赫尔《期权、期货及其他衍生产品》(第8版)复习笔记及课后习题详解 (58).docx
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1、CHAPTER21BasicNumericalProceduresPracticeQuestionsProblem21.1.WhichofthefollowingcanbeestimatedforanAmericanoptionbyconstructingasinglebinomialtree:delta,gamma,vega,theta,rho?Delta,gamma,andthetacanbedeterminedfromasinglebinomialtree.Vegaisdeterminedbymakingasmallchangetothevolatilityandrecomputingt
2、heoptionpriceusinganewtree.Rhoiscalculatedbymakingasmallchangetotheinterestrateandrecomputingtheoptionpriceusinganewtree.Problem21.2.Calculatethepriceofathree-tnonthAmericanputoptiononanon-dividend-payingstockwhenthestockpriceis$60,thestrikepriceis$60,therisk-freeinterestrateis10%perannum,andthevola
3、tilityis45%perannum.Useabinomialtreewithatimeintervalofonemonth.Inthiscase,S0=60,K=60,=0.1,=0.45,T=0.25,andz=0.0833.Alsou=e而=e=1,1387J=1=0.8782u=e0,x00833=1.0084adp=-=0.4998u-d1-p=0.5002TheoutputfromDerivaGemforthisexampleisshownintheFigureS21.1.Thecalculatedpriceoftheoptionis$5.16.Growthfactorperst
4、ep,a=1.0084Probabilityofupmove,p=0.499788.59328/0Upstepsize,u=1.1387Downstepsize,d=0.878277.8008468.323137 68.32313夕.79934605.1627811S 52.69079、60? 3.6265348.6033827 46.27?sj 52.69079 7.30920613.7287 40.6351419.36486Node Time: 0.00000.08330.16670.2500Figure 521.1: TreeforProblem21.2Problem21.3.Expla
5、inhowthecontrolvariatetechniqueisimplementedwhenatreeisusedtovalueAmericanoptions.Thecontrolvariatetechniqueisimplementedby1. ValuinganAmericanoptionusingabinomialtreeintheusualway(=fA).2. ValuingtheEuropeanoptionwiththesameparametersastheAmericanoptionusingthesametree(=fE).3. ValuingtheEuropeanopti
6、onusingBlack-Scholes-Merton(=y嬴).ThepriceoftheAmericanoptionisestimatedas/+jw-Problem21.4.Calculatethepriceofanine-monthAmericancalloptiononcornfutureswhenthecurrentfuturespriceis198cents,thestrikepriceis200cents,therisk-freeinterestrateis8%perannum,andthevolatilityis30%perannum.Useabinomialtreewith
7、atimeintervalofthreemonths.Inthiscase与=198,K=200,r=0.08,=0.3,T=0.75,andZ=0.25.Also-=/3庇=1.1618J=1=0.8607a=1?=0.4626u-d1-p=0.5373TheoutputfromDerivaGemforthisexampleisshownintheFigureS21.2.Thecalculatedpriceoftheoptionis20.34cents.Growthfactorperstep,a=1.0000NodeTime:0.00000.25000.500.75Figure 521.2:
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