内容案例上学期beamlecture risk

BEAM047FUNDAMENTALSOFFINANCIALTom Lecture9:RiskManagement:AnIntroductiontoTimevalueandintrinsicvalueThefactorsaffectingoptionprices价ThebinomialoptionpricingTheBlack-ScholesoptionpricingmodelFuturesRiskBrigham,E.,andHouston,J.,(2010)FundamentalsofFinancial(CustomEdition).SouthWesternCengageLearning.Chapter ShouldfirmshedgetheriskoftheircashInprinciple,welldiversifiedinvestorscareonlyaboutmarketrisk,sinceitisonlythisthatcontributestotheriskoftheiroveralldiversifiedportfolio,andsocorporatehedgingshouldnotaffectthevalueoftheirinvestmentInpractice,however,therearevariousreasonswhyafirmmaychoosetohedgeidiosyncraticrisk非系统风险,includingreducingthevolatility波动性of e,reducingtheprobabilityoffinancialdistress,increaseddebtcapacity借债能力,bettermanagerialcompensation补偿systems,andcomparative比较advantagesinThemostimportanttoolusedbycorporationsforhedgingriskisthederivativesecurity(orjustderivative),includingforward远期合同contract,futurescontract合同,option,swap互惠,etcItisasecuritywhosevaluedependsonthevalueofoneormoreunderlyingassetsincludingfinancialsecurities(e.g.stocks,interestrates),agriculturalproducts(e.g.wheat,cotton),preciousmetals贵金属(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投机,sometimeswithdisastrousconsequences(forexample,LTCM,BaringsBank)LTCM——Long-TermCapital Long-TermCapitalManagement(LTCM)wasthelargesthedgefund对冲基金($126billioninassets)thatnearlycollapsed价格inlateTheprincipalshareholderswereNobelprize-winningeconomistsMyronScholesandRobertMerton.LTCM'sinvestmentstrategieswerebaseduponhedgingagainstarangeofvolatilityinforeigncurrenciesandbonds.WhenRussiadeclared声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场易、柜台卖买(i.e.directlynegotiatedbetweentwoindividuals)oronanexchangeThemostimportantexchangesforderivativestradingareIntercontinentalExchange(whichincludesLIFFE,TheLondonInternationalFinancialFuturesandOptionsExchange),TheChicagoBoardofTrade(CBOT)DerivativesmarketshavegrownmorerapidlythanothermajormarketsinrecentOptionsAnoptiononastockisasecuritythatgivestheholdertherighttobuyorselloneshareofthestockonorbeforeaparticulardateforapredeterminedprice预先约定价格.Acalloption看涨givestheholdertherighttobuythestockwhileaputoption看跌thatgivestheholdertherighttoselltheTheexerciseprice(orstrikeprice执行价isthepriceatwhichtheholdercanbuyorselltheunderlyingstockTheexpirationdateisthedateonorbeforewhichtheholdercanbuyorselltheunderlyingstock;conventionaloptionsaregenerallywrittenforafewmonths;Long-termEquityiciPationSecurities(LEAPS)havematuritiesofuptothreeyearsAEuropeanoption欧式canonlybeexercisedon以控制theexpiration期满whileanAmericanoptioncanbeexercisedatanytimebeforematurity到期Acoveredoption有抵iswhereaninvestorwhosellsacalloptionownstheunderlyingstock,whileanakedoption无担保isonewheretheinvestordoesn’townthestockAnout-of-the-money外在价值optionisonewhichwouldresultinalossifexercisedimmedia y,whileanin-the-money内在价值optionisonewhichwouldresultinaprofitTimevalueandintrinsic AcalloptiongivestherighttobuyanassetwhosevalueiscurrentlyS,fortheexercisepriceXIftheoptionwereexercisednowitwouldgenerateanimmediatecashflowofS–X thentheoptionwouldnotbeexercisedandsoitsvaluewouldbezeroTheintrinsicvalueofacalloptionisthereforeequalThereissomechancethatthevalueoftheassetwillrisebeforetheexpiryoftheoption;thereisalsosomechancethatitwillfall,butthevalueoftheoptionisboundedfrombelowThereforethevalueoftheoptionwillgenerallyexceeditsintrinsicvalue;thedifferenceisthetimevalueoftheoptionOptionpremium溢价=intrinsicvalue+timeAtexpiry期满timevalueiszeroandthevalueoftheoptionisexactlyequaltoitsintrinsicvalueSimilarly,theintrinsicvalueofaputoptionisequaltomax(XProfits/LossesonBasicXX

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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无argumentscanthenbeusedtofindtheequilibriumpriceoftheoptionatearliernodesofthetreeByworkingbackwards逆向作业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复杂的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远期交货ratherthanspotdelivery当场;themarketforspotdeliveryiscalledthespotmarket,whilethemarketforforwarddeliveryiscalledtheforwardmarketThecontractisusuallysettledonthedeliverydate;forwardcontractsareusuallymadeoncommodities商品suchaswheatorsugar,butmayalsobemadeonfinancialassets,suchasforeigncurrenciesorinterestratesAninvestorwhoentersintoaforwardcontacttobuytheassetislonginthecontract,whileaninvestorwhoentersintoaforwardcontracttoselltheassetisshortinthecontractExpostitisalwaysthecasethatitwouldhavebeenbetterforoneofthecounterpartiesnottohaveenteredintotheforwardcontractbuttohavewaitedandthendealtinthespotmarketinsteadNevertheless,exante,itmustbedesirableforbothcounterpartiestoenterintotheforwardcontractTheforward Wefirstassumethattherearenotransactionscosts费用,nostoragecosts仓储成本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潜伏价格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倒转)ThemarginaccountToeliminate清除thecosttotheexchangearisingfromdefaulted违约contracts,thecounterparties反方toafuturescontracteachopensamarginaccountwiththeirbroker人,inwhichtheyceadeposit,orinitialmargin原始押金,thatisequaltothe dailylossthatacontractislikelytooccurAttheendofeachday’strading,anylossaccruingtoacontractmustbepaidintothemarginaccount,andanyprofitcanbewithdrawn取出Theminimumallowablemarginisknownasthemaintenancemarginlevel保证金比率,andpaymentsintothemarginaccountareknownasvariationmargin盈亏保证金,ormaintenanceThemargin Thisisknownasmarking-to-Ifacounterpartyfailstomark-to-marketthenthecontractisclosed;thatday’slosseswillbecoveredbythebalanceofthemarginaccount保证金账户;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权益转易.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伦敦银行间拆放款利率+0.15%(toGE)and6.0%(toKO).Intotal,itmakes6.2%LIBOR+0.3%)(LIBOR+0.156.0%=0.35%,whichisbetterthanInpractice,onlythenettedamountswouldbepaid,whichreducesthecreditriskforallpartiesAlso,JPMmayonlyenterintoaswapcontractwithoneofthefirms,buthedgetheresultinginterestrateriskthatitwouldfacebybuyingorsellinginterestratefuturesWeknowafinancialsecurityisassociatedwithastreamoffuturecashflows.Astructurednoteisadebtobligationthatisderivedfrom来someotherdebtExamplesofstructurednotesincludecolla lizedmortgageobligations房屋抵押(bondswhosecashflowsareguaranteed担保bypoolsofmortgages抵押thatarec