期权、期货及其他衍生产品 课件 hull-OFOD11-ppt-28

Options,Futures,andOtherDerivativesEleventhEditionChapter28MartingalesandMeasuresCopyright©2022,2018,2012PearsonEducation,Inc.AllRightsReservedDerivativesDependentonaSingleUnderlyingVariable(Equations28.1to28.3)Consideravariable(notnecessarilythepriceofatradedsecurity)thatfollowstheprocessImaginetwoderivativesdependentonwithpricesFormingaRisklessPortfolio

(Equations28.4and28.5)Wecansetuparisklessportfolioconsistingof1stderivativeand2ndderivativeMarketPriceofRisk(Equation28.6)Thisshowsthatisthesameforallderivativesdependentonthesameunderlyingvariable,θWerefertoasthemarketpriceofriskforanddenoteitbyExtensionoftheAnalysistoSeveralUnderlyingVariables(Equations28.12and28.13)IffdependsonseveralunderlyingvariableswithMartingalesAmartingaleisastochasticprocesswithzerodriftAvariablefollowingamartingalehasthepropertythatitsexpectedfuturevalueequalsitsvaluetodayAlternativeWorldsInthetraditionalrisk-neutralworldInaworldwherethemarketpriceofriskTheEquivalentMartingaleMeasureResultIfwesetequaltothevolatilityofasecurityg,thenIto’slemmashowsthatf/gisamartingaleforallderivativesecuritypricesfNumeraireWewillrefertoaworldwherethemarketpriceofriskisthevolatilityofgas“aworlddefinedbynumeraireg”IfdenotesexpectationsinaworlddefinedbynumerairegAlternativeChoicesfortheNumeraireSecuritygMoneyMarketAccountZero-couponbondpriceAnnuityfactorMoneyMarketAccountastheNumeraireThemoneymarketaccountisanaccountthatstartsat$1andisalwaysinvestedattheshort-termrisk-freeinterestrate.TheprocessforthevalueoftheaccountisThishaszerovolatility.Usingthemoneymarketaccountasthenumeraireleadstothetraditionalrisk-neutralworldwhereMoneyMarketAccountSincetheequationbecomesWheredenotesexpectationsinthetraditionalrisk-neutralworldZero-CouponBondMaturingattimeTasNumeraireTheequationbecomeswhereP(0,T)isthepriceofazero-couponbondmaturingattimedenotesexpectationsinaworldwherethebondpriceisthenumeraireForwardPricesInaworlddefinedbynumeraireP(0,T),theexpectedvalueofasecurityattimeTisitsforwardprice.InterestRatesInaworlddefinedbynumerairetheexpectedvalueofaninterestratelastingbetweentimesistheforwardinterestrate.AnnuityFactorastheNumeraireTheequationbecomesAnnuityFactorsandSwapRatesSupposethats(t)istheswapratecorrespondingtotheannuityfactorA.Then:ExtensiontoSeveralIndependentFactors(1of2)Inthetraditionalrisk-neutralworldForotherworldsthatareinternallyconsistentExtensiontoSeveralIndependentFactors(2of2)WedefineaworlddefinedbynumerairegasworldwhereAsintheone-factorcase,f/gisamartingaleandtherestoftheresultshold.ApplicationsExtensionofBlack’smodeltocasewhereinterestratesarestochasticValuationofanoptiontoexchangeoneassetforanotherBlack’sModel(Equations28.28and28.29)ByworkinginaworlddefinedbynumeraireP(0,T),itcanbeseenthatBlack’smodelistruewheninterestratesarestochasticprovidingtheforwardpriceoftheunderlyingassetishasaconstantvolatilityOptiontoExchangeanAssetWorthUforOneWorthV(Section28.7)ThiscanbevaluedbyworkinginaworlddefinedbynumeraireUValueisChangeofNumeraire(Section28.8)Whenwechangethenumerairesecurityfromgtoh,thedriftofavariablevincreasesbyisthevolatilityofisthevolatilityofisthecorrelationbetweenCopyrightThisworkisprotectedbyUnitedStatescopyrightlawsandis

providedsolelyfortheuseofinstructorsinteachingtheircoursesandassessingstudentlearning.Disseminationorsaleofanypartofthiswork(includingontheWorldWideWeb)willdestroytheintegrityoftheworkandisnotpermitted.Theworkandmaterialsfromitshouldneverbemadeavailabletostudentsexceptbyinstructorsusingthe