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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:
8、 TreeforProblem21.4Problem21.5.Consideranoptionthatpaysofftheamountbywhichthefinalstockpriceexceedstheaveragestockpriceachievedduringthelifeoftheoption.Canthisbevaluedusingthebinomialtreeapproach?Explainyouranswer.Abinomialtreecannotbeusedinthewaydescribedinthischapter.Thisisanexampleofwhatisknownas
9、ahistory-dependentoption.Thepayoffdependsonthepathfollowedbythestockpriceaswellasitsfinalvalue.Theoptioncannotbevaluedbystartingattheendofthetreeandworkingbackwardsincethepayoffatthefinalbranchesisnotknownunambiguously.Chapter27describesanextensionofthebinomialtreeapproachthatcanbeusedtohandleoption
10、swherethepayoffdependsontheaveragevalueofthestockprice.Problem21.6.tiForadividend-payingstock,thetreeforthestockpricedoesnotrecombine;butthetreeforthestockpricelessthepresentvalueoffuturedividendsdoesrecombine.,Explainthisstatement.SupposeadividendequaltoDispaidduringacertaintimeinterval.IfSisthesto
11、ckpriceatthebeginningofthetimeinterval,itwillbeeitherSu-DorSd-Dattheendofthetimeinterval.Attheendofthenexttimeinterval,itwillbeoneof(Su-D)u,(Su-D)d,(Sd-D)uand(Sd-D)d.Since(SU-D)ddoesnotequal(Sd-D)uthetreedoesnotrecombine.IfSisequaltothestockpricelessthepresentvalueoffuturedividends,thisproblemisavoi
12、ded.Problem21.7.ShowthattheprobabilitiesinaCox,RossyandRubinsteinbinomialtreearenegativewhentheconditioninfootnote8holds.Withtheusualnotationa-d,u-al-P=;u-aIfau,oneofthetwoprobabilitiesisnegative.Thishappenswhene(r-q)trThisinturnhappenswhen(q-r)4tor(r-q)4tHencenegativeprobabilitiesoccurwhen(r-)71Thi
13、sistheconditioninfootnote8.Problem21.8.Usestratifiedsamplingwith100trialstoimprovetheestimateofinBusinessSnapshot21.1andTable21.1.InTable21.1cellsAl,A2,A3,.,AlOOarerandomnumbersbetween0and1defininghowfartotherightinthesquarethedartlands.CellsBl,B2,B3,.,B100arerandomnumbersbetween0and1defininghowhigh
14、upinthesquarethedartlands.ForstratifiedsamplingwecouldchooseequallyspacedvaluesfortheA,sandtheB,sandconsidereverypossiblecombination.Togenerate100samplesweneedtenequallyspacedvaluesfortheA,sandtheB,ssothatthereare1010=100combinations.Theequallyspacedvaluesshouldbe0.05,0.15,0.25,.,0.95.Wecouldtherefo
15、resettheA,sandB,sasfollows:Al=A2=A3=.=AlO=0.05All=A12=A13=.=A20=0.15A91=A92=A93=.=Al=0.95andBI=BlI=B21=.=B91=0.05B2=B12=B22=.=B92=0.15BIO=B20=B30=.=BlOO=0.95Wegetavalueforequalto3.2,whichisclosertothetruevaluethanthevalueof3.04obtainedwithrandomsamplinginTable21.1.BecausesamplesarenotrandomWecannoteasilycalculateastandarderroroftheestimate.Problem21.9.ExplainwhytheMonteCarlosimulationapproachcannoteasilybeusedforAmericanstylederivatives.InMonteCarlosimulationsamplevaluesforthederivativesecurityinaris
