l2学习建议本门课程难度比较大计算公式很多一定要着重理解

BriefBriefStudySession1-10%-StudySession5%-StudySession5%-StudySession5-StudySession7-StudySession9-StudySession12-FixedStudySessionStudySession5%-5%- SS14:DerivativeInvestments:ValuationandReading40:PricingandValuationofForwardReading41:ValuationofContingentReading42:Derivatives学习建议ReviewofDerivativesinLevelReviewthebasicsofderivativeReviewthefundamentalofderivativeContractsenteredintoatonepointintimethatrequirebothpartiestoengageinatransactionatalaterpointintime(theexpiration)ontermsagreeduponatthestart.Forward,future,andContingentDerivativesinwhichtheoutcomeorpayoffisdependentontheoutcomeorpayoffofanunderlyingasset.Anover-the-counterderivativecontractinwhichtwopartiesagreethatoneparty,thebuyer,willpurchaseanunderlyingassetfromtheotherparty,theseller,atalaterdateatafixedprice(forwardprice)theyagreeonwhenthecontractissigned.Inadditiontothe(forward)price,thetwopartiesalsoagreeonseveralothermatters,suchastheidentityandthequantityoftheunderlying.Futurescontractsarespecializedforwardcontractsthathavebeenstandardizedandtradeonafutureexchange.Futurecontractshavespecificunderlyingassets,timestoexpiration,deliveryandsettlementconditions,andTheexchangeoffersafacilityintheformofaphysicallocationand/oranelectronicsystemaswellasliquidityprovidedbyauthorizedmarketmakers.Anover-the-counterderivativecontractinwhichtwopartiesagreetoexchangeaseriesofcashflowswherebyonepartypaysavariableseriesthatwillbedeterminedbyanunderlyingassetorrateandtheotherpartypayseither(1)avariableseriesdeterminedbyadifferentunderlyingassetorrateor(2)afixedseries.Aswapisaseriesof(off-market)PriceofforwardThefixedpriceorrateatwhichtheunderlyingwillbepurchasedatalaterdate.Generallymaynotchangeasthe(expected)priceoftheunderlyingassetchanges.Thedifferenceof“withtheposition”from“withouttheMayincreaseordecreaseasthe(expected)priceoftheunderlyingassetchanges.Aderivativecontractinwhichoneparty,thebuyer,paysasumofmoneytotheotherparty,thesellerorwriter,andreceivestherighttoeitherbuyorsellanunderlyingassetatafixedpriceeitheronaspecificexpirationdateoratanytimepriortotheexpirationdate.Anoptionisaright,butnotanDefaultinoptionsispossibleonlyfromtheshorttotheOptionOptionpremium(c0,p0):paymenttosellerfromCalloption:righttoPutoption:righttoExerciseprice/strikeprice(X):thefixedpriceatwhichtheunderlyingassetcanbepurchased.Americanoption:exercisableatorpriortoEuropeanoption:exercisableonlyatArbitrageisatypeoftransactionundertakenwhentwoassetsorportfoliosproduceidenticalresultsbutsellfordifferentprices.LawofoneAssetsthatproduceidenticalfuturecashflowsregardlessoffutureeventsshouldhavethesameprice;Traderwillexploitthearbitrageopportunityquickly(buylowandsellhigh),thenmakethepricesconverge.Creationofanassetorportfoliofromanotherasset,portfolio,and/orderivative.Anassetandahedgingpositionofderivativeontheassetcanbecombinedtoproduceapositionequivalenttoarisk-freeasset.Asset+Derivative=Risk-freeAsset-Risk-freeasset=-Derivative-Risk-freeasset=-A“-”signindicatesashortposition,orborrowingatNoarbitrageDeterminethepriceofaderivativebyassumingthattherearenoarbitrageopportunities(noarbitragepricing).Thederivativepricecanthenbeinferredfromthecharacteristicsoftheunderlyingandthederivative,andtherisk-freerate.Ready!PricingandValuationofForwardDescribehowforwardcontractsispricedandCalculateandinterprettheno-arbitragevalueofforwardcontract.PricingofIftheunderlyingassetgeneratesnoperiodiccashflow,theforwardpricecanbecalculatedasfollows:F0(T)=S0:spotr:riskfreeCarryarbitrageWhentheforwardcontractisoverpriced,F0(T)>Cash-and-CarryArbitrageisAtinitiation,borrowingmoneyS0atrisk-freerate,buying(long)thespotasset,andselling(short)theforwardatF0(T);Initialinvestmentatinitiation:Atexpiration,settlingtheshortpositiononforwardcontractbydeliveringtheasset.Profitatexpiration:F0(T)-CarryarbitrageWhenforwardcontractisunderpriced,F0(T)<ReverseCash-and-CarryArbitrageisAtinitiation,borrowingandselling(short)thespotasset,investingtheproceedS0atrisk-freerate,andbuying(long)theforwardatF0(T).Initialinvestmentatinitiation:Atexpiration,payingF0(T)tosettlethelongpositiononforwardcontract,anddeliveringthespotassettoclosetheshortpositiononspotasset.Profitatexpiration:S0(1+r)T-PricingofIftheunderlyingassetgeneratesperiodiccashflow,theforwardpricecanbecalculatedas:F0(T)=(S0-γ+γ:benefitofcarryingthespotasset,inpresentvalueθ:costofcarryingthespotasset,inpresentvalueγ-θ:netcostofInthefinancialworld,wegenerallydefinevalueasthevaluetothelongposition.Atinitiation,theforwardcontracthaszeroNeitherpartytoaforwardtransactionpaystoenterthecontractatinitiation.V0(T)=ValuationofforwardDuringitslife(t<T),thevalueofaforwardcontractis:Vt(T)=(St-γt+θt)-F0(T)(1+r)-(T-t)ϴt:presentvalueofthecostofholdinganasset(ttoγt:presentvalueofthebenefitofholdinganasset(ttoAtexpiration,thevalueofaforwardcontractis:VT(T)=ST-F0(T)AssumethatatTime0weenteredintoaone-yearforwardcontractwithpriceF0(T)=105.Ninemonthslater,atTimet=0.75,theobservedpriceofthestockisS0.75=110andtheinterestrateis5%.Calculatethevalueoftheexistingforwardcontractexpiringinthreemonths.Vt(T)=St-F0(T)(1+r)-(T-t)=110–105(1+5%)-0.25=Importance:Pricingandvaluationofforwardcontractonunderlyingwith/withoutcashflows.Exam是forwardpricingandvaluation的一般形式，对后面的学PricingandValuationofEquityandCurrencyDescribehowequityandcurrencyforwardcontractsispricedandvalued;Calculateandinterprettheno-arbitragevalueofequityandcurrencyforwardcontract.PricingandvaluationofequityIftheunderlyingisastockandhasdiscretedividends,thenforwardpricecanbecalculatedas:F0(T)=(S0- or:F0(T)=S0×(1+r)T-PVD:presentvalueofexpectedFVD:futurevalueofexpectedThevalueofequityforwardcanbecalculatedas:Vt(T)=(St-PVDt)-F0(T)(1+r)-(T-t)SupposeNestléstockistradingforCHF70andpaysaCHF2.20dividendinonemonth.Further,assumetheSwissone-monthrisk-freerateis1.0%,quotedonanannualcompoundingbasis.Assumethatthestockgoesex-dividendthesamedaythesinglestockforwardcontractexpires.Thus,thesinglestockforwardcontractexpiresinonemonth.Calculatetheone-monthforwardpriceforNestléstock.TTF0(T)=

1+r)

=70·(1+0.01)12-2.2=Supposeweboughtaone-yearforwardcontractat102andtherearenowthreemonthstoexpiration.Theunderlyingiscurrentlytradingfor110,andinterestratesare5%onanannualcompoundingbasis.Iftherearenoothercarrycashflows,calculatetheforwardvalueoftheexistingcontract. T) 1r- =110- 1+

=PricingandvaluationofequityindexfForequityindex,theforwardpriceisusuallycalculatedasifthedividendsarepaidcontinuously:fF0(T)=

·e(Rc-δcff

:continuouslycompoundedrisk-free:continuouslycompoundeddividendThevalueofequityindexforwardcanbecalculatedfV(T)=S·e-δc·(T-t)-F(T)·e-Rc·(T-f ThecontinuouslycompoundeddividendyieldontheEUROSTOXX50is3%,andthecurrentstockindexlevelis3,500.Thecontinuouslycompoundedannualinterestrateis0.15%.Calculatethethreemonthforwardprice.ffF0(T)=

·e(Rc-dc)·T=3500·e(0.15%-3%)·0.25=PricingandValuationofCurrencyPricingofcurrencyThepriceofcurrencyforwardcanbecalculatedbycoveredinterestrateparity(IRP):1+ F0(T)=S0·1+RDC FCF0(T)andS0arequotedbydirectquotation:RDC:interestrateofdomesticRFC:interestrateofforeignForcontinuouslycompoundedrisk-freeF(T)= -Rc·e ValuationofcurrencyThevalueofcurrencyforwardcanbecalculated V(T)= -(T-t)-F cForcontinuouslycompoundedrisk-freecV(T)=

-Rc·(T-t)-F(T)·e-

AcorporationsoldEuro(€)againstBritishpound(£)forwardataforwardrateof£0.8for€1atTime0.ThecurrentspotmarketatTimetissuchthat€1isworth£0.75,andtheannuallycompoundedrisk-freeratesare0.80%fortheBritishpoundand0.40%fortheEuro.AssumeatTimettherearethreemonthsuntiltheforwardcontractexpiration.CalculatetheforwardpriceFt(£/€,T)atTimetandthevalueofforeignexchangeforwardcontractatTimet.TheforwardpriceFt(£/€,T)atTime1+ 1+0.8%1+Ft(T)=St1+FC

=ThevalueofforeignexchangeforwardcontractatTime

Importance:PricingandvaluationofequityPricingandvaluationofcurrencyExamPricingandValuationofDescribehowinterestrateforwardcontractsispricedandvalued;Calculateandinterprettheno-arbitragevalueofinterestrateforwardcontract.ForwardrateagreementAFRAisanover-the-counter(OTC)forwardcontractinwhichtheunderlyingisaninterestrate(e.g.Libor).Longpositioncanbeviewedastheobligationtotakealoanatthecontractrate(i.e.,borrowatthefixedrate,floatingreceiver);gainswhenreferencerateincrease;Shortpositioncanbeviewedastheobligationtomakealoanatthecontractrate(i.e.,lendatthefixedrate,fixedreceiver);gainswhenreferenceratedecrease.ThenotationofThenotationofFRAistypically“a×ba:thenumberofmonthsuntilthecontractb:thenumberofmonthsuntiltheunderlyingloanisExample:3×93×9 3months

9monthsTheusesofLocktheinterestrateorhedgetheriskofborrowingorlendingatsomefuturedate.Onepartywillpaytheotherpartythedifference(basedonnotionalvalue)betweentheinterestratespecifiedintheFRAandthemarketinterestrateatcontractIfforwardrate<spotrate,thelongreceivesIfforwardrate>spotrate,theshortreceivesPricingofThe“forwardprice”inFRAisactuallyaforwardrate,itcanbecalculatedfromthespotrates.FRArateisjusttheunbiasedestimateoftheforwardRecalltheforwardratemodelin“FixedIncomeLevelButweusesimpleinterestformoneymarketNote:Liborratesareadd-onrateandquotedon30/360daybasisinannualPricingofFRAForwardratemodelsshowhowforwardratescanbeextrapolatedfromspotrates.a ·30·b=a

30·a

30·b-

·

·1+FR

1+

·30·01+Sa

1+FR·30·b-a)BasedonmarketquotesonCanadiandollar(C$)Libor,thesix-monthC$Liborandthenine-monthC$Liborarepresentlyat1.5%and1.75%,respectively.Assumea30/360-daycountconvention.Calculatethe6×9FRAfixedrate.=So,FRArate=ValuationofFRAatexpiration(t=Althoughtheinterestontheunderlyingloancomesattheendoftheloan,theFRAissettledattheexpirationofFRA.For“a×bFRA”,the“interestsaving”duetotheFRApositioncomesat“Timeb”,butissettledat“Timea”;Sothe“interestsaving”needtobediscountedto“Timetocalculatethevalueof

DaysVt

Example:1·4Specificationof1·4Term=30Notionalamount=$1Underlyingrate=90-dayForwardrate=Att=30days,90-dayLIBOR=8%,clarifythepayment(value)ofthisFRA.Solution:1·4Underlyingfloatingrate>fixedrate,solongpositionreceivespayment. rate:

ExpiryofPayment=

Interest(8%-7%)x90/360x=DiscountatLIBORfor90$2,500/[1+(8%xIn30days,aUKcompanyexpectstomakeabankdeposit£10Mforaperiodof90daysat90-dayLiborset30daysfromtoday.Thecompanyisconcernedaboutadecreaseininterestrates.Itsfinancialadvisersuggeststhatitnegotiatetoday,atTime0,a1×4FRA,aninstrumentthatexpiresin30daysandisbasedon90-dayLibor.Thecompanyentersintoa£10Mnotionalamount1×4receive-fixedFRAthatisadvancedset,advancedsettled.ExampleAfter30days,90-dayLiborinBritishpoundsis0.55%.IftheFRAwasinitiallypricedat0.60%,thepaymentreceivedbytheUKcompanytosettleitwillbeclosestto?BecausetheUKcompanyreceivesfixedintheFRA,itbenefitsfromadeclineinrates.[10M×(0.006–0.0055)×0.25]/[1+=ValuationofFRApriortoexpiration(t<Step1:calculatethenewFRArate1+S t=1+S ·1+ a- Step2:calculatethevalueofFRANP·(FRt-FR0)·Daysfromatob Vt

1+

Daysfromttob· · 0Initiation

FRA

UnderlyingWeenteredalong6×9FRAatarateof0.86%,withnotionalamountofC$10M.The6-monthspotC$Liborwas0.628%,and9-monthC$Liborwas0.712%.After90dayshavepassed,the3-monthC$Liboris1.25%andthe6-monthC$Liboris1.35%.Calculatethevalueofthereceive-floating6×9FRA.Step[1+(1.25%×90/360)]×[1+(newFRA=[1+So,newFRArate=StepVt=10M×(1.46%-=Importance:PricingandvaluationofExam常考点：FRAvaluePricingandValuationofFixed-IncomeDescribehowfixedincomeforwardcontractsispricedandvalued;Calculateandinterprettheno-arbitragevalueoffixedincomeforwardcontract.PricingandvaluationoffixedincomeSimilartoequityforward,theforwardpriceoffixedincomeforwardcanbecalculatedas:F0(T)=(S0- or:F0(T)=S0×(1+r)T-PVC:presentvalueofexpectedcouponFVC:futurevalueofexpectedcouponThevalueoffixedincomeforwardcanbecalculatedas:Vt(T)=(St-PVCt)-F0(T)×(1+r)-(T-t)Or:Vt(T)=[Ft(T)-F0(T)]×(1+r)-(T-Onemonthago,wepurchasedfiveeuro-bondforwardcontractswithtwomonthstoexpirationandacontractnotionalof€100,000eachatapriceof145(quotedasapercentageofpar).Theeuro-bondforwardcontractnowhasonemonthtoexpirationandthecurrentforwardpriceis148.Assumetherisk-freerateis0.1%,calculatethevalueoftheeuro-bondforwardposition.Vt(T)=[Ft(T)-F0(T)]×(1+r)-(T-=(148-145)×(1+0.1%)-1/12=So,thevalueoftheforwardpositionis:0.029997×€100,000×5=€14998.5PricingoffixedincomeIntermsoffixedincomefutures,thereareseveraluniqueThepricesofbondsareoftenquotedwithoutaccruedinterest(i.e.flatprice,cleanprice).Bondfuturescontractsoftenhavemorethanonebondthatcanbedeliveredbytheshort(deliveryoption),andconversionfactor(CF)isusedinanefforttomakealldeliverablebondsroughlyequalinprice.Pricepaid=FuturesPricingoffixedincomefuturesWhenmultiplebondscanbedeliveredforafuturescontractwithparticularmaturity,acheapest-to-deliver(CTD)bondtypicallyemergesafteradjustingfortheconversionfactor.PricingoffixedincomefuturesCalculationofaccruedinterestAI=T

T T PricingoffixedincomefuturesThequotedpriceoffixedincomefuturescanbecalculatedQuotedfuturesprice=[(S0-PVC)×(1+r)T-AIT]/CFor:Quotedfuturesprice=[S0×(1+r)T-AIT-FVC]/CFAIT:theaccruedinterestatmaturityofthefuturesS0:bondfullS0=Quotedprice+AI0:theaccruedinterestatinitiationofthefutureCF:theconversionSupposetheunderlyingofEuro-bondfuturesisaGermanbondthatisquotedat€108andhasaccruedinterestof€0.083.Theeuro-bondfuturescontractmaturesinonemonth.Atexpiration,theunderlyingbondwillhaveaccruedinterestof€0.25andhavenocouponpaymentsdueuntilthefuturescontractexpires.Assumetheconversionfactoroftheunderlyingbondis0.729535andthecurrentone-monthrisk-freerateis0.1%,calculatethepriceoftheEuro-bondfutures.AccordingtotheCF=0.729535;T=1/12;FVC=0;r=S0=€108+€0.083=AIT=Sothefuturesprice[108.083×(1+0.1%)1/12-0.25]/0.729535=PricingandValuationofForwardandAbriefTheforwardorfuturespriceissimplythevalueoftheunderlyingadjustedforanycarrycashflows;TheforwardvalueissimplythepresentvalueofthedifferenceinforwardpricesatanintermediatetimeintheThefuturesvalueiszeroaftermarkingtomarketbecauseprofitsandlossesaresettleddaily.Thetimevalueofmoneymakesitnotequivalenttoforwardvalue,butthedifferencestendtobesmall.Importance:PricingandvaluationoffixedincomePricingandvaluationoffixedincomeExam常考点：fixedincomeforwardpricevaluePricingandValuationofInterestRateDescribehowinterestrateswapispricedandCalculateandinterprettheno-arbitragevalueofinterestrateswap.TherearethreekindsofIfAloansmoneytoBforafixedrateofinterestandBloansthesameamounttoAforfloatingrateofinterest.CurrencyIftheloansareintwodifferentEquityIfoneofthereturnsstreamsisbasedonastockportfolioorindexreturn.PlainVanillainterestrateswapisaninterestrateswapinwhichonepartypaysafixedrate(fixed-ratepayer)andtheotherpaysafloatingrate(floating-ratepayer).Notionalamountisnotexchangedatthebeginningortheendoftheswap,becausebothloansareinsamecurrencyandamount;Onsettlementdates,interestpaymentsareFloatingratepaymentsaretypicallymadeinPricingofinterestratePrinciple:thefixedrateinswap(FS,swaprate)shouldmakesthecontractvaluezeroatinitiation. Areceive-floating,pay-fixedswapisequivalenttobeinglongafloating-ratebondandshortafixed-ratebond;Ifbothbondsarepricedatpar,theinitialcashflowsarezeroandtheparpaymentsattheendoffseteachother;So,thecouponrateoffixed-ratebondshouldequaltheswaprate.Exampleofreceive-floating,pay-fixedinterestrateS0- S1- Sn-1-12nSwap 12n=Longrate

Shortfixed-ratebond

- - -FS- PricingofPlainVanillainterestrateAtinitiation,thefloating-ratebondhasavalueequaltoitsparvalue,whatweshoulddoistofindafixed-ratebondwithavalueequaltothesameparvalueatinitiation.PricingofPlainVanillainterestrateswapAssumeFastheperiodiccouponpaymentofthen-periodfixed-ratebondwithparvalueof$1.1=

+F·D2+F·D3+...+F·Dn+Dn=discountfactororPVfactor,thepriceofzero-couponbondwithparvalueof$1andmaturityofnThen,weF 1-D1+D2+D3+...+MaturityPresentvalue12345Supposewearepricingafive-yearLibor-basedinterestMaturityPresentvalue12345F 1-D1+D2+D3+...+ 1-0.990099+0.977876+0.965136+0.951529+=ValuationofPlainVanillainterestrateThevalueofaswapisthedifferenceofvaluebetweenthefloating-ratebondandthefixed-ratebondatanytimeduringthelifeoftheswap.Vt(T)=PVFloating-ratebond–PVFixed-ratebondForfixed-ratereceiver(floating-rateVt(T)=PVFixed-ratebond–PVFloating-rateNote:thevalueofafloatingratebondwillbeequaltotheparvalueateachsettlementdate.Ateachsettlementdate,thecouponrateofafloatingratewillberesettothemarketrate,sothebondwillbepriceatpar.Twoyearsago,weenteredaannual-reset€100M7-yearreceive-fixedinterestrateswapwithfixedswaprateof2%.TheestimatedPVfactorsaregiveninthefollowingtable.Weknowthecurrentequilibriumfixedswaprateis1.3%.Calculatethevalueforthepartyreceivingthefixedrate.MaturityPV12345Becausethevalueofthefloatingratebondisequaltothenewfixedratebond,sothevalueoftheswapisthedifferenceofvaluebetweentheoldfixedratebondandthenewfixedratebond:(Fold-Fnew)·(D1+D2+D3+D4+D5=(2%-1.3%)·Importance:PricingandvaluationofinterestrateExamPricingandValuationofCurrencyandEquityDescribehowcurrencyandequityswapispricedandvalued;Calculateandinterprettheno-arbitragevalueofcurrencyandequityswap.CurrencyCurrencyswapinvolvestwodifferentTheprincipleamountofcurrencyswapisexchangedatthebeginningaccordingtotheexchangerate,andreturnedatOnsettlementdates,interestpaymentsarenotFloatingratepaymentsaretypicallymadeinCurrencyswapTherearefourpossiblestructuresforcurrencyReceivefixedandpayReceivefloatingandpayReceivefixedandpayReceivefloatingandpayPricingandvaluationofcurrencyThepricingandvaluationofcurrencyswaparesimilartothatofinterestrateswap:ThefixedrateinacurrencyswapissimplytheswapratecalculatedfromthespotratesofthecorrespondingF 1-D1+D2+D3+...+Thevalueofaswapisthedifferenceofvaluebetweenthetwoequivalentbonds.EquityThereare3typesofequityswaps,andthereareno“pricing”problemforthelasttwotype.EquityreturnforfixedEquityreturnforfloatingEquityreturnforanotherequityTheequitylegofanequityswapcanbeanindividualstock,apublishedstockindex,oracustomportfolio;andtheequitylegcashflowcanbewithorwithoutdividends.EquityswapNotionalamountisnotexchangedatthebeginningortheendoftheswap.Onsettlementdates,paymentsarePricingofequityThepricingofequityswapissimilartothatofinterestrateswap,wecanusethesameformulatocalculatethefixedF 1-D1+D2+D3+...+ValuationofequityValuationofequityswapisalsosimilartothatofinterestrateswap,andequalsthedifferenceofvaluebetweenthetwolegsoftheequityswap.Forreceivefixedrate,payequityreturnswap:Vt(T)=PVFixed-ratebond–(St/St-)×NPSt:thecurrentequitySt-:theequitypriceobservedatthelastresetValuationofequityswapForreceivefloatingrate,payequityreturnswap:Vt(T)=PVFloating-ratebond–(St/St-)×NPForreceiveequity(1)return,payanotherequity(2)Vt(T)=(S1,t/S1,t-)×NP–(S2,t/S2,t-)×NPor:Vt(T)=(R1–R2)×NPR:equityreturnafterthelastresetAninvestorpaysthestockAreturnandreceivesstockBreturnina$1millionquarterly-payswap.Afteronemonth,stockAisup2%andstockBisdown1%.Calculatethevalueoftheswaptotheinvestor.Vt(T)=(-2%–1%)×$1,000,000=Importance:PricingandvaluationofcurrencyPricingandvaluationofequityExamBinomialOptionValuationModelDescribeandinterpretthebinomialoptionvaluationmodel;Identifyanarbitrageopportunityinvolvingoptionsanddescribetherelatedarbitrage.BinomialoptionvaluationBinomialmodelisbasedontheideathat,overthenextperiod,somevaluewillchangetooneoftwopossibleToconstructabinomialmodel,weneedtoknowthebeginningassetvalue(S0),thesizeofthetwopossiblechanges(U,D),andtheprobabilitiesofeachofthesechangesoccurring(πU,πD).One-periodbinomialmodelUProb.=UProb.=

S+=S-=

C+=Max(0,S+−X)P+=Max(0,X−C−=Max(0,S−−X)P−=Max(0,X− πU=(1+Rf-D)/(U-D),risk-neutralprobabilityofanup-πD=1-πU,risk-neutralprobabilityofandown-One-periodbinomialmodelWithone-periodbinomialmodel,thevalueofanoptiononstockcanbecalculatedas:Step1:Calculatethepayoffoftheoptionatmaturityinboththeup-move(C+,P+)anddown-movestates(C-,P-);Step2:Calculatetheexpectedvalueoftheoptioninoneperiodastheprobability-weightedaverageofthepayoffsineachstate;Steps3:Discountthisexpectedvaluebacktotodayattherisk-freerate.One-periodbinomialmodelValueofancallC0=(πU×C++πD×C−)/(1+RfValueofanputP0=(πU×P++πD×P−)/(1+RfAnon-dividend-payingstockiscurrentlytradingat€100.Acalloptiononthestockhasoneyeartomatureandexercisepriceof€100.Assumetherisk-freeinterestrateis5.15%andasingle-periodbinomialoptionvaluationmodelwhereU=1.35andD=0.74,calculatethecalloptionvalue.S+=US0=1.35×100=135;S–=DS0=0.74×100=C+=Max(0,S+–X)=35;C–=Max(0,S-–X)=πU=(1+Rf-D)/(U-=(1+5.15%-0.74)/(1.35-0.74)=πD=1–πU=ValueofthecallC0=(πU×C++πD×C−)/(1+Rf=(35×0.511+0.489×0)/1.0515=Iftheoptionmarketpriceisdifferentfromthecalculatedpricefromthebinomialvaluationmodel,anarbitrageopportunityexist:Ifmarketprice>calculatedprice,selltheoptionandwe -h:hedgeratio,ordelta,h=DS=S+-Ifmarketprice<calculatedprice,buytheoptionandhsharesofthestockforeachoptionweAnon-dividend-payingstockiscurrentlytradingat€100.Acalloptiononthestockhasoneyeartomatureandexercisepriceof€100.Assumetherisk-freeinterestrateis5.15%andasingle-periodbinomialoptionvaluationmodelwhereu=1.35andd=0.74.CalculatethehedgeDescribethearbitrageopportunityifthemarketvalueofthecalloptionis€12.S+=US0=135;S-=DS0=C+=Max(0,S+–X)=35;C-=Max(0,S-–X)=Hedgeratio:h=(C+-C−)/(S+-S-)=35/61=Becausethemarketpriceofoption(€12)islowerthanthecalculatedarbitrage-freeprice(€17.01,calculatedinpreviousexample),anarbitrageprofitcanbeearnedbybuythecalloptionandsell0.574shareofthestockforeachoptionweImportance:BinomialoptionvaluationmodelanditsArbitrageopportunitiesinvolvingHedgeExam常考点：one-periodoptionvalue的计算,hedgeratio的计BinomialOptionValuationModelCalculatetheno-arbitragevaluesofEuropeanandAmericanoptionsusingatwo-periodbinomialmodel;Calculateandinterpretthevalueofaninterestrateoptionusingatwo-periodbinomialmodel.Two-periodbinomialmodelforEuropeanUsingthetwo-periodbinomialmodeltovalueanoptionissimilar,butwithmoresteps:Step1:calculatethethreepossiblevaluesofstockatS++=UUS0;S+−=S−+=UDS0=S0;S−−=Step2:calculatethepayoffoftheoptionatC++=Max(0,S++−X);P++=Max(0,X−S++C+−=C−+=Max(0,S0−X);P+−=P−+=Max(0,X−C−−=Max(0,S−−−X);P−−=Max(0,X−S−−Two-periodbinomialmodelforEuropeanoptionStep3:calculatetheoptionvalueatT=1(C+orP+,C-orP-)bydiscountingtheexpectedpayoffatT=2backoneperiodatrisk-freerate;C+=(πU×C+++πD×C+−)/(1+RfC−=(πU×C+−+πD×C−−)/(1+RfP+=(πU×P+++πD×P+−)/(1+RfP−=(πU×P+−+πD×P−−)/(1+RfTwo-periodbinomialmodelforEuropeanoptionSteps4:calculatetheoptionvalueatT=0(C0orP0)bydiscountingtheexpectedoptionvalueatT=1backoneperiodatrisk-freerate.C=(πU×C++πD×C−)/(1+RfP=(πU×P++πD×P−)/(1+RfTwo-periodbinomialmodelforEuropeanoptionS++=C++=Max(0,S++−X

P++=Max(0,X−S++C+,

S+-=S-+=

=

S-=

C+−=C−+=Max(0,S0−P+−=P−+=Max(0,X−C−,

S--=C−-=Max(0,S−-−XP−-=Max(0,X−S−- Youobservea€50priceforanon-dividend-payingstock.AnEuropean-stylecalloptionhastwoyearstomatureandantheexercisepriceof€50.Assumetherisk-freerateis5%,U=1.356andD=CalculatethecurrentcalloptionCalculatethecurrentputoptionStep1:S++=91.94;S+−=S−+=50.44;S−−=StepC++=41.95;P++=C+−=C−+=0.44;P+−=P−+=C−−=0;P−−=AnswerStepπU=[(1+0.05)-0.744]/(1.356-0.744)=πD=1-πU=C+=(0.5×41.94+0.5×0.44)/1.05=C−=(0.5×0.44+0.5×0)/1.05=P+=(0.5×0+0.5×0)/1.05=P−=(0.5×0+0.5×22.32)/1.05=StepC0=(0.5×20.18+0.5×0.22)/1.05=P0=(0.5×0+0.5×10.63)/1.05=S++=S++=C++=S0=P+=P++=S+-=S-+=C+−=C−+=C0=P0=S-=P+−=P−+=C−=P−=S--=C−-=P−-=Recalltheput-callparity:S0+P0=C0+PV(X)TheputoptionvaluecanbecomputedsimplybyputcallP0=C0+PV(X)–=9.71+(50/1.052)–=Two-periodbinomialmodelforAmericanNondividend-payingAmericancalloptionsonstockwillnotbeexercisedearlybecausethevalueofacalloptionwillbegreaterthanitsexercisevalue.WorthmorealivethanTwo-periodbinomialmodelforAmericanoptionDeepin-the-moneyputoptionorcalloptionondividend-payingstockmaybenefitfromearlyexercise:Fordeepin-the-moneyputoption,theexercisevaluecanbeinvestedattherisk-freerate,andearninterestthatexceedthetimevalueoftheput;Forcalloptionondividend-payingstock,thestockpricefallsatex-dividenddate,anditmaybevaluabletoexercisetheoptionbeforethepricefalling.Two-periodbinomialmodelforAmericanoptionWhenvaluetheAmericanoptionsthatmaybeexercisedearly,weneedtodetermineiftheoptionwillbeexercisedateachnode:Iftheexercisevalueisgreaterthanthecalculatedarbitrage-freevalue,earlyexerciseisvaluable;Usethehigherbetweenexercisevalueandthecalculatedpriceateachnode.Anon-dividend-payingstockiscurrentlytradingat$72,aputoptiononthisstockhasaexercisepriceof$75andamaturityof2years.Supposetheinterestrateis3%,U=1.356andD=0.541,πU=0.6andπD=CalculatetheputoptionvalueifitisEuropean-CalculatetheputoptionvalueifitisAmerican-IfitisEuropeanP+=

P++P+−=P−+=P0

P−=

P−-= IfitisAmericanP+=P+=

P++P+−=P−+=P0

P−=P−= P−-= BinomialinterestrateAinterestratemodelthatassumesinterestratesatanypointoftime(node)haveanequalprobabilityoftakingoneoftwopossiblevaluesinthenextperiod,anupperpath(U)andalowerpath(L).Theinterestratesateachnodeareone-periodforwardratescorrespondingtothenodalperiod.E.g.:interestratei2,LUatnode2istheratethatwilloccurifinitialinteresti0atnode0followsthelowerpathtonode1,andthenfollowstheupperpathtonode2.iUi0L

Node Node NodeBinomialvaluationmodelforinterestrateThevaluationofinterestrateoptionissimilartothatofstockoption,exceptthatthepayoffatmaturityisdifferent:Callpayoff=Max(0,underlyingrate–exercisePutpayoff=Max(0,exerciserate–underlyingAEuropeancalloptionontheoneyearinterestrate(theunderlying)hasanexerciserateof3.25%withmaturityoftwoyearsand