)-(-t)=Ft(T)-F0(T)(l+r)-(-t)Atexpiration(t=T):V(T)=S-F0(T)(ED×(T三UT(E)×(I)OLL'，gdIS)U(1>PBMOJEoUU-PX匚O-ro>一UA,27<x-=，T匕<×s"-L)A-PJBM0JxpwAI-nbo-ro>111(Eszl2>dIS)M(Iy>:PJPMOJo-ro>H二UOJ)peMoMoUo4e=e>三三0Sesras-SValuationofForwardValuationofforward(Cont.)Thevalueofcurrencyforward:Vt(T)=St×(l+RpcM-F0(T)X(1+RdcMForcontinuouslycompoundedrisk-freerate:Vt(T)=St×e-Rfc(T-1)-F0(T)×-rdc-1(ValuationofForwardValuationofforward(Cont.)ThevalueofFRAatexpiration(t=a):' DayS)I 360 JUsesimpleinterestformoneymarketinstrument.NPx(Underlyingrate-Forwardrate)x01Underlyingrate×aDaySII360IbValuationofForwardValuationofforward(Cont.)ThevalueofFRApriortoexpiration(t<a):Step1:calculatethenewFRArate(FR)t1Sblbt1Sa：t1FRtaStep2:calculatethevalueofFRAas:NPx(FR,一FR。)(DayS既atob)j吓DaySfrOmttObj%C360J0tabPractice1TheU.S.risk-freerateis2.96%,theJapaneseyenrisk-freerateis1.00%,andthespotexchangeratebetweentheUnitedStatesandJapanis$0.00757peryen.Bothratesarecontinuouslycompounded.Thepriceofa180-dayforwardcontractontheyenandthevalueoftheforwardposition90daysintothecontractwhenthespotrateis$0.00797areclosestto:Forward PriceA) $0.00764B) $0.00764C) $0.00750Value After 90 Days$0.00212$0.00037$0.00212Answer:BF(T)=$0.00757×e(0.0296-o.ooo)×(iso/365)=$0.00764oV(T)=0.00797-o.o.(90/365)-0.00764xe-0.0296.(90/355)=$0.0003706TgI9sO8lnm6(8oss(V:。1Issop-<一O-ra>IUnoHJ-Hog-"!ABP06IU8n。El£6IBp&JdPUe(mo-l)SBHPuo>UBGPWU-Uopuol£UoPcSBq<M-El-MraJProM0J(9×m)CO三EOTSeuo三sodMo-e001-*TdroSAeP。6ZB3=3dAnswer:B10,000,000×(5.2%-5.0%)×'JV7-136°/4935.8390/f90A15.2%×360!PricingofSwapPricingofswapPrinciple:thefixedrateinswap(FS,swaprate)shouldmakesthecontractvaluezeroatinitiation.Methodology:Areceive-floating,pay-fixedinterestrateswapisequivalenttobeinglongafloating-ratebondandshortafixed-ratebond;Ifbothbondsarepricedatpar,theinitialcashflowsarezeroandtheparpaymentsattheendoffseteachother.PVFiXedratebondPVFlOatingratebond-PdVdIUePricingandValuationofSwapPricingofswap(Cont.)S0-FSS1-FSSrv-FSSwapI1110tlt2t11Longfloating-、.0ratebond1S+Parn-1Shortfixed-FSratebond1-FS-FS-Par0tt2tnPricingofSwapPricingofswap(Cont.)Sothecouponrateoffixed-ratebondshouldequaltheswaprate.-unF=D1+D/3nwherein:Dn=discountfactororPVfactor,thepriceofzero-couponbondwithparvalueof$1andmaturityofnperiods.ValuationofSwapValuationofswapValuationofplainvanillainterestrateswapForfixed-ratereceiver(floating-ratepayer):V(T)=PVFiXedratebondor:,、，、/Floating-ratebond、Vt(T)=(F0-Ft)XNPXfD1+D2+÷Dn)Forfixed-ratepayer(floating-ratereceiver):V(T)=PVFloating-ratebond,、，.，Fixed-ratebond,or：Vt(T)=(Ft-F0)×NP×(D1+D2+.+Dn)Note:thevalueofafloatingratebondwillbeequaltotheparvalueateachsettlementdate.ValuationofSwapValuationofswap(Cont.)ValuationofequityswapForreceivefixedrate,payequityreturnswap:Vt(T)=PVFixed.ratebond-(S/S)×NPtt-St:thecurrentequityprice;St.:theequitypriceobservedatthelastresetdate.Forreceivefloatingrate,payequityreturnswap:Vt(T)=PVnoawratebond-(S/S)×NPtt-ValuationofSwapValuationofswap(Cont.)Forreceiveequity(1)return,payanotherequity(2)returnswap:Vt(T)=(S1SmJ×NP-(S2S2JXNPor:Vt(T)=(R1-R2)XNPR:equityreturnafterthelastresetdate.Practice1Supposewearepricingafive-yearLibor-basedinterestrateswapwithannualresets(30/360daycount).Theestimatedpresentvaluefactorsaregiveninthefollowingtable.Maturity(years)PresentValueFactors10.99009920.97787630.96513640.95152950.937467Practice1Thefixedrateoftheswapwillbeclosestto:A.1.0%.B.1.3%.C.1.6%.Answer:B1.3%1-0.937467F二0.990099+0.977876+0.965136+0.951529+0.937467muooPOEUoI-aNsao-ss.-se-sPOELIo-Inra>Uo0-B-IUoIl三MEoccouOUoABn-e>三三m三sIou's三三MBinomialOptionValuationModelOne-periodbinomialmodelDiagramillustration:S0Co = (TlUXC+%XC-)/(1+ Rf?C+=Max(0,S+-X)P+=Max(O,X-S+)C-=Max(O,S-X)P-=Max(0,X-S-)P0=(u×P÷+0×P-)(l+Rf)11=(1+RD)(U-D)risk-neutralprobabilityofup-move;Rty1-R(UriSk-neutralprobabilityofdown-move.Two-periodbinomialmodelforEuropeanoptionC-=Max(0,S-X)P-=Max(0,X-S-)BinomialOptionValuationModelTwo-periodbinomialmodelforEuropeanoption(Cont.)CalculationofC+andP+,C-andP-:C÷=(<C÷+RC+-)/(1+RPC-=(r3<C÷-+11RC-)(1+RFP÷=(h王P+RRP÷-)/(1+RP-=(r卞P+-+R存P-)/(1+RPCalculationofC©ndPC(<C+RC-)(1+RP5=(nP+n侪P-)(1+RBinomialOptionValuationModelTwo-periodbinomialmodelforAmericanoptionDeepin-the-moneyputoptionorcalloptionondividendpayingstockmaybenefitfromearlyexercise.WhenvaluetheAmericanoptionsthatmaybeexercisedearly,weneedtodetermineiftheoptionwillbeexercisedateachnode:Usethehigherbetweenexercisevalueandthecalculatedpriceateachnode.BinomialOptionValuationModelBinomialvaluationmodelforinterestrateoptionThevaluationofinterestrateoptionissimilartothatofstockoption,exceptthatthepayoffatmaturityisdifferent:Callpayoff=Max(0,underlyingrate-exerciserate)×NPPutpayoff=Ma×(07exerciserate-underlyingrate)XNPNote:wemustusebinomialinterestratetreemodelforvaluationofinterestrateoption.Practice1Astockispricedat40andtheperiodicrisk-freerateofinterestis8%.Thevalueofatwo-periodEuropeancalloptionwithastrikepriceof37onashareofstockusingabinomialmodelwithanupfactorof1.20andadownfactorof0.833isclosestto:A)$9.25.B)$3.57.C)$9.13.Practice1Answer:C=(1+0.08-0.833)/(1.20-0.833)=0.673;11u=1-0.673=0.327;dS+=57.60,C+=20.60;S+-=S-+二40;C+-二C-+二3;S-=27.76;C-=0;C+=20.6(0.673)+3(0.327)/1.08=13.745;C-=3(0.673)+O(0.327)/1.08=1.869;C=13.745(0.673)+1.869(0.327)/1.08=9.13.OC-=OT=OT=IT=2Black-Scholes-MertonModelAssumptionsoftheBSMmodelTheoptionsareEuropean-Style;TheunderlyingassetpricefollowsageometricBrownianmotion,andmovessmoothlyfromvaluetovalue;Therisk-freeinterestrateisknownandconstant;borrowingandlendingattherisk-freerateisallowed;Thevolatilityoftheunderlyingassetreturnisknownandconstant;Themarketisfrictionless.Black-Scholes-MertonModelBSMmodelFormulasfortheBSMmodelare:C0=S0N(d1)-Xe,N(d2)P=Xe-R；TN(-d)-SN(-d)0201C0=PVISoeR*TN(d)-XN(d2)也=PV1XN(-d2)-S0epfxN(-c)1InterpretationofBSMmodel:N(d)N(-distherisk-neutralprobabilitythatacall/putoptionwillbeexercisedatexpiration;Optionvalue=PVofexpectedpayoffatexpiration;Calloptioncanberegardedasleveragedstockinvestment,putoptioncanberegardedaslongbond+shortstock.Black-Scholes-MertonModelBSMmodel(Con.)BSMmodelwithcarryingbenefitsorcosts:SubstituteS百YXTfOrS.0BlackmodelforEuropeanoptionsonfutures:SubstituteF(T)×eRf.forS.oBlackmodelforinterestrateoption:-RrN一N-MC=e12FRA×N(d)-XN(d)NP×。1aI12JSwaption:optionwithunderlyingofinterestrateswap.Payerswaption:expectstheinterestratetoincrease;Receiverswaption:expectstheinterestratetodecrease.ionGreeksOptionGreeksDelta（八）;thesensitivityofoptionpriceagainsttheunderlyingprice(三);Fornon-dividendpayingstock:OptionGreeksDeltahedgeDelta-neutralportfolio:theportfoliovaluedoesnotchangewithvariationoftheunderlyingassetprice.Numberofoptionsneededtodeltahedge:IMstock.stockN-McaltDeltacaPutDeltaputLongstocks,shortcalloptions;Longstocks,longputoptions;Longcalloptions,longputoptions.OptionGreeksArbitrageopportunityinvolvingoptionsDelta()isalsothehedgeratio(h).Ifmarketprice>calculatedprice,selltheoptionandbuyhsharesofthestockforeachoptionwesold;Ifmarketprice<calculatedprice,buytheoptionandsellhsharesofthestockforeachoptionwebought.OptionGreeksOptionGreeks(Cont.)Gamma():thesensitivityoftheoption'sdeltaagainsttheunderlyingassetprice.Gammaforacallandputoptionwithidenticalfeaturesarethesame,andbotharepositive;Gammaislargestwhentheoptionisat-the-money;Iftheoptionisdeepin-orout-of-the-money,gammaapproacheszero.Gammarisk:stockpricesjumpratherthanmovecontinuouslyandsmoothly.OptionGreeksOptionGreeks(Cont.)Theta():thesensitivityoftheoptionpriceagainsttimepassage(t).Thetaisusuallynegativeforbothcallandputoption;Withexcerptiontodeepin-the-moneyputoption.Vega（八）:thesensitivityofoptionpriceagainstthevolatilityoftheunderlyingassetprice.VegaforcalloptionisequaltoVegaforputoptionwithidenticalfeatures,andbotharepositive;Vegaishighwhenoptionsareatornearthemoney.OptionGreeksRhoRho(p):thesensitivityofoptionpriceagainsttherisk-freerate.Rhoispositiveforcalloption,andnegativeforputoption.Practice1Inordertoformadynamichedgeusingstockandcallswithadeltaof0.2,aninvestorcouldbuy10,000sharesofstockand:A)write2,000calls.B)write50,000calls.C)buy50z000calls.eg1UelSU-IXQJULlIJ>0-s01sJoroooocJoa-一lSJJO=一M-8ooossJoUo-sodto-<.-0LllU-SE(JU-IsoroJoJ00sAq&d-see。U二=M-raouroCQL>><1.3"一3dIngRiskExposuresInterestrateswapFixed-ratereceiver:increaseduration;Fixed-ratepayer:decreaseduration.Interestratefutures1.onginterestratefutures:increaseduration;Shortinterestratefutures:decreaseduration.Stockindexfutures1.ongstockindexfutures:increasetheequityexposure;Shortstockindexfutures:decreasetheequityexposure.IngRiskExposuresEquityswapEquityswapcanbeusedtomodifyexposuretoequitymarkettemporarilywithoutactuallydisposingtheequityportfolio.CurrencyswapCurrencyswapareusuallyusedbycompaniestoreducetheirfundingcosts.Currencyforward/futuresCurrencyforward/futurescanbeusedtomanageforeignexchangeraterisk.etic EquivalenciesSyntheticassetwithoptions1.ongasset=longcall+shortput(S=C-P)Shortasset=shortcall+longput(-S=-C+P)Syntheticoptions1.ongcall=longasset+longput(C=S+P)1.ongput=Shortasset+longcall(P=-S+C)Syntheticassetwithforward/futures1.ongasset=longfutures+risk-freeasset(cash)DerivativeStrategiesCoveredcall(S-C)Investmentobjectives:IncomegenerationImprovingonthemarketTargetpricerealizationRiskofcoveredcall:Keepsthedownsideriskofthestockposition;Givesuptheupsidepotentialofthestockposition.DerivativeStrategiesProtectiveput(S+P)Investmentobjectives:Provideprotectionorinsuranceagainstapricedecline.Riskofprotectiveput:Theputpremiumwillreducetheportfolioreturn.DerivativeStrategiesCoveredcallvs.(longasset+shortforward)Fromtheaspectofdelta,acoveredcallpositionisequivalenttoapositionof(longastock+shortforwardfordetalcaunit).Bothofthemhavedeltaof(1-deltacJtProtectiveputvs.(longasset+shortforward)Fromtheaspectofdelta,aprotectiveputpositionisequivalenttoapositionof(longastock+shortforwardfordetalputunit).Bothofthemhavedeltaof(1+deltaptDerivativeStrategiesSpreadstrategyBullspread:longanoptionandshortanotherwithahigherexerciseprice;BullcallspreadBullputspreadBearspread:longanoptionandshortanotherwithalowerexerciseprice;BearcallspreadBearputspreadDerivativeStrategiesCollarStructure:longput+shortcall+underlyingassetInvestmentobjective:buyaprotectiveputandsellacalltooffsetthepremium.Zero-costcollar:thepremiumsforcallandputareequal.Straddle1.ongstraddle:Longcall+longput,withthesameexerciseprice,onthesameunderlyingasset.1.ongvolatility.DerivativeStrategiesChoiceofderivativestrategiesForexpectationofmarketdirection,typically:1.ongcall/put:strongbullish/bearishexpectation;1.ongcall+shortput:averagebullishexpectation;Shortcall+longput:averagebearishexpectation;Writingcall/put:weakbearish/bullishexpectation.Forexpectationofvolatility,typically:1.ongstraddle:highvolatilityexpectation;Shortstraddle:lowvolatilityexpectation.