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19.1OPTIONS、FUTURES&OTHERDERIVATIVESSTUDENTPRESENTATIONTUTOR：PROFYE指导老师：叶永刚教授STUDENTNAME：HEHAO学生姓名：贺昊NO.：20032105077319.2ExtensionoftheTheoreticalFrameworkforPricingDerivatives;MartingalesandMeasuresChapter1919.3DerivativesDependentonaSingleUnderlyingVariableConsideravariable,,(notnecessarilythepriceofatradedsecurity)thatfollowstheprocessImaginetwoderivativesdependentonwithprices?and?Suppose?ƒƒƒ12dmdtsdzddtdzddtdz.1111222219.4FormingaRisklessPortfolioWecansetuparisklessportfolio,consistingof+ƒofthe1stderivativeandƒofthe2ndderivativeƒ)?ƒ)?=(ƒƒƒƒ)221211212122121211((t19.5Sincetheportfolioisriskless:=Thisgives:or11rtrrrr12212122MarketPriceofRisk(Page500)•Thisshowsthat(–r)/isthesameforallderivativesdependentonthesameunderlyingvariable,•Wereferto(–r)/asthemarketpriceofriskforanddenoteitbyl19.6DifferentialEquationforƒ(Equation19.10,page501)UsingIto’slemmatoobtainexpressionsforandintermsofmands.Theequationl=rbecomeslƒƒ?ƒ?2tmssr()22219.7Risk-NeutralValuation•Thedifferentialequationshowsthatislikeastockpricepayingadividendyieldofr–m+ls•Thisanalogyshowsthatwecanvalueƒinarisk-neutralworldprovidingthedriftrateofisreducedfrommtom–ls19.8ExtensionoftheAnalysistoSeveralUnderlyingVariables(Equations19.12and19.13,page503)?ƒddtdzriiiniiinl1119.9DerivativesDependentonCommodityPrices(Page506)Foracommoditythefuturespricegivestheexpectedvalueinthetraditionalrisk-neutralworld19.10Martingales(Page507)•Amartingaleisastochasticprocesswithzerodrfit•Amartingalehasthepropertythatitsexpectedfuturevalueequalsitsvaluetoday19.11AlternativeWorldsInthetraditionalrisk-neutralworldInaworldwherethemarketpriceofriskisdfrfdtfdzdfrfdtfdzll()19.12AKeyResult(Page509)Ifwesetequaltothevolatilityofasecurity,thenIto'slemmashowsthatisamartingaleforallderivativesecuritypricesandareassumedtoprovidenoincomeduringtheperiodunderconsideration)lgfgffg(19.13ForwardRiskNeutralityWerefertoaworldwherethemarketpriceofriskisthevolatilityofgasaworldthatisforwardriskneutralwithrespecttog.IfEgdenotesaworldthatisFRNwrtgfgEfggTT0019.14AleternativeChoicesfortheNumeraireSecurityg•MoneyMarketAccount•Zero-couponbondprice•Annuityfactor19.15MoneyMarketAccountastheNumeraire•Themoneymarketaccountisanaccountthatstartsat$1andisalwaysinvestedattheshort-termrisk-freeinterestrate•Theprocessforthevalueoftheaccountisdg=rgdt•Thishaszerovolatility.Usingthemoneymarketaccountasthenumeraireleadstothetraditionalrisk-neutralworld19.16MoneyMarketAccountcontinuedSince=1and=,theequationbecomeswheredenotesexpectationsinthetraditionalrisk-neutralworldggefgEfgfEefETrdtgTTrdtTTT00000019.17Zero-CouponBondMaturingattimeTasNumeraireTheequationbecomeswhere(,)isthezero-couponbondpriceanddenotesexpectationsinaworldthatisFRNwrtthebondpricefgEfgfPTEfPTEgTTTTT00000(,)[]19.18ForwardPricesInaworldthatisFRNwrtP(0,T),theexpectedvalueofasecurityattimeTisitsforwardprice19.19InterestRatesInaworldthatisFRNwrtP(0,T2)theexpectedvalueofaninterestratelastingbetweentimesT1andT2istheforwardinterestrate19.20AnnuityFactorastheNumeraireTheequationbecomesfgEfgfAtEfATgTTnNATnN000,,()()19.21AnnuityFactorsandSwapRates•SupposethatSn,NistheswapratecorrespondingtotheannuityfactorAn,N•InaworldthatisFRNwrttheannuityfactor,theexpectedswaprateequaltheforwardswaprate19.22ExtensiontoSeveralIndependentFactors(Page513)Inarisk-neutralworldForotherworldsthatareinternallyconsistentdftrtftdttftdzdgtrtgtdttgtdzdftrttftdttftdzdgtrttfiiimgiiimifiimfiiimigiim()()()()()()()()()()()()()()()()()()()ll11111gtdttgtdzgiiim()()()119.23ExtensiontoSeveralIndependentFactorscontinuedWedefineaworldthatisFRNwrtasworldwhereAsintheone-factorcase,isamartingaleandtherestoftheresultshold.igfggil19.24Applications(Section19.7,page514)•ValuationofaEuropeancalloptionwheninterestratesarestochastic•Valuationofanoptiontoexchangeoneassetforanother19.25ChangeofNumeraire(Section19.8,page517)Whenwechangethenumerairesecurityfromto,thedriftofavariableincreasesbywhereisthevolatilityofisthevolatilityofandisthecorrelationbetweenandghvvqfgqvqvqvq,,,19.26Quantos(Section19.9,page518)•Quantosarederivativeswherethepayoffisdefinedusingvariablesmeasuredinonecurrencyandpaidinanothercurrency•Example:contractprovidingapayoffofST–Kdollars($)whereSistheNikkeistockindex(ayennumber)19.27DiffSwap•Diffswapsareatypeofquanto•Afloatingrateisobservedinonecurrencyandappliedtoaprincipalinanothercurrency19.28QuantoscontinuedWhenwemovefromthetraditionalriskneutralworldincurrencytothetraditionalriskneutralworldincurrency,thegrowthrateofavariableincreasesbywhereisthevolatilityof,isthevolatilityoftheexchangerate(unitsofYperunitofX),andisthecoefficientofcorrelationbetweenthetwoYXVVVSVS19.29QuantoscontinuedWhenwemovefromaforwardriskneutralworldincurrencytoaforwardriskneutralworldincurrencybothbeingwrttozero-couponbondsmaturingattime,thegrowthrateofavariableincreasesbywhereisthevolatilityoftheforwardvalueof,isth