內(nèi)容案例上學(xué)期beamlecture risk

BEAM047FUNDAMENTALSOFFINANCIALTom Lecture9:RiskManagement:AnIntroductiontoTimevalueandintrinsicvalueThefactorsaffectingoptionprices價ThebinomialoptionpricingTheBlack-ScholesoptionpricingmodelFuturesRiskBrigham,E.,andHouston,J.,(2010)FundamentalsofFinancial(CustomEdition).SouthWesternCengageLearning.Chapter ShouldfirmshedgetheriskoftheircashInprinciple,welldiversifiedinvestorscareonlyaboutmarketrisk,sinceitisonlythisthatcontributestotheriskoftheiroveralldiversifiedportfolio,andsocorporatehedgingshouldnotaffectthevalueoftheirinvestmentInpractice,however,therearevariousreasonswhyafirmmaychoosetohedgeidiosyncraticrisk非系統(tǒng)風(fēng)險,includingreducingthevolatility波動性of e,reducingtheprobabilityoffinancialdistress,increaseddebtcapacity借債能力,bettermanagerialcompensation補償systems,andcomparative比較advantagesinThemostimportanttoolusedbycorporationsforhedgingriskisthederivativesecurity(orjustderivative),includingforward遠(yuǎn)期合同contract,futurescontract合同,option,swap互惠,etcItisasecuritywhosevaluedependsonthevalueofoneormoreunderlyingassetsincludingfinancialsecurities(e.g.stocks,interestrates),agriculturalproducts(e.g.wheat,cotton),preciousmetals貴金屬(e.g.gold,silver),rawmaterials原材料(e.g.steel,stic),theweather(e.g.temperature,snowfall),macroeconomicindicators(e.g.inflation),volatilityandotherderivatives(e.g.futuresonoptions,optionsonswaps)Inadditiontohedging,derivativescanalsobeusedforspeculation投機,sometimeswithdisastrousconsequences(forexample,LTCM,BaringsBank)LTCM——Long-TermCapital Long-TermCapitalManagement(LTCM)wasthelargesthedgefund對沖基金($126billioninassets)thatnearlycollapsed價格inlateTheprincipalshareholderswereNobelprize-winningeconomistsMyronScholesandRobertMerton.LTCM'sinvestmentstrategieswerebaseduponhedgingagainstarangeofvolatilityinforeigncurrenciesandbonds.WhenRussiadeclared聲itwasdefaultingonitsbonds,LTCM'sriskytradesbroughtitclosetobankruptcy.TheFederalReservehadtotakestepstobailitBaringsBank(1762- ‘OnJanuary171995theHanshinearthquakestruckKobe,Japan,killing6,425peopleandcausinginsuredlossesof$2.716bn. Asaresult,BaringstraderNickLeesonfacedseriousproblems. Inhisroleasawriterofso-called“straddles”...ontheNikkei225index,hehadbeenspeculatingthattheindexwouldnotmovesignificantlyoutofitstradingrange... TheearthquakeresultedintheNikkei255shedding11%andtheconsequencesforLeesonarewellknown: on26February1995,Barings,theUK’soldestmerchantbankinggroup,wascedinMasteringRisk,20June,2000,Part9WarrenBuffetonExcerptsfromtheBerkshireHathawayannualreportfor‘Iviewderivativesastimebombs,bothforthepartiesthatdealinthemandtheeconomicsystem.’‘Inmyview,derivativesarefinancialweaponsofmassdestruction,carryingdangersthat,whilenowlatent,arepotentiallylethal.’ A Derivativescanbetradedover-the-counter場易、柜臺賣買(i.e.directlynegotiatedbetweentwoindividuals)oronanexchangeThemostimportantexchangesforderivativestradingareIntercontinentalExchange(whichincludesLIFFE,TheLondonInternationalFinancialFuturesandOptionsExchange),TheChicagoBoardofTrade(CBOT)DerivativesmarketshavegrownmorerapidlythanothermajormarketsinrecentOptionsAnoptiononastockisasecuritythatgivestheholdertherighttobuyorselloneshareofthestockonorbeforeaparticulardateforapredeterminedprice預(yù)先約定價格.Acalloption看漲givestheholdertherighttobuythestockwhileaputoption看跌thatgivestheholdertherighttoselltheTheexerciseprice(orstrikeprice執(zhí)行價isthepriceatwhichtheholdercanbuyorselltheunderlyingstockTheexpirationdateisthedateonorbeforewhichtheholdercanbuyorselltheunderlyingstock;conventionaloptionsaregenerallywrittenforafewmonths;Long-termEquityiciPationSecurities(LEAPS)havematuritiesofuptothreeyearsAEuropeanoption歐式canonlybeexercisedon以控制theexpiration期滿whileanAmericanoptioncanbeexercisedatanytimebeforematurity到期Acoveredoption有抵iswhereaninvestorwhosellsacalloptionownstheunderlyingstock,whileanakedoption無擔(dān)保isonewheretheinvestordoesn’townthestockAnout-of-the-money外在價值optionisonewhichwouldresultinalossifexercisedimmedia y,whileanin-the-money內(nèi)在價值optionisonewhichwouldresultinaprofitTimevalueandintrinsic AcalloptiongivestherighttobuyanassetwhosevalueiscurrentlyS,fortheexercisepriceXIftheoptionwereexercisednowitwouldgenerateanimmediatecashflowofS–X thentheoptionwouldnotbeexercisedandsoitsvaluewouldbezeroTheintrinsicvalueofacalloptionisthereforeequalThereissomechancethatthevalueoftheassetwillrisebeforetheexpiryoftheoption;thereisalsosomechancethatitwillfall,butthevalueoftheoptionisboundedfrombelowThereforethevalueoftheoptionwillgenerallyexceeditsintrinsicvalue;thedifferenceisthetimevalueoftheoptionOptionpremium溢價=intrinsicvalue+timeAtexpiry期滿timevalueiszeroandthevalueoftheoptionisexactlyequaltoitsintrinsicvalueSimilarly,theintrinsicvalueofaputoptionisequaltomax(XProfits/LossesonBasicXX

Buya

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Buya

Sella

SellaXXXXAstraddle:buyacallandbuyaThefactorsaffectingoption Shareprice:thehighertheshareprice,thehigherthepriceofacallandthelowerthepriceofaputoptionExerciseprice:thehighertheexerciseprice,thelowerthepriceofacalloptionandthehigherthepriceofaputoptionTimetomaturity到期thelongerthetimetomaturity,thehigherthepriceofbothcallandputoptionsThefactorsaffectingoption Volatilityofthesharepricethehigherthevolatilityoftheshareprice,thehigherthepriceofbothcallandputoptions rateofinterest:thehighertherisk rateofinterest,thehigherthepriceofcalloptionsandthelowerthepriceofputoptionsThedividendspaidbythesharethehigherthedividendspaidbytheshare,thelowerthepriceofacalloptionandthehigherthepriceofaputoptionThefactorsaffectingoption AsummaryofthefactorsaffectingoptionpricesisasTheobjectiveofoptionpricingistoestablishthefunctionalformofThetwomostimportantoptionpricingmodelsarethebinomialmodelandtheBlack-ScholesmodelThebinomialoptionpricing Thebinomialoption-pricingmodelisprobablythemostwidelyusedoption-pricingmodelThebinomialmodelassumesthatovershortperiodsoftime,thesharepricemoveseitherupordownbyfixedproportions固定比例Theresulting‘tree’forthesharepriceimpliesaunique‘tree’fortheoptionpriceThenatureoftheoptioncontractdeterminesthevalueoftheoptionattheterminalnodesofthetreeThebinomialoptionpricing No-arbitrage無argumentscanthenbeusedtofindtheequilibriumpriceoftheoptionatearliernodesofthetreeByworkingbackwards逆向作業(yè)wecanthereforededucetheequilibriumpriceoftheoptionatthecurrentdateThebinomialmodelcanbeeasilyprogrammedinExcelorVisualBasicandcanbeadaptedtonumerous,andoftenquitecomplicated,option-pricingproblemsTheoneperiodbinomialConsiderasimpleoneperiodmodel:Todayisdate0andtomorrowisdate1Thereisanon-dividendpayingstockwhosepriceattime0isequaltoSandaEuropeancalloptiononthestockwhose(unknown)valueattime0isCAtdate1,thestockpricewilleitherrisetoSu=S(1+U)orwillfallTheperperiod(simple)interestrateisr,whereThecalloptionexpiresatdate1andhasanexercisepriceForexample,assumeS=U=D=-r=X=Theoneperiodbinomial WecanconstructabinomialtreeforthestockpriceasfollowsTheoptionpricenextperiodisgivenbytheirexpirationvalues

Ourobjectiveistofindtheequilibriumpriceoftheoption,CCreatea portfoliothatislong sharesandshortin1Thecurrentvalueofthisportfolio

Nextperiodtheportfoliowillbe

Now,considerthevalue that ,i.e.so

Cd

CdS(1U)S(1 SuSThisportfolioisrisklessandmustthereforeearntheriskrate,S

1ThiscanbesolvedtoyieldthecurrentoptionqCu(1q)Cd

qrC1

UD,Thevaluesoftheoptionatdate1,expiration

Cd,aregivenbyThevaluesqand1–qareknownasrisk-neutralprobabilitiesTheoneperiodbinomial Theoneperiodbinomialmodelcanbeeasilyextendedtothemulti-period多周期Workingbackwards,eachperiodtheoptionpriceiscalculatedasthepresentvalueoftheexpectedoptionpricethefollowingperiod,usingtheriskneutralprobabilitiesandthe ReplicatingportfolioCombinethestockandarisk- bondtoreplicatethecalloption’scashflows.HoldΔsharesofstockandhaveBpoundsinvestedinthebondearningrisklessraterorborrowBpoundswithrisklessrater. Cu=max(0,Su-X) Cd=max(0,Sd–Thecurrentvalueoftheportfolio=Letthevalueoftheportfoliooneperiodlatermatchtwo esofthecalloption.i.e.ΔS(1+U)+B(1+r)= ΔS(1+D)+B(1+r)=Solvethe cucd,

BS

ThepresentvalueofthecallmusthavethesameValueasthereplicatedC

BTheBlack-Scholes Thebinomialmodelassumesthatineach(discrete分散的period,thesharepricemoveseitherupordownbydiscreteamountsBlackandScholes(1973)derivedanoptionpricingmodelthatassumesinsteadthatthesharepricemovescontinuouslyTheadvantageofthisformulationisthatitpermitsthederivationofaclosedformsolutiontothepriceofanoptionthatisveryeasytoTheBlack-Scholes AlthoughderivedforEuropeancallandputoptionsonnon-dividendpayingstocks,itoftenprovidesanadequate充足的approximationtothepriceofmorecomplicated復(fù)雜的AcentralassumptionoftheBlack-ScholesmodelisthatstockarelognormallyThismeansthattheinstantaneousreturnfromholdingasharestockbetweentimetandtimet

isnormallydistributed,

dt)TheBlack-ScholesConsideranon-dividendpayingstockwhosepriceislognormallyTheBlack-ScholesmodelyieldsthefollowingformulaeforthepriceofEuropeancallandputoptionsonthestock Tln(S/X)(r2/ ln(S/(XerT))2T/ ln(S/(XerT T d1 d2d1

ln(S/(XerT))T TT TwhereN(.)denotesthecumulativestandardnormalTheBlack-ScholesTheBlack-ScholesformulaeforcallandputpricesareeasilyimplementedinExcelAforwardcontractisacontractmadeattime0,tobuyorsellanassetorgoodatsometimeinthefuture,T,ontermsagreedattime0Thetermsincludethety,quality,priceanddeliveryAforwardcontractisthereforeacontractforforwarddelivery遠(yuǎn)期交貨ratherthanspotdelivery當(dāng)場;themarketforspotdeliveryiscalledthespotmarket,whilethemarketforforwarddeliveryiscalledtheforwardmarketThecontractisusuallysettledonthedeliverydate;forwardcontractsareusuallymadeoncommodities商品suchaswheatorsugar,butmayalsobemadeonfinancialassets,suchasforeigncurrenciesorinterestratesAninvestorwhoentersintoaforwardcontacttobuytheassetislonginthecontract,whileaninvestorwhoentersintoaforwardcontracttoselltheassetisshortinthecontractExpostitisalwaysthecasethatitwouldhavebeenbetterforoneofthecounterpartiesnottohaveenteredintotheforwardcontractbuttohavewaitedandthendealtinthespotmarketinsteadNevertheless,exante,itmustbedesirableforbothcounterpartiestoenterintotheforwardcontractTheforward Wefirstassumethattherearenotransactionscosts費用,nostoragecosts倉儲成本nocreditriskandthattheassetsorgoodsarecontinuouslydivisible;thesimpleriskrateofinterestisassumedtobeaconstant,r;weassumethatthereisnocosttoenteringintoaforwardcontractThecurrentspotpriceoftheassetis

andweareinterestedinforwardpriceattime0fordeliveryattime

F(0,TSupposethataninvestorbuysoneunitoftheassetinthespotatthespotprice,S0,andsimultaneouslyentersintoacontracttodelivertheassetattimeT,whenhereceivesthe

F(0,TSuchaninvestmenthasaguaranteedcashflow,andsoitsreturnmustbeexactlythesameasthereturnonariskasset,rThereturnontheinvestmentisequaltoSincethecashflowsarecertain,thisreturnmustbeequaltotheriskrateinordertoavoidarbitrageopportunities,(F(0,T)S0)/

Theforwardpriceisthereforegiven0F(0,T)S(1r)T0Asthematuritydateapproaches,theforwardpriceconvergestotheunderlyingprice潛伏價格Theforward Forexample,consideracontractforforwarddeliveryofonetonneofcocoaon31April20x8.Todayis1November20x7andthecurrentspotpriceis$955.00pertonne.Therisk rateofinterestis5.35%perannum.Supposethatyouboughtonetonneofcocoafor$955.00,andsimultaneouslyenteredintoashortforwardcontracttodeliveronetonneofcocoainsixmonths’timeattheforwardpriceofF(0,0.5)Thereturnonthisinvestmentisequalto(F(0,0.5)Sincethisisarisk return,itshouldbeequaltotherisk overthesameperiod(F(0,0.5)–$955.00)/$955.00=(1+5.35%)0.5–AndsotheforwardpriceisequalF(0, ==Futurescontracts合Forwardcontractshavetheadvantageofbeingtailor-tosuitindividualHowever,theyhaveanumberofdisadvantages:(1)theycannotbecancelledwithouttheagreementofbothcounterparties中的反方(2)theforwardcontractcannotbetransferredtoathirdparty,(3)thereisalwaysthepossibilitythatoneofthecounterpartieswillFuturescontractsoffermanyoftheadvantagesofforwardcontracts,butwithoutthedisadvantagesdescribedaboveAfuturescontractisacontractmadeattime0,tobuyorsellanassetorgoodatsometimeinthefutureT,ontermsagreedattimeT.Likeaforwardcontract,afuturescontractisthereforeacontractforforwarddeliveryratherthanspotdeliveryUnlikeaforwardcontract,however,thetermsofthefuturescontractarestandardisedFuturescontractsaretradedonexchangesthatareregulated,andofferprotectionagainstdefaultApositioninafuturescontractcanbeterminatedeitherbyholdingituntildeliveryorbytakinganequalandoppositepositioninthefuturesmarket(whichisknownasareversingtrade倒轉(zhuǎn))ThemarginaccountToeliminate清除thecosttotheexchangearisingfromdefaulted違約contracts,thecounterparties反方toafuturescontracteachopensamarginaccountwiththeirbroker人,inwhichtheyceadeposit,orinitialmargin原始押金,thatisequaltothe dailylossthatacontractislikelytooccurAttheendofeachday’strading,anylossaccruingtoacontractmustbepaidintothemarginaccount,andanyprofitcanbewithdrawn取出Theminimumallowablemarginisknownasthemaintenancemarginlevel保證金比率,andpaymentsintothemarginaccountareknownasvariationmargin盈虧保證金,ormaintenanceThemargin Thisisknownasmarking-to-Ifacounterpartyfailstomark-to-marketthenthecontractisclosed;thatday’slosseswillbecoveredbythebalanceofthemarginaccount保證金賬戶;sincefuturescontractsaremarked-to-market,thelargestlossthatcanoccurthroughdefaultisthechangeinthefuturespriceinasingledaytoensurethatthisiscoveredbythesurplusinthemarginaccount,futuresexchangesusuallyoperateasystemofpricelimitsThemarginisknownastheperformancebondinsomeForwardpricesandfutures Sincefuturescontractsaremarked-to-market,theirvalueonanyparticulardayisequaltozeroThedeliverypriceofafuturescontractandthespotpricewillgenerallydiffer,buttheyareobviouslyrelatedSimilartoaforwardcontract,asthedeliverydateapproaches,thespotandfuturespriceswillconvergeItisreasonabletoassumethatsinceforwardsandfuturesarebothcontractsforfuturedelivery,theirpriceswillbesimilarForwardpricesandfutures However,forwardandfuturescontractsarefundamentallydifferentsincetheformerinvolvesasinglecashflowatthedeliverydate,whilethelatterinvolvesdailycashflowsintheformofinitialandvariationNevertheless,ifinterestratesareconstant,futuresandforwardpricesmustbeidenticalinordertoprecludearbitrageopportunities.Swapsarecontractualagreementsbetweenindividualsorcompaniestoexchangecashflows,e.g.InterestrateswapsCurrencyswapsCreditdefaultswapConsiderinterestrateswapsfortwocompanies,andtwosourcesoffinance,namelyfixedandfloatingrateloans.Thefixedrateloanisavailableataknownrateofinterestthatisfixedforthematurityoftheloan.Thefloatingrateloanisavailableatvariablerate,usuallyquotedasLIBORplusafixedSupposethatforsomereason,oneofthecompanieshasaadvantageinthefixedratemarket,whiletheotherhasacomparativeadvantageinthefloatingratemarketIfthecompaniesenterintoainterestrateswapcontractthentheyagreetoeachtakeoutaloaninthemarketinwhichtheyhaveacomparativeadvantage,butthentoexchangeallthecashflowsassociatedwiththeloanOften,thecompaniesinvolveddonotactuallytakeouttheloans.Theymayjustwishtoexchangecashflows.Inordertoachievethis,thecompanieswillsetanotionalprincipal權(quán)益轉(zhuǎn)易.ThisisusedsolelytodeterminethepatternofinterestpaymentsthatfollowCurrencyswapsareagreementstoexchangecashflowsdenominatedinonecurrencywithcashflowsdenominatedinAcreditdefaultswap(CDS)isaswapagreementthattheselleroftheCDSwillcompensatethebuyerintheeventofaloandefault.ThebuyeroftheCDSmakesaseriesofpaymentstothesellerand,inexchange,receivesapayoffiftheloandefaults.Swapsareusuallyarrangedthroughanintermediarybankwhowillearnafixedpercentagecommission.Supposethaton15December,GeneralElectric(GEC)通用borrows$25mforoneyearatLIBOR+0.25%.However,GECwouldactuallypreferafixedrateloanandapproachesJPMorgan(JPM)toarrangeaswap.GEC’sbestfixedrateis6.4%Simultaneously,Coca-Cola(KO)borrows$25mforoneyearatfixed.However,KOwouldactuallypreferafloatingrateloan,andalsoapproachesJPMtoarrangeaswap.KO’sbestfloatingrateisJPMoffersGECafixedrateof6.2%,andoffersKOafloatingrateloanatLIBOR+0.3% GEpaysLIBOR+0.25%(toitsbank)plus6.2%(toJPM),butreceivesLIBOR+0.15%(fromJPM).Intotal,itthereforepays(LIBOR+0.25%)+6.2%–(LIBOR+0.15%)=6.3%,whichisbetterthanitsbestfixedrate,whichisKOpays6.1%(toitsbank)plusLIBOR+0.3%(toJPM),butreceives6.0%(fromJPM).Intotal,itthereforepays6.1%+(LIBOR+0.3%)–6.0%=LIBOR+0.4%,whichisbetterthanitsbestfloatingratewhichisLIBOR+0.5%JPMreceives6.2%(fromGE)plusLIBOR+0.3%(fromKO)butpaysLIBOR倫敦銀行間拆放款利率+0.15%(toGE)and6.0%(toKO).Intotal,itmakes6.2%LIBOR+0.3%)(LIBOR+0.156.0%=0.35%,whichisbetterthanInpractice,onlythenettedamountswouldbepaid,whichreducesthecreditriskforallpartiesAlso,JPMmayonlyenterintoaswapcontractwithoneofthefirms,buthedgetheresultinginterestrateriskthatitwouldfacebybuyingorsellinginterestratefuturesWeknowafinancialsecurityisassociatedwithastreamoffuturecashflows.Astructurednoteisadebtobligationthatisderivedfrom來someotherdebtExamplesofstructurednotesincludecolla lizedmortgageobligations房屋抵押(bondswhosecashflowsareguaranteed擔(dān)保bypoolsofmortgages抵押thatarec