衍生产品市场-课后答案-第十章

Chapter10BinomialOptionPricing„Question10.11.Since25uC=and0dC=wehave25500.50.Δ==Tosolvethebondamount,onecoulduseEquation(10.2);however,onceweknowtheoption’s,Δfindingthereplicatingbondpositionisasimplealgebraexercise(i.e.,nomemorization!).Wewillreplicatetheup-valueofthecall:040451302540384316eBBe.−..×+=⇒=−=−.Thevalueoftheoptionisthevalueofthereplicatingportfolio:05100384316115684C=.×−.=.2.Since0uP=and25dP=−wehave2550050.−Δ==−.Wewillreplicatetheup-valueoftheput:04045130065624513eBBe.−.−.×+=⇒==.Thevalueoftheoptionisthevalueofthereplicatingportfolio:05100624513124513P=−.×+.=.„Question10.21.Wewilluserisk-neutralpricingforthisproblem.Ourrisk-neutralprobabilityofanupmovementis()04*804816138rhedepudδ−.−−.===.−.−.Theno-arbitragevalueoftheEuropeancallis:**040[(1)]3504816161958rhudCeCpCpe−−.=+−=××.=.Forthereplicatingstrategy,theoption’sdeltais3550070.Δ=/=.Wecanusetheoptionvalueresulttosolveforthebondamount(onceagain,nomemorizationrequired).Theoptionvalueisthetotalcostofreplication.Thecostof0.7sharesis$70.Sincetheoptionvalueis$16.1958,wemustborrow$70$16.1958$53.8042,−=i.e.,$53.8042.B=−2.Ifweobserveapriceof$17,thentheoptionpriceistoohighrelativetoitstheoreticalvalue.Weselltheoptionfor$17andsyntheticallycreateacalloptionfor$16.196.Inordertodoso,webuy0.7unitsoftheshareandborrow$53.804.Thesetransactionsyieldnoriskandaprofitof$0.804.Chapter10BinomialOptionPricing1133.Ifweobserveapriceof$15.50,thentheoptionpriceistoolowrelativetoitstheoreticalvalue.Webuytheoptionandsyntheticallycreateashortpositionintheoption(we’llreceive$16.1958).Inordertodoso,weshort0.7sharesandlend$53.8042.Thesetransactionsyieldnoriskandaprofitof$0.696.„Question10.31.Wewilluserisk-neutralpricingforthisproblem.Ourrisk-neutralprobabilityofanupmovementis()04*804816138rhedepudδ−.−−.===.−.−.Theno-arbitragevalueoftheEuropeanputis:**040[(1)]150518474708rhudPePpPpe−−.=+−=××.=.Forthereplicatingstrategy,theoption’sdeltais1550030.Δ=−/=−.Wecanusetheoptionvalueresulttosolveforthebondamount(onceagain,nomemorizationrequired).Theoptionvalueisthetotalcostofreplication.Byshorting0.3shares,wereceive$30.Sincetheoptionvalueis$7.4708,wemustlend$37.4708,i.e.,B=$37.4708.2.Ifweobserveapriceof$8,thentheoptionpriceistoohighrelativetoitstheoreticalvalue.Weselltheoptionfor$8andsyntheticallycreatealongputoptionfor$7.471.Inordertodoso,weshort0.3unitsoftheshareandlend$37.4708.Thesetransactionsyieldnoriskandaprofitof$0.5292.3.Ifweobserveapriceof$6,thentheputoptionpriceistoolowrelativetoitstheoreticalvalue.Webuytheoptionandsyntheticallycreateashortpositionintheoption(we’llreceive$7.4708).Inordertodoso,webuy0.3unitsoftheshareandborrow$37.4708.Thesetransactionsyieldnoriskandaprofitof$1.4708.„Question10.4Thestockpricesevolveaccordingtothefollowingpicture:Sincewehavetwobinomialsteps,andatimetoexpirationofoneyear,hisequalto0.5.Therefore,wecancalculatewiththeusualformulasfortherespectivenodes:00,1001,801,1300.6910.225149.12713.83591.27519.9944.16538.725dutStStSBBBCCC======Δ=Δ=Δ==−=−=−===114McDonald�FundamentalsofDerivativesMarkets„Question10.5080:S=0,80tS==,64tS=1=,104tS=1=Δ0.465100.7731B−28.59620−61.7980C8.6078018.6020090:S=0,90tS==,72tS=1=,117tS=1=Δ0.587200.9761B−40.61800−87.7777C12.2266026.42230110:S=0,110tS==,88tS=1=,143tS=1=Δ0.77720.44091B−57.0897−29.8229−91.2750C28.40608.977151.72500120:S=0,120tS==,96tS=1=,156tS=1=Δ0.84890.62081B−65.0523−45.8104−91.2750C36.818613.789664.7250Astheinitialstockpriceincreases,the95-strikecalloptionisincreasinglyinthemoney.Witheverythingelseequal,itismorelikelythattheoptionfinishesinthemoney.Ahedger,e.g.,amarketmaker,mustthereforebuymoresharesinitiallytobeabletocovertheobligationshewillhavetomeetatexpiration.Thisnumberofsharesinthereplicatingportfolioismeasuredbydelta.Theinitialcalldeltathusincreaseswhentheinitialstockpriceincreases.„Question10.6Thestockpricesevolveaccordingtothefollowingpicture:Chapter10BinomialOptionPricing115Sincewehavetwobinomialsteps,andatimetoexpirationofoneyear,hisequalto0.5.Therefore,wecancalculatewiththeusualformulasfortherespectivenodes:00,1001,801,1300.30880.775038.56977.439607.689715.43960dutStStSBBBPPP======Δ=−Δ=−Δ=======„Question10.7080:S=0,80tS==,64tS=1=,104tS=1=Δ−0.5350−1−0.2269B59.099891.27529.4770P16.303927.2755.8770090:S=0,90tS==,72tS=1=,117tS=1=Δ−0.4128−1−0.0239B47.078191.2753.4973P9.922619.2750.69730110:S=0,110tS==,88tS=1=,143tS=1=Δ−0.2228−0.55910B30.606461.45210P6.102212.252100120:S=0,120tS==,96tS=1=,156tS=1=Δ−0.1511−0.37920B22.643745.46460P4.51469.064600130:S=0,130tS==,104tS=1=,169tS=1=Δ−0.0904−0.22690B14.681129.47700P2.92715.87700116McDonald�FundamentalsofDerivativesMarketsAstheinitialstockpriceincreases,the95-strikeputoptionisincreasinglyoutofthemoney.Witheverythingelseequal,itismorelikelythattheoptionfinishesoutofthemoney.Ahedger,e.g.,amarketmaker,mustthereforesellfewersharesinitiallytobeabletocovertheobligationshewillhavetomeetatexpiration.Thisnumberofsharesinthereplicatingportfolioismeasuredbydelta.Theinitialputdeltathustendstowardszerowhentheinitialstockpriceincreases.„Question10.8Wemustcomparetheresultsoftheequiva