Chapter12OptionsonStockIndices,Currencies,an

2019/9/301OptionsonStockIndices,Currencies,andFuturesChapter122019/9/302EuropeanOptionsonStocksPayingContinuousDividendsWegetthesameprobabilitydistributionforthestockpriceattimeTineachofthefollowingcases:1.ThestockstartsatpriceS0andprovidesacontinuousdividendyield=q2.ThestockstartsatpriceS0e–qTandprovidesnoincome2019/9/303EuropeanOptionsonStocksPayingContinuousDividendscontinuedWecanvalueEuropeanoptionsbyreducingthestockpricetoS0e–qTandthenbehavingasthoughthereisnodividend2019/9/304ExtensionofChapter7Results(Equations12.1to12.3)cSeXeqTrT0LowerBoundforcalls:LowerBoundforputspXeSerTqT0PutCallParitycXepSerTqT02019/9/305ExtensionofChapter11Results(Equations12.4and12.5)cSeNdXeNdpXeNdSeNddSXrqTTdSXrqTTqTrTrTqT01220110202222()()()()ln(/)(/)ln(/)(/)where2019/9/306TheBinomialModel(Risk-neutralworld)S0uƒuS0dƒdS0ƒf=e-rT[pfu+(1-p)fd]2019/9/307TheBinomialModelcontinuedInarisk-neutralworldthestockpricegrowsatr-qratherthanatrwhenthereisadividendyieldatrateqTheprobability,p,ofanupmovementmustthereforesatisfypS0u+(1-p)S0d=S0e(r-q)TsothatpedudrqT()2019/9/308IndexOptionsOptioncontractsareon100×theindexThemostpopularunderlyingindicesaretheS&P100(American)OEXtheS&P500(European)SPXContractsaresettledincash2019/9/309IndexOptionExampleConsideracalloptiononanindexwithastrikepriceof560Suppose1contractisexercisedwhentheindexlevelis580Whatisthepayoff?2019/9/3010UsingIndexOptionsforPortfolioInsuranceSupposethevalueoftheindexisS0andthestrikepriceisXIfaportfoliohasabof1.0,theportfolioinsuranceisobtainedbybuying1putoptioncontractontheindexforeach100S0dollarsheldIfthebisnot1.0,theportfoliomanagerbuysbputoptionsforeach100S0dollarsheldInbothcases,Xischosentogivetheappropriateinsurancelevel2019/9/3011Example1Portfoliohasabetaof1.0Itiscurrentlyworth$5millionTheindexcurrentlystandsat1000Whattradeisnecessarytoprovideinsuranceagainsttheportfoliovaluefallingbelow$4.8million?2019/9/3012Example2Portfoliohasabetaof2.0Itiscurrentlyworth$1millionandindexstandsat1000Therisk-freerateis12%perannum(每3个月计一次复利）Thedividendyieldonboththeportfolioandtheindexis4%(每3个月计一次复利）Howmanyputoptioncontractsshouldbepurchasedforportfolioinsurance?2019/9/3013Ifindexrisesto1040,itprovidesa40/1000or4%returnin3monthsTotalreturn(incl.dividends)=5%Excessreturnoverrisk-freerate=2%Excessreturnforportfolio=4%IncreaseinPortfolioValue=4+3-1=6%Portfoliovalue=$1.06millionCalculatingRelationBetweenIndexLevelandPortfolioValuein3months)(fmfirrrrb2019/9/3014DeterminingtheStrikePriceValueofIndexin3monthsExpectedPortfolioValuein3months($millions)1,0801.141,0401.061,0000.989600.909200.82Anoptionwithastrikepriceof960willprovideprotectionagainsta10%declineintheportfoliovalue2019/9/3015ValuingEuropeanIndexOptionsWecanusetheformulaforanoptiononastockpayingacontinuousdividendyieldSetS0=currentindexlevelSetq=averagedividendyieldexpectedduringthelifeoftheoption2019/9/3016CurrencyOptionsCurrencyoptionstradeonthePhiladelphiaExchange(PHLX)Therealsoexistsanactiveover-the-counter(OTC)marketCurrencyoptionsareusedbycorporationstobuyinsurancewhentheyhaveanFXexposure2019/9/3017TheForeignInterestRateWedenotetheforeigninterestratebyrfWhenaU.S.companybuysoneunitoftheforeigncurrencyithasaninvestmentofS0dollarsThereturnfrominvestingattheforeignrateisdollarsThisshowsthattheforeigncurrencyprovidesa“dividendyield”atraterf0)(SetTrf2019/9/3018ValuingEuropeanCurrencyOptionsAforeigncurrencyisanassetthatprovidesacontinuous“dividendyield”equaltorfWecanusetheformulaforanoptiononastockpayingacontinuousdividendyield:SetS0=currentexchangerateSetq=rƒ2019/9/3019FormulasforEuropeanCurrencyOptionscSeNdXeNdpXeNdSeNddSXrrfTTdSXrrfTTrTrTrTrTff01220110202222()()()()ln(/)(/)ln(/)(/)where2019/9/3020AlternativeFormulasFSerrTf00()UsingceFNdXNdpeXNdFNddFXTTddTrTrT[()()][()()]ln(/)/0122011022122019/9/3021MechanicsofCallFuturesOptionsMostofFuturesoptionsareAmerican.Thematuritydateisusuallyon,orafewdaysbefore,theearliestdeliverydateoftheunderlyingfuturescontract.Whenacallfuturesoptionisexercisedtheholderacquires1.Alongpositioninthefutures2.Acashamountequaltotheexcessofthefuturespriceoverthestrikeprice2019/9/3022MechanicsofPutFuturesOptionWhenaputfuturesoptionisexercisedtheholderacquires1.Ashortpositioninthefutures2.Acashamountequaltotheexcessofthestrikepriceoverthefuturesprice2019/9/3023ThePayoffsIfthefuturespositionisclosedoutimmediately:Payofffromcall=F0-XPayofffromput=X-F0whereF0isfuturespriceattimeofexercise2019/9/3024WhyFuturesOptioninsteadofSpotOption?FuturesismoreliquidandeasiertogetthepriceinformationCanbesettledincashLowertransactioncost2019/9/3025Put-CallParityforFuturesOptionConsiderthefollowingtwoportfolios:1.EuropeancallplusXe-rTofcash2.EuropeanputpluslongfuturespluscashequaltoF0e-rTTheymustbeworththesameattimeTsothatc+Xe-rT=p+F0e-rT2019/9/3026FuturesPrice=$33OptionPrice=$4FuturesPrice=$28OptionPrice=$0Futuresprice=$30OptionPrice=?BinomialTreeExampleA1-monthcalloptiononfutureshasastrikepriceof29.2019/9/3027ConsiderthePortfolio:longDfuturesshort1cal