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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.BecausesamplesarenotrandomWecannote
16、asilycalculateastandarderroroftheestimate.Problem21.9.ExplainwhytheMonteCarlosimulationapproachcannoteasilybeusedforAmericanstylederivatives.InMonteCarlosimulationsamplevaluesforthederivativesecurityinarisk-neutralworldareobtainedbysimulatingpathsfortheunderlyingvariables.Oneachsimulationrun,valuesf
17、ortheunderlyingvariablesarefirstdeterminedattime/,thenattime2,thenattime3r,etc.Attimezr(z=0,1,2?)itisnotpossibletodeterminewhetherearlyexerciseisoptimalsincetherangeofpathswhichmightoccuraftertime/havenotbeeninvestigated.Inshort,MonteCarlosimulationworksbymovingforwardfromtimettotimeT.Othernumerical
18、procedureswhichaccommodateearlyexerciseworkbymovingbackwardsfromtimeTtotimet.Problem21.10.Anine-monthAmericanputoptiononanon-dividend-payingstockhasastrikepriceof$49.Thestockpriceis$50,therisk-freerateis5%perannum,andthevolatilityis30%perannum.Useathree-stepbinomialtreetocalculatetheoptionprice.Inth
19、iscase,50=50,K=49,r=0.05,-0.30,T=0.75,andr=0.25.Also=Q=e3。辰=.i6i8J=1=0.8607u,0.050.25= 1.0126a-dPF=0.50431-p=0.4957TheoutputfromDerivaGemforthisexampleisshownintheFigureS21.3.Thecalculatedpriceoftheoptionis$4.29.Using100stepsthepriceobtainedis$3.91Bolded values are a result of exercise0.250.500.75Gr
20、owth factor per step, a = 1.0126Probability of up move, p = 0.5043Node Time:0.00Figure 521.3: TreeforProblem21.10Problem21.11.Useathree-time-steptreetovalueanine-monthAmericancalloptiononwheatfutures.Thecurrentfuturespriceis400cents,thestrikepriceis420cents,therisk-freerateis6%,andthevolatilityis35%
21、perannum.Estimatethedeltaoftheoptionfromyourtree.InthiscaseF0=400,K=420,r=0.06,=0.35,T=0.75,andr=0.25.Also产防=1.1912J=1=0.8395ua=lp=伫=0.4564u-d1-p=0.5436TheoutputfromDerivaGemforthisexampleisshownintheFigureS21.4.Thecalculatedpriceoftheoptionis42.07cents.Using100timestepsthepriceobtainedis38.64.Theop
22、tion,sdeltaiscalculatedfromthetreeis(79.971-11.419)/(476.498-335.783)=0.487When100stepsareusedtheestimateoftheoption,sdeltais0.483.Ateachnode:0.25000.50000.75Node Time: o.ooUpper value = UnderIyingAsset Price Lower value = Option PriceBolded values are a result of exerciseGrowth factor per step, a =
23、 1.0000Probabilityof upmove, p =0.4564Figure 521.4: TreeforProblem21.11Problem21.12.Athree-monthAmericancalloptiononastockhasastrikepriceof$20.Thestockpriceis$20,therisk-freerateis3%perannum,andthevolatilityis25%perannum.Adividendof$2isexpectedin1.5months.Useathree-stepbinomialtreetocalculatetheopti
24、onprice.Inthiscasethepresentvalueofthedividendis2003x,25=1.9925.WefirstbuildatreeforS0=20-1.9925=18.0075,K=20,r=0.03,=0.25,andT=0.25with/=0.08333.ThisgivesFigureS21.5.Fornodesbetweentimesand1.5monthswethenaddthepresentvalueofthedividendtothestockprice.TheresultisthetreeinFigureS21.6.Thepriceoftheopt
25、ioncalculatedfromthetreeis0.674.When100stepsareusedthepriceobtainedis0.690.TreeshowsstockpriceslessPVofdividendat0.125yearsGrowthfactorperstep,a=1.25Probabilityofupmove,p=0.4993NodeTime:0.000.08330.16670.25Figure 521.5: FirstTreeforProblem21.12Ateachnode:Uppervalue=UnderlyingAssetPrice1.owervalue=Op
26、tionPriceBoldedvaluesarearesultofexerciseProbabilityofupmove,p=0.499322.36045Z36045320.80358175614* 15.58720 T 14.50 面0NodeTime:0.00000.08330.16670.25Figure 521.6: FinalTreeforProblem21.12Problem21.13.Aone-yearAmericanputoptiononanon-dividend-payingstockhasanexercisepriceof$18.Thecurrentstockpriceis
27、$20,therisk-freeinterestrateis15%perannum,andthevolatilityofthestockis40%perannum.UsetheDerivaGetnsoftwarewithfourthree-monthtimestepstoestimatethevalueoftheoption.Displaythetreeandverifythattheoptionpricesatthefinalandpenultimatenodesarecorrect.UseDerivaGemtovaluetheEuropeanversionoftheoption.Useth
28、econtrolvariatetechniquetoimproveyourestimateofthepriceoftheAmericanoption.InthiscaseS0=20,K=18,r=0.15,=0.40,T=1,and/=0.25.Theparametersforthetreeareu=e=e04y=.224J=1m=0.8187.0382a-d1.0382-0.8187八一Up=0.545u-d1.2214-0.8187ThetreeproducedbyDerivaGemfortheAmericanoptionisshowninFigureS21.7.Theestimatedv
29、alueoftheAmericanoptionis$1.29.Ateachnode:Uppervalue三UnderlyingAssetPriceLowervalue=OptionPrice44.51082-0BoldedvaluesarearesultofeerdseGrowthfactorperstepza=1038229.836492.4 乃 954 13,4064-59359913.40644.59359910.976237.02376716374622.012886207)24.428060.386S0224.42806201.28786129.83649036.442至Probab
30、ilityofupmove,p=0.5451Upstepsizezu三1.2214Downstepsize,d三0.81879.013421NodeTime:0.000.250.500.75L00Figure 521.7: TreetoevaluateAmericanoptionforProblem21.13At each node:Upper value = Underlying Asset PriceLower value = Option PriceBolded values are a result of exerciseGrowth factor per step, a = 1038
31、2Probability of up move, p = 0.5451Up step sizeru = 1.2214Down step size, d = 0.81873644238 044.5108229.83649024.428060.38650220 1.143973-20 0.88203424.4280602 1475873.84423313.40644.593599Node Time:0.000.250.7510.976236.36126716.374622Q128868.9865799.0134210.501.00Figure 521.8: TreetoevaluateEurope
32、anoptioninProblem21.13AsshowninFigureS21.8,thesametreecanbeusedtovalueaEuropeanputoptionwiththesameparameters.TheestimatedvalueoftheEuropeanoptionis$1.14.TheoptionparametersareS0=20,K=18,r=0.15,=0.40andT=1ln(2018)+0.15+0.4022C0.40=U.o3o4J2=J1-0.40=0.4384N(-d1)=0.2009;N(-d2)=0.3306ThetrueEuropeanputp
33、riceistherefore18产0.3306-200.2009=1.10ThiscanalsobeobtainedfromDerivaGem.ThecontrolvariateestimateoftheAmericanputpriceistherefore1.29+1.10-1.14=$1.25.Problem21.14Atwo-monthAmericanputoptiononastockindexhasanexercisepriceof480.Thecurrentleveloftheindexis484,therisk-freeinterestrateis10%perannum,thed
34、ividendyieldontheindexis3%perannum,andthevolatilityoftheindexis25%perannum.Dividethelifeoftheoptionintofourhalf-monthperiodsandusethebinomialtreeapproachtoestimatethevalueoftheoption.InthiscaseS0=484,K=480,r=0.10,=0.25q=0.03,T=0.1667,andr=0.04167U=e=e025三7=1.0524J=1=0.9502u。二八*L(X)292a-d1.0029-0.950
35、2，、p=0.516u-d1.0524-0.9502ThetreeproducedbyDerivaGemisshownintheFigureS21.9.Theestimatedpriceoftheoptionis$14.93.At each node:Upper value = Underlying Asset PriceLower value = Option PriceBolded values are a result of exerciseGrowth factor per step, a = 1.0029Probability of up move, p =0.5159Up step
36、 size, u 1.0524Down step size, d = 0.9502564.0698-0509.3401593.602-056.869 0509.34010536.0069098643248425.8430345992067129342.96083415.296164.70388* J0392L 42.9608385.36521NodeTime:0.00000.04170.08330.12500.1667Figure 521.9: TreetoevaluateoptioninProblem21.14Problem21.15Howcanthecontrolvariateapproa
37、chtoimprovetheestimateofthedeltaofanAmericanoptionwhenthebinomialtreeapproachisused?FirstthedeltaoftheAmericanoptionisestimatedintheusualwayfromthetree.Denotethisby*.ThenthedeltaofaEuropeanoptionwhichhasthesameparametersastheAmericanoptioniscalculatedinthesamewayusingthesametree.DenotethisbyNB.Final
38、lythetrueEuropeandelta,iscalculatedusingtheformulasinChapter19.Thecontrolvariateestimateofdeltaisthen:5+金Problem21.16.SupposethatMonteCarlosimulationisbeingusedtoevaluateaEuropeancalloptiononanon-dividend-payingstockwhenthevolatilityisstochastic.Howcouldthecontrolvariateandantitheticvariabletechniqu
39、ebeusedtoimprovenumericalefficiency?Explainwhyitisnecessarytocalculatesixvaluesoftheoptionineachsimulationtrialwhenboththecontrolvariateandtheantitheticvariabletechniqueareused.Inthiscaseasimulationrequirestwosetsofsamplesfromstandardizednormaldistributions.Thefirstistogeneratethevolatilitymovements
40、.Thesecondistogeneratethestockpricemovementsoncethevolatilitymovementsareknown.Thecontrolvariatetechniqueinvolvescarryingoutasecondsimulationontheassumptionthatthevolatilityisconstant.Thesamerandomnumberstreamisusedtogeneratestockpricemovementsasinthefirstsimulation.Animprovedestimateoftheoptionpric
41、eiswheref；istheoptionvaluefromthefirstsimulation(whenthevolatilityisstochastic),fistheoptionvaluefromthesecondsimulation(whenthevolatilityisconstant)andfBisthetrueBlack-Scholes-Mertonvaluewhenthevolatilityisconstant.Tousetheantitheticvariabletechnique,twosetsofsamplesfromstandardizednormaldistributi
42、onsmustbeusedforeachofvolatilityandstockprice.DenotethevolatilitysamplesbyV、andV21andthestockpricesamplesbySiandS2.VlisantithetictoV2)andS、isantithetictoS2/.Thusif/%/=+0.83,+0.41,0.21?then/%/=-0.83,-0.41,+0.21?Similarlyfor/S1)andS2.Anefficientwayofproceedingistocarryoutsixsimulationsinparallel:Simul
43、ation 1: UseSimulation 2: UseISJIS2 /withvolatilityconstantwithvolatilityconstantSimulation 3: UseSimulation 4: UseSimulation 5: UseSimulation 6: UseISJ and VJISJ and V2) IS2 and VJ IS2 / and V2Iffjistheoptionpricefromsimulationi,simulations3and4provideanestimate0.5(/fti)fortheoptionprice.Whenthecon
44、trolvariatetechniqueisusedwecombinethisestimatewiththeresultofsimulation1toobtain0.5(6+f4)-f+fBasanestimateofthepricewherefBis,asabove,theBIack-Scholes-Mertonoptionprice.Similarlysimulations2,5and6provideanestimate0.5(兵+f6)-f2+fOverallthebestestimateis:0505(A)-Z+05(A)-1Problem21.17.Explainhowequatio
45、ns(21.27)to(21.30)changewhentheimplicitfinitedifferencemethodisbeingusedtoevaluateanAmericancalloptiononacurrency.ForanAmericancalloptiononacurrency2+(rf)S*S2皂=rftfS2S2Withthenotationinthetextthisbecomesg+（r）心”4为32 fi,j+l - 2fi.j + fi,jT52forJ=1,2?M-Iand/=0,1?TV-I.Rearrangingtermsweobtainaje+bjfij+c
46、jp+=fzwhere%(r-rf)jt-2j2tbi=l+2/2r+zJJ1z.*12,4cj=-r-rf)jM-j-tEquations(21.28),(21.29)and(21.30)become=max7S-K,0j=(WZo=Oi=0,l度fiM=MAS-K/=0,IMProblem21.18.AnAmericanputoptiononanon-dividend-payingstockhasfourmonthstomaturity.Theexercisepriceis$21,thestockpriceis$20,therisk-freerateofinterestis10%perannum,andthevolatilityis30%perannum.Usetheexplicitversionofthefinitedifferenceapproachtovaluetheoption.Usestockpriceintervalsof$4andtimeintervalsofonemon
