lecture_2Basicconcepts(衍生金融工具-人民银行研究院

BasicconceptsWhatisaderivative?ContractAderivativeisacontractbetweentwopartiesthatspecifiesconditions–inparticular,datesandtheresultingvalueofunderlyingvariables–underwhichpaymentsorpayoffsaretobemadebetweentheparties.Example1:Socialsecurityisaderivativewhichrequiresaseriesofpaymentsfromanindividualtothegovernmentbeforeage65,andpayoffsafterage65fromthegovernmenttotheindividualaslongastheindividualremainsalive.Example2:Earthquakeinsuranceisaderivativeinwhichanindividualmakesregularannualpaymentsinexchangeforapotentiallymuchlargerpayofffromtheinsurancecompanyshouldanearthquakedestroyhisproperty.Derivativeisalsoknownascontingentclaimssincetheirpayoffsarecontingentupontheoutcomeofunderlyingvariables.PayofftableEarthquakeinsurancepolicyRichterScaleDamagePayoff0-4.95.0-5.45.5-5.96.0-6.97.0-8.9NoneSlightSmallMediumLarge$075010,00025,00050,000SubjectiveprobabilityRichterScaleSubjectiveprobability0-4.95.0-5.45.5-5.96.0-6.97.085%1031.50.5%StatisticsofpayoffPayofffrominsurance(X1,X2,…Xn)Subjectiveprobabilities(Q1,Q2,,…Qn)Expectedpayoff,EX=Q1X1+Q2X2+…+QnXnVarianceofpayoff,Var(X)=Q1[X1-E(X)]2+Q2[X2-E(X)]2+…+Qn[Xn-E(X)]2Covarianceofpayoff,Cov(X,Y)=Q1[X1-E(X)][Y1-E(Y)]+Q2[X2-E(X)][Y2-E(Y)]+…+Qn[Xn-E(X)][Yn-E(Y)]Inourinsuranceexample,E(X)=1,000,Var(X)23,931,250,Std=4,892.InsurancepremiumRiskfreerateis5%.Forthegivenexpectedvalueayearformnow,howmuchareyouwillingtopaytobytheinsurance?1,000?1,000/1.05=952.38,adjustfortimevalueofmoney?Peoplearerisk-averse,theywouldwanttopaymore,thepremium!Risk-aversionAnotherdollarwhenyouarealreadyrichissimplynotasvaluabletoyou(intermsofyourwelfareorutility)asanextradollarwhenyouarepoor.SayyourentirewealthisHKD1,000,000.Takingtheextremecase,thechanceofmakingaprofitofanother1milliondollarsisnotworthitifitcomeswithanequalchanceoflosing1milliondollars.Risk–neutral(riskadjusted)probabilitiesAverysimplewaytoadjustforrisk-aversionistoweightdollarslessthanwehavein“rich”statesandmorethanwehavein“poor”states.Risk-neutralprobability=subjectiveprobability*riskaversionadjustmentTheexpectedvalueunderRisk-neutralprobability=1,160Homeowner’spresentvalue:1,160/1.05=1,104.76RichterScalePayoffSubjectiveprobabilityRisk-aversionadjustmentRisk-neutralprobabilityRisk-neutralprobability*payoff0-4.95.0-5.45.5-5.96.0-6.97.0-8.9$075010,00025,00050,00085%1031.50.5%.9939.99761.04721.14301.2787.845.100.031.017.007075310425350Earthquakeinsurance(allriskdiversifiablefrominsurancecompanyside)Eventhroughthehomeowneriswillingtopayapremiumashighas1,104.76,theinsurancecompanyiswillingtosellthepolicyforalower952.38premium.Competitionamonginsurancecompaniesservestodrivethepremiumdownto952.38.Sotheinsuranceisagooddealforthehomeowner!Thereasonwhytheinsurancecompanyiscontenttocharge952.38isthat,bysellingmanypoliciesindifferentpartsofthecountry,itcandiversifyawayalmostallitsrisk–unlikethehomeownerwhohasonehouse.Theinsurancecompanycanthenactasarisk-neutralinvestorandchangeapremiumequaltothetime-discountexpectedvalue.Nationalcatastropheinsurance(noriskdiversifiable)Anationalcatastropheisaneventsuchasaneconomicdepression,(oranuclearwarfare),thatnegativelyaffectseveryindividualintheeconomysimultaneously.Nodiversificationispossibleinthefaceofsuchanevent.Thelawoflargenumbersdoesnotapplysinceallindividualoutcomesareperfectlycorrelated.Inthatcase,evencompetitiveinsurancecompanieswouldchargethemaximumpremiumof1,104.76.Moregenerally,dependingontheamountofriskthatcanbediversifiedawaybytheinsurancecompanies,thepremiumchargeswillfallbetween952.38and1,104.76.InverseproblemProblem:Knowingthemarket’sriskneutralprobabilities,determinethemarketpriceofderivatives.Inverseproblems:Knowingthemarketpriceofderivatives,determinethemarket’srisk-neutralprobabilities.Theartofmodernderivativesvaluationistolearnasmuchaspossibleaboutthemarket’sriskneutralprobabilitiesfromasfewderivativesaspossible.State-contingentclaimsandcompetemarketsConsider:1.Asset:payoff=[123],availablepayoffs=[a2a3a]2.Cash:payoff=[111]Totalavailablepayoffs=[a+c2a+c3a+c]Onecanpurchase[012]since[123]–[111]==[012],butcan’tpurchase[100]Consider:1.Asset:payoff=[123],availablepayoffs=[a2a3a]2.Cash:payoff=[111],availablepayoffs=[a+c2a+c3a+c]3.Derivative:Payoff=[110],availablepayoffs=[a+c+d2a+c+d3a+c]Onecannowcreate“state-contingentclaims”[100]=-[123]+3[111]–[110][010]=[123]-3[111]+2[110][001]=0[123]+[111]–[110]arbitrarypayoffsnowpossible(”completemarkets”)[xyz]=x[100]+y[010]+z[001]InverseproblemAsset:S=(1*P1+2*P2+3P3)/(1+r)Cash:1/(1+r)=(P1+P2+P3)/(1+r)Derivatives:C=(P1+P2)/(1+r)P1=3-(1+r)(S+C)P2=(1+r)(S+2C)–3P3=1-(1+r)CFundamentaltheoremsoffinancialeconomics1.Riskneutralprobabilitiesexistifonlyiftherearenorisklessarbitrageopportunities2.Theriskneutralprobabilitiesareuniqueifandonlyifthemarketiscomplete3.Undercertainconditions,theabilitytorevisetheportfolioofavailablesecuritiesovertimecanmakeupforthemissingsecuritiesandeffectivelycompletethemarket.DynamicreplicationAvailablesecurities:[123]and[111]only.Canwecreate[110]tocompletethem