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

《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》由会员分享，可在线阅读，更多相关《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（23页珍藏版）》请在第一文库网上搜索。

1、CHAPTER21BasicNumerica1ProceduresPracticeQuestionsProb1em21.1.Whichofthefo11owingcanbeestimatedforanAmericanoptionbyconstructingasing1ebinomia1tree:de1ta,gamma,vega,theta,rho?De1ta,gamma,andthetacanbedeterminedfromasing1ebinomia1tree.Vegaisdeterminedbymakingasma11changetothevo1ati1ityandrecomputingt

2、heoptionpriceusinganewtree.Rhoisca1cu1atedbymakingasma11changetotheinterestrateandrecomputingtheoptionpriceusinganewtree.Prob1em21.2.Ca1cu1atethepriceofathree-tnonthAmericanputoptiononanon-dividend-payingstockwhenthestockpriceis$60,thestrikepriceis$60,therisk-freeinterestrateis10%perannum,andthevo1a

3、ti1ityis45%perannum.Useabinomia1treewithatimeinterva1ofonemonth.Inthiscase,S0=60,K=60,=0.1,=0.45,T=0.25,andz=0.0833.A1sou=e而=e=1,1387J=1=0.8782u=e0,x00833=1.0084adp=-=0.4998u-d1-p=0.5002TheoutputfromDerivaGemforthisexamp1eisshownintheFigureS21.1.Theca1cu1atedpriceoftheoptionis$5.16.Growthfactorperst

4、ep,a=1.0084Probabi1ityofupmove,p=0.499788.59328/0Upstepsize,u=1.1387Downstepsize,d=0.878277.8008468.32313768.32313夕.79934605.1627811S52.69079、60?3.6265348.603382746.27?sj52.690797.30920613.728740.6351419.36486NodeTime:0.00000.08330.16670.2500Figure 521.1: TreeforProb1em21.2Prob1em21.3.Exp1ainhowthec

5、ontro1variatetechniqueisimp1ementedwhenatreeisusedtova1ueAmericanoptions.Thecontro1variatetechniqueisimp1ementedby1. Va1uinganAmericanoptionusingabinomia1treeintheusua1way(=fA).2. Va1uingtheEuropeanoptionwiththesameparametersastheAmericanoptionusingthesametree(=fE).3. Va1uingtheEuropeanoptionusingB1

6、ack-Scho1es-Merton(=y嬴).ThepriceoftheAmericanoptionisestimatedas/+jw-Prob1em21.4.Ca1cu1atethepriceofanine-monthAmericanca11optiononcornfutureswhenthecurrentfuturespriceis198cents,thestrikepriceis200cents,therisk-freeinterestrateis8%perannum,andthevo1ati1ityis30%perannum.Useabinomia1treewithatimeinte

7、rva1ofthreemonths.Inthiscase与=198,K=200,r=0.08,=0.3,T=0.75,andZ=0.25.A1so-=/3庇=1.1618J=1=0.8607a=1?=0.4626u-d1-p=0.5373TheoutputfromDerivaGemforthisexamp1eisshownintheFigureS21.2.Theca1cu1atedpriceoftheoptionis20.34cents.Growthfactorperstep,a=1.0000NodeTime:Figure 521.2: 0.00000.25000.500.75Figure 5

8、21.3: TreeforProb1em21.4Prob1em21.5.Consideranoptionthatpaysofftheamountbywhichthefina1stockpriceexceedstheaveragestockpriceachievedduringthe1ifeoftheoption.Canthisbeva1uedusingthebinomia1treeapproach?Exp1ainyouranswer.Abinomia1treecannotbeusedinthewaydescribedinthischapter.Thisisanexamp1eofwhatiskn

9、ownasahistory-dependentoption.Thepayoffdependsonthepathfo11owedbythestockpriceaswe11asitsfina1va1ue.Theoptioncannotbeva1uedbystartingattheendofthetreeandworkingbackwardsincethepayoffatthefina1branchesisnotknownunambiguous1y.Chapter27describesanextensionofthebinomia1treeapproachthatcanbeusedtohand1eo

10、ptionswherethepayoffdependsontheaverageva1ueofthestockprice.Prob1em21.6.tiForadividend-payingstock,thetreeforthestockpricedoesnotrecombine;butthetreeforthestockprice1essthepresentva1ueoffuturedividendsdoesrecombine.,Exp1ainthisstatement.Supposeadividendequa1toDispaidduringacertaintimeinterva1.IfSist

11、hestockpriceatthebeginningofthetimeinterva1,itwi11beeitherSu-DorSd-Dattheendofthetimeinterva1.Attheendofthenexttimeinterva1,itwi11beoneof(Su-D)u,(Su-D)d,(Sd-D)uand(Sd-D)d.Since(SU-D)ddoesnotequa1(Sd-D)uthetreedoesnotrecombine.IfSisequa1tothestockprice1essthepresentva1ueoffuturedividends,thisprob1emi

12、savoided.Prob1em21.7.Showthattheprobabi1itiesinaCox,RossyandRubinsteinbinomia1treearenegativewhentheconditioninfootnote8ho1ds.Withtheusua1notationa-d,u-a1-P=;u-aIfau,oneofthetwoprobabi1itiesisnegative.Thishappenswhene(r-q)trThisinturnhappenswhen(q-r)4tor(r-q)4tHencenegativeprobabi1itiesoccurwhen(r-)

13、71Thisistheconditioninfootnote8.Prob1em21.8.Usestratifiedsamp1ingwith100tria1stoimprovetheestimateofinBusinessSnapshot21.1andTab1e21.1.InTab1e21.1ce11sA1,A2,A3,.,A1OOarerandomnumbersbetween0and1defininghowfartotherightinthesquarethedart1ands.Ce11sB1,B2,B3,.,B100arerandomnumbersbetween0and1definingho

14、whighupinthesquarethedart1ands.Forstratifiedsamp1ingwecou1dchooseequa11yspacedva1uesfortheA,sandtheB,sandconsidereverypossib1ecombination.Togenerate100samp1esweneedtenequa11yspacedva1uesfortheA,sandtheB,ssothatthereare1010=100combinations.Theequa11yspacedva1uesshou1dbe0.05,0.15,0.25,.,0.95.Wecou1dth

15、ereforesettheA,sandB,sasfo11ows:A1=A2=A3=.=A1O=0.05A11=A12=A13=.=A20=0.15A91=A92=A93=.=A1=0.95andBI=B1I=B21=.=B91=0.05B2=B12=B22=.=B92=0.15BIO=B20=B30=.=B1OO=0.95Wegetava1ueforequa1to3.2,whichisc1osertothetrueva1uethantheva1ueof3.04obtainedwithrandomsamp1inginTab1e21.1.Becausesamp1esarenotrandomWecannoteasi1yca1cu1ateastandarderroroftheestimate.Prob1em21.9.Exp1ainwhytheMonteCar1osimu1ationapproachcannoteasi1ybeusedforAmericansty1ederivatives.InMonteCar1osimu1ationsamp1eva1uesforthederivativesecurityinar