衍生产品市场-课后答案-第五章

Chapter5FinancialForwardsandFutures„Question5.1Fourdifferentwaystosellashareofstockthathasaprice0Sattime0.DescriptionGetPaidatTimeLoseOwnershipofSecurityatTimeReceivePaymentofOutrightSale000Sattime0SecuritySaleandT00rTSeattimeTLoanSaleShortPrepaidForward0T?ContractShortForwardTT?rTe×Contract„Question5.21.Theownerofthestockisentitledtoreceivedividends.Aswewillgetthestockonlyinoneyear,thevalueoftheprepaidforwardcontractistoday’sstockprice,lessthepresentvalueofthefourdividendpayments:31240.060,1$50$1$50$0.985$0.970$0.956$0.942$50$3.853$46.147iPTiFe−×==−=−−−−=−=∑2.Theforwardpriceisequivalenttothefuturevalueoftheprepaidforward.Withaninterestrateof6%andanexpirationoftheforwardinoneyear,wehave:0.0610.0610,0,$46.147$46.1471.0618$49.00PTTFFee××=×=×=×=Notethatthisisequivalenttotakingthefuturevalueoftheinitialstockpriceandsubtractingthefuturevalue(at6%)ofthedividendsreceived,asinEquation(5.6)ofthetext.60McDonald�FundamentalsofDerivativesMarkets„Question5.31.Theownerofthestockisentitledtoreceivedividends.Wehavetooffsettheeffectofthecontinuousincomestreaminformofthedividendyieldbytailingtheposition:0.0810,$50$500.9231$46.1558PTFe−×==×=WeseethatthevalueisverysimilartothevalueoftheprepaidforwardcontractwithdiscretedividendsthatwehavecalculatedinQuestion5.2.InQuestion5.2.,wereceivedfourcashdividends,withpaymentsspreadoutthroughtheentireyear,totaling$4.Thisimpliesatotalannualdividendyieldofapproximately$4$500.08.÷=2.Theforwardpriceisequivalenttothefuturevalueoftheprepaidforward.Withaninterestrateof6%andanexpirationoftheforwardinoneyearwethushave:0.0610.0610,0,$46.1558$46.15581.0618$49.01PTTFFee××=×=×=×=WecouldalsouseEquation(5.7)ofthetext,i.e.,(.06.08)0,$50$49.01.TFe−==„Question5.41.Weusethecontinuouslycompoundedinterestrateandthetimetoexpirationinyears(6monthsis0.5year)inEquation(5.7).Wehave:0.050.50,0$35$351.0253$35.886.rTTFSee××=×=×=×=2.Theannualizedforwardpremiumiscalculatedas:0,011$35.50annualizedforwardpremiumlnln0.02840.5$35TFTS⎛⎞⎛⎞===⎜⎟⎜⎟⎝⎠⎝⎠Noticethisislessthantheinterestrate,hencetheindexmustpayadividend.3.Wecoulduse()0,0rTTFSeδ−=andsolveforδ.However,itiseasiertousethepreviousresultconcerningtheannualizedforwardpremium.Theforwardpremiumissimplythedifferencebetweentherisk-freerateandthedividendyield:()0,000()11annualizedforwardpremiumlnln11ln()()rTTrTFSeTTSSerTTTrδδδδ−−⎛⎞⎛⎞×==⎜⎟⎜⎟⎝⎠⎝⎠==−=−Therefore,wecansolve:0028400500216δδ.=.−⇒=.Theannualizeddividendyieldis2.16%.Chapter5FinancialForwardsandFutures61„Question5.51.Weusethevaluationformula,()00,rTTFSeδ−,=withacontinuouslycompoundedinterestrateof5,r%=adividendyieldof0,δ=andtimetoexpiration075.T=.Remembertimeisinyears,hence9monthsis3/4ofayear.Wehave:00507500$1100$110010382$114202.rTTFSee×.×.,=×=,×=,×.=,.2.Weengageinareversecashandcarrystrategy.Inparticular,wedothefollowing:DescriptionTodayIn9monthsLongforward,resultingfromcustomerpurchase00,TTSF−Sellshorttheindex0S+TS−Lend0S+0S−0rTSe×Total000,rTTSeF×−Specifically,withthenumbersgivenintheexercise,andassumingtheforwardpriceisthenoarbitragepricewedeterminedinPart(1),wehave:DescriptionTodayIn9monthsLongforward,resultingfromcustomerpurchase0$1,142.02TS−Sellshorttheindex$1,100TS−Lend$1,100−$1,1000.050.75$1,100$1,142.02e××=Total00Therefore,themarketmakerisperfectlyhedged.Shedoesnothaveanyriskinthefuture,becauseshehassuccessfullycreatedasyntheticshortpositionintheforwardcontract.3.Now,wewillengageincashandcarryarbitrage:DescriptionTodayIn9monthsShortforward,resultingfromcustomerpurchase00,TTFS−Buytheindex0S−TSBorrow0S+0S+0rTSe−×Total00,0rTTFSe−×62McDonald�FundamentalsofDerivativesMarketsSpecifically,withthenumbersoftheexercise,wehave:DescriptionTodayIn9monthsShortforward,resultingfromcustomerpurchase0$1,142.02TS−Buytheindex−$1,100TSBorrow$1,100$1,1000.050.75$1,100$1,142.02e×−×=−Total00Again,themarketmakerisperfectlyhedged.Hedoesnothaveanyindexpriceriskinthefuture,becausehehassuccessfullycreatedasyntheticlongpositionintheforwardcontractthatperfectlyoffsetshisobligationfromthesoldforwardcontract.„Question5.61.Weusethevaluationformula,()00,rTTFSeδ−,=withacontinuouslycompoundedinterestrateof5,r%=adividendyieldof015,δ=.andtimetoexpiration075.T=.Wehave:()(0050015)07500$1100$110010266$112926.rTTFSeeδ−×.−.×.,=×=,×=,×.=,.2.Weengageinareversecashandcarrystrategy.Wemusttailourshortindexpositionto0157509888.Teeδ−−.×.==.NoticeshortingtheindexrequirespayingthecontinuousdividendsHenceshorting0.9888unitsrequireshavingtobuybackmore(i.e.,1unit)oftheindex.Thespecificsofthereversecashandcarryare:DescriptionTodayIn9monthsLongforward,resultingfromcustomerpurchase00,TTSF−Sellshorttailedpositionoftheindex0TSeδ−+TS−Lend0TSeδ−0TSeδ−−()0rTSeδ−×Total0()00,rTTSeFδ−×−Usingthegivennumbers,wehave:DescriptionTodayIn9monthsLongforward,resultingfromcustomerpurchase0$1,129.26TS−Sellshorttailedpositionoftheindex$1,100.9888×1087.69=TS−Lend$1,087.69$1,087.69−0.050.75$1,087.69$1,129.26e××=Total00Therefore,themarketmakerisperfectlyhedged.Hedoesnothaveanyriskinthefuture,becausehehassuccessfullycreatedasyntheticshortpositionintheforwardcontract.Chapter5FinancialForwar