HullOFOD9eSolutionsCh13第九版期權(quán)、期貨及其他衍生品課后答案

實用文檔HullOFOD9eSolutionsCh13第九版期權(quán)、期貨及其他衍生品課后答案CHAPTER13BinomialTreesPracticeQuestionsProblem13.1.Astockpriceiscurrently$40.Itisknownthatattheendofonemonthitwillbeeither$42or$38.Therisk-freeinterestrateis8%perannumwithcontinuouscompounding.Whatisthevalueofaone-monthEuropeancalloptionwithastrikepriceof$39?Consideraportfolioconsistingof1-:Calloption+?:SharesIfthestockpricerisesto$42,theportfolioisworth423?-.Ifthestockpricefallsto$38,itisworth38?.Thesearethesamewhen42338?-=?or075?=..Thevalueoftheportfolioinonemonthis28.5forbothstockprices.Itsvaluetodaymustbethepresentvalueof28.5,or0080083332852831e-.?..=..Thismeansthat402831f-+?=.wherefisthecallprice.Because075?=.,thecallpriceis400752831$169?.-.=..Asanalternativeapproach,wecancalculatetheprobability,p,ofanupmovementinarisk-neutralworld.Thismustsatisfy:0080083334238(1)40ppe.?.+-=sothat00800833344038pe.?.=-or05669p=..Thevalueoftheoptionisthenitsexpectedpayoffdiscountedattherisk-freerate:008008333[305669004331]169e-.?.?.+?.=.or$1.69.Thisagreeswiththepreviouscalculation.Problem13.2.Explaintheno-arbitrageandrisk-neutralvaluationapproachestovaluingaEuropeanoptionusingaone-stepbinomialtree.Intheno-arbitrageapproach,wesetuparisklessportfolioconsistingofapositionintheoptionandapositioninthestock.Bysettingthereturnontheportfolioequaltotherisk-freeinterestrate,weareabletovaluetheoption.Whenweuserisk-neutralvaluation,wefirstchooseprobabilitiesforthebranchesofthetreesothattheexpectedreturnonthestockequalstherisk-freeinterestrate.Wethenvaluetheoptionbycalculatingitsexpectedpayoffanddiscountingthisexpectedpayoffattherisk-freeinterestrate.Problem13.3.Whatismeantbythedeltaofastockoption?Thedeltaofastockoptionmeasuresthesensitivityoftheoptionpricetothepriceofthestockwhensmallchangesareconsidered.Specifically,itistheratioofthechangeinthepriceofthestockoptiontothechangeinthepriceoftheunderlyingstock.Problem13.4.Astockpriceiscurrently$50.Itisknownthatattheendofsixmonthsitwillbeeither$45or$55.Therisk-freeinterestrateis10%perannumwithcontinuouscompounding.Whatisthevalueofasix-monthEuropeanputoptionwithastrikepriceof$50?Consideraportfolioconsistingof1-:Putoption+?:SharesIfthestockpricerisesto$55,thisisworth55?.Ifthestockpricefallsto$45,theportfolioisworth455?-.Thesearethesamewhen45555?-=?or050?=-..Thevalueoftheportfolioinsixmonthsis275-.forbothstockprices.Itsvaluetodaymustbethepresentvalueof275-.,or010********e-.?.-.=-..Thismeansthat502616f-+?=-.wherefistheputprice.Because050?=-.,theputpriceis$1.16.Asanalternativeapproachwecancalculatetheprobability,p,ofanupmovementinarisk-neutralworld.Thismustsatisfy:01055545(1)50ppe.?.+-=sothat010*******pe.?.=-or07564p=..Thevalueoftheoptionisthenitsexpectedpayoffdiscountedattherisk-freerate:0105[007564502436]116e-.?.?.+?.=.or$1.16.Thisagreeswiththepreviouscalculation.Problem13.5.Astockpriceiscurrently$100.Overeachofthenexttwosix-monthperiodsitisexpectedtogoupby10%ordownby10%.Therisk-freeinterestrateis8%perannumwithcontinuouscompounding.Whatisthevalueofaone-yearEuropeancalloptionwithastrikepriceof$100?Inthiscase110u=.,090d=.,05t?=.,and008r=.,sothat0080509007041110090ep.?.-.==..-.ThetreeforstockpricemovementsisshowninFigureS13.1.Wecanworkbackfromtheendofthetreetothebeginning,asindicatedinthediagram,togivethevalueoftheoptionas$9.61.Theoptionvaluecanalsobecalculateddirectlyfromequation(13.10):22200805[0704121207041029590029590]961e-?.?..?+?.?.?+.?=.or$9.61.FigureS13.1:TreeforProblem13.5Problem13.6.ForthesituationconsideredinProblem13.5,whatisthevalueofaone-yearEuropeanputoptionwithastrikepriceof$100?VerifythattheEuropeancallandEuropeanputpricessatisfyput–callparity.FigureS13.2showshowwecanvaluetheputoptionusingthesametreeasinProblem13.5.Thevalueoftheoptionis$1.92.Theoptionvaluecanalsobecalculateddirectlyfromequation(13.10):20080522[0704102070410295910295919]192e-?.?..?+?.?.?+.?=.or$1.92.Thestockpriceplustheputpriceis10019210192$+.=..Thepresentvalueofthestrikepriceplusthecallpriceis008110096110192e$-.?+.=..Thesearethesame,verifyingthatput–callparityholds.FigureS13.2:TreeforProblem13.6Problem13.7.Whataretheformulasforuanddintermsofvolatility?ue=andde-=Problem13.8.ConsiderthesituationinwhichstockpricemovementsduringthelifeofaEuropeanoptionaregovernedbyatwo-stepbinomialtree.Explainwhyitisnotpossibletosetupapositioninthestockandtheoptionthatremainsrisklessforthewholeofthelifeoftheoption.Therisklessportfolioconsistsofashortpositionintheoptionandalongpositionin?shares.Because?changesduringthelifeoftheoption,thisrisklessportfoliomustalsochange.Problem13.9.Astockpriceiscurrently$50.Itisknownthatattheendoftwomonthsitwillbeeither$53or$48.Therisk-freeinterestrateis10%perannumwithcontinuouscompounding.Whatisthevalueofatwo-monthEuropeancalloptionwithastrikepriceof$49?Useno-arbitragearguments.Attheendoftwomonthsthevalueoftheoptionwillbeeither$4(ifthestockpriceis$53)or$0(ifthestockpriceis$48).Consideraportfolioconsistingof:shares1option+?:-:Thevalueoftheportfolioiseither48?or534?-intwomonths.If48534?=?-i.e.,08?=.thevalueoftheportfolioiscertaintobe38.4.Forthisvalueof?theportfolioisthereforeriskless.Thecurrentvalueoftheportfoliois:0850f.?-wherefisthevalueoftheoption.Sincetheportfoliomustearntherisk-freerateofinterest010212(0850)384fe.?/.?-=.i.e.,223f=.Thevalueoftheoptionistherefore$2.23.Thiscanalsobecalculateddirectlyfromequations(13.2)and(13.3).106u=.,096d=.sothat01021209605681106096ep.?/-.==..-.and010212056814223fe-.?/=?.?=.Problem13.10.Astockpriceiscurrently$80.Itisknownthatattheendoffourmonthsitwillbeeither$75or$85.Therisk-freeinterestrateis5%perannumwithcontinuouscompounding.Whatisthevalueofafour-monthEuropeanputoptionwithastrikepriceof$80?Useno-arbitragearguments.Attheendoffourmonthsthevalueoftheoptionwillbeeither$5(ifthestockpriceis$75)or$0(ifthestockpriceis$85).Consideraportfolioconsistingof:shares1option-?:+:(Note:Thedelta,?ofaputoptionisnegative.Wehaveconstructedtheportfoliosothatitis+1optionand-?sharesratherthan1-optionand+?sharessothattheinitialinvestmentispositive.)Thevalueoftheportfolioiseither85-?or755-?+infourmonths.If85755-?=-?+i.e.,05?=-.thevalueoftheportfolioiscertaintobe42.5.Forthisvalueof?theportfolioisthereforeriskless.Thecurrentvalueoftheportfoliois:0580f.?+wherefisthevalueoftheoption.Sincetheportfolioisriskless005412(0580)425fe.?/.?+=.i.e.,180f=.Thevalueoftheoptionistherefore$1.80.Thiscanalsobecalculateddirectlyfromequations(13.2)and(13.3).10625u=.,09375d=.sothat00541209375063451062509375ep.?/-.==..-.103655p-=.and005412036555180fe-.?/=?.?=.Problem13.11.Astockpriceiscurrently$40.Itisknownthatattheendofthreemonthsitwillbeeither$45or$35.Therisk-freerateofinterestwithquarterlycompoundingis8%perannum.Calculatethevalueofathree-monthEuropeanputoptiononthestockwithanexercisepriceof$40.Verifythatno-arbitrageargumentsandrisk-neutralvaluationargumentsgivethesameanswers.Attheendofthreemonthsthevalueoftheoptioniseither$5(ifthestockpriceis$35)or$0(ifthestockpriceis$45).Consideraportfolioconsistingof:shares1option-?:+:(Note:Thedelta,?,ofaputoptionisnegative.Wehaveconstructedtheportfoliosothatitis+1optionand-?sharesratherthan1-optionand+?sharessothattheinitialinvestmentispositive.)Thevalueoftheportfolioiseither355-?+or45-?.If:35545-?+=-?i.e.,05?=-.thevalueoftheportfolioiscertaintobe22.5.Forthisvalueof?theportfolioisthereforeriskless.Thecurrentvalueoftheportfoliois40f-?+wherefisthevalueoftheoption.Sincetheportfoliomustearntherisk-freerateofinterest(4005)102225f?.+?.=.Hence206f=.i.e.,thevalueoftheoptionis$2.06.Thiscanalsobecalculatedusingrisk-neutralvaluation.Supposethatpistheprobabilityofanupwardstockpricemovementinarisk-neutralworld.Wemusthave4535(1)40102pp+-=?.i.e.,1058p=.or:058p=.Theexpectedvalueoftheoptioninarisk-neutralworldis:00585042210?.+?.=.Thishasapresentvalueof210206102.=..Thisisconsistentwiththeno-arbitrageanswer.Problem13.12.Astockpriceiscurrently$50.Overeachofthenexttwothree-monthperiodsitisexpectedtogoupby6%ordownby5%.Therisk-freeinterestrateis5%perannumwithcontinuouscompounding.Whatisthevalueofasix-monthEuropeancalloptionwithastrikepriceof$51?AtreedescribingthebehaviorofthestockpriceisshowninFigureS13.3.Therisk-neutralprobabilityofanupmove,p,isgivenby00531209505689106095ep.?/-.==..-.Thereisapayofffromtheoptionof561851518.-=.forthehighestfinalnode(whichcorrespondstotwoupmoves)zeroinallothercases.Thevalueoftheoptionistherefore2005612518056891635e-.?/.?.?=.ThiscanalsobecalculatedbyworkingbackthroughthetreeasindicatedinFigureS13.3.Thevalueofthecalloptionisthelowernumberateachnodeinthefigure.FigureS13.3:TreeforProblem13.12Problem13.13.ForthesituationconsideredinProblem13.12,whatisthevalueofasix-monthEuropeanputoptionwithastrikepriceof$51?VerifythattheEuropeancallandEuropeanputpricessatisfyput–callparity.IftheputoptionwereAmerican,woulditeverbeoptimaltoexerciseitearlyatanyofthenodesonthetree?ThetreeforvaluingtheputoptionisshowninFigureS13.4.Wegetapayoffof-.=.if-.=.ifthemiddlefinalnodeisreachedandapayoffof51451255875515035065thelowestfinalnodeisreached.Thevalueoftheoptionistherefore2005612.??.?.+.?.=.(06520568904311587504311)1376e-.?/ThiscanalsobecalculatedbyworkingbackthroughthetreeasindicatedinFigureS13.4.Thevalueoftheputplusthestockpriceis.+=.137********Thevalueofthecallplusthepresentvalueofthestrikepriceis005612e-.?/.+=.16355151376Thisverifiesthatput–callparityholdsTotestwhetheritworthexercisingtheoptionearlywecomparethevaluecalculatedfortheoptionateachnodewiththepayofffromimmediateexercise.AtnodeCthepayofffrom-.=..Becausethisisgreaterthan2.8664,theoptionshouldimmediateexerciseis5147535beexercisedatthisnode.TheoptionshouldnotbeexercisedateithernodeAornodeB.FigureS13.4:TreeforProblem13.13Problem13.14.Astockpriceiscurrently$25.Itisknownthatattheendoftwomonthsitwillbeeither$23or$27.Therisk-freeinterestrateis10%perannumwithcontinuouscompounding.SupposeTSisthestockpriceattheendoftwomonths.Whatisthevalueofaderivativethatpaysoff2TSatthistime?Attheendoftwomonthsthevalueofthederivativewillbeeither529(ifthestockpriceis23)or729(ifthestockpriceis27).Consideraportfolioconsistingof:shares1derivative+?:-:Thevalueoftheportfolioiseither27729?-or23529?-intwomonths.If2772923529?-=?-i.e.,50?=thevalueoftheportfolioiscertaintobe621.Forthisvalueof?theportfolioisthereforeriskless.Thecurrentvalueoftheportfoliois:5025f?-wherefisthevalueofthederivative.Sincetheportfoliomustearntherisk-freerateofinterest010212(5025)621fe.?/?-=i.e.,6393f=.Thevalueoftheoptionistherefore$639.3.Thiscanalsobecalculateddirectlyfromequations(13.2)and(13.3).108u=.,092d=.sothat01021209206050108092ep.?/-.==..-.and010212(0605072903950529)6393fe-.?/=.?+.?=.Problem13.15.Calculateu,d,andpwhenabinomialtreeisconstructedtovalueanoptiononaforeigncurrency.Thetreestepsizeisonemonth,thedomesticinterestrateis5%perannum,theforeigninterestrateis8%perannum,andthevolatilityis12%perannum.Inthiscase(005008)11209975ae.-.?/==.010352ue.==.109660du=/=.0997509660045531035209660p.-.==..-.Problem13.16.Thevolatilityofanon-dividend-payingstockwhosepriceis$78,is30%.Therisk-freerateis3%perannum(continuouslycompounded)forallmaturities.Calculatevaluesforu,d,andpwhenatwo-monthtimestepisused.Whatisthevalueofafour-monthEuropeancalloptionwithastrikepriceof$80givenbyatwo-stepbinomialtree.Supposeatradersells1,000options(10contracts).Whatpositioninthestockisnecessarytohedgethetrader’spositionatthetimeofthetrade?4898.08847.01303.18847.08847.0/11303.112/230.01667.030.0=--=====??epudeuThetreeisgiveninFigureS13.5.Thevalueoftheoptionis$4.67.Theinitialdeltais9.58/(88.16–69.01)whichisalmostexactly0.5sothat500sharesshouldbepurchased.FigureS13.5:TreeforProblem13.16Problem13.17.Astockindexiscurrently1,500.Itsvolatilityis18%.Therisk-freerateis4%perannum(continuouslycompounded)forallmaturitiesandthedividendyieldontheindexis2.5%.Calculatevaluesforu,d,andpwhenasix-monthtimestepisused.Whatisthevaluea12-monthAmericanputoptionwithastrikepriceof1,480givenbyatwo-stepbinomialtree.4977.08805.01357.18805.08805.0/11357.15.0)025.004.0(5.018.0=--=====?-?epudeuThetreeisshowninFigureS13.6.Theoptionisexercisedatthelowernodeatthesix-monthpoint.Itisworth78.41.FigureS13.6:TreeforProblem13.17Problem13.18.Thefuturespriceofacommodityis$90.Useathree-steptreetovalue(a)anine-monthAmericancalloptionwithstrikeprice$93and(b)anine-monthAmericanputoptionwithstrikeprice$93.Thevolatilityis28%andtherisk-freerate(allmaturities)is3%withcontinuouscompounding.4651.08694.01503.18694.018694.0/11503.125.028.0=--=====?uudeuThetreeforvaluingthecallisinFigureS13.7aandthatforvaluingtheputisinFigureS13.7b.Thevaluesare7.94and10.88,respectively.824637FigureS13.7a:CallFigureS13.7b:PutFurtherQuestionsProblem13.19.Thecurrentpriceofanon-dividend-payingbiotechstockis$140withavolatilityof25%.Therisk-freerateis4%.Forathree-monthtimestep:(a)Whatisthepercentageupmovement?(b)Whatisthepercentagedownmovement?(c)Whatistheprobabilityofanupmovementinarisk-neutralworld?(d)Whatistheprobabilityofadownmovementinarisk-neutralworld?Useatwo-steptreetovalueasix-monthEuropeancalloptionandasix-monthEuropeanputoption.Inbothcasesthestrikepriceis$150.(a)25.025.0?=eu=1.1331.Thepercentageupmovementis13.31%(b)d=1/u=0.8825.Thepercentagedownmovementis11.75%(c)Theprobabilityofanupmovementis5089.0)8825.1331.1/()8825.()25.004.0=--?e(d)Theprobabilityofadownmovementis0.4911.ThetreeforvaluingthecallisinFigureS13.8aandthatforvaluingtheputisinFigureS13.8b.Thevaluesare7.56and14.58,respectively.FigureS13.8a:CallFigureS13.8b:PutProblem13.20.InProblem13.19,supposethatatradersells10,000Europeancalloptions.Howmanysharesofthestockareneededtohedgethepositionforthefirstandsecondthree-monthperiod?Forthesecondtimeperiod,considerboththecasewherethestockpricemovesupduringthefirstperiodandthecasewhereitmovesdownduringthefirstperiod.Thedeltaforthefirstperiodis15/(158.64–123.55)=0.4273.Thetradershouldtakealongpositionin4,273shares.Ifthereisanupmovementthedeltaforthesecondperiodis29.76/(179.76–140)=0.7485.Thetradershouldincreasetheholdingto7,485shares.Ifthereisadownmovementthetradershoulddecreasetheholdingtozero.Problem13.21.Astockpriceiscurrently$50.Itisknownthatattheendofsixmonthsitwillbeeither$60or$42.Therisk-freerateofinterestwithcontinuouscompoundingis12%perannum.Calculatethevalueofasix-monthEuropeancalloptiononthestockwithanexercisepriceof$48.Verifythatno-arbitrageargumentsandrisk-neutralvaluationargumentsgivethesameanswers.Attheendofsixmonthsthevalueoftheoptionwillbeeither$12(ifthestockpriceis$60)or$0(ifthestockpriceis$42).Consideraportfolioconsistingof:shares1option+?:-:Thevalueoftheportfolioiseither42?or6012?-insixmonths.If426012?=?-i.e.,06667?=.thevalueoftheportfolioiscertaintobe28.Forthisvalueof?theportfolioisthereforeriskless.Thecurrentvalueoftheportfoliois:0666750f.?-wherefisthevalueoftheoption.Sincetheportfoliomustearntherisk-freerateofinterest01205(0666750)28fe.?..?-=i.e.,696f=.Thevalueoftheoptionistherefore$6.96.Thiscanalsobecalculatedusingrisk-neutralvaluation.Supposethatpistheprobabilityofanupwardstockpricemovementinarisk-neutralworld.Wemusthave0066042(1)50ppe.+-=?i.e.,181109p=.or:06161p=.Theexpectedvalueoftheoptioninarisk-neutralworldis:120616100383973932?.+?.=.Thishasapresentvalueof00673932696e-..=.Hencetheaboveanswerisconsistentwithrisk-neutralvaluation.Problem13.22.Astockpriceiscurrently$40.Overeachofthenexttwothree-monthperiodsitisexpectedtogoupby10%ordownby10%.Therisk-freeinterestrateis12%perannumwithcontinuouscompounding.a.Whatisthevalueofasix-monthEuropeanputoptionwithastrikepriceof$42?b.Whatisthevalueofasix-monthAmericanputoptionwithastrikepriceof$42?a.AtreedescribingthebehaviorofthestockpriceisshowninFigureS13.9.Therisk-neutralprobabilityofanupmove,p,isgivenby012312090065231109ep.?/-.==..-.Calculatingtheexpectedpayoffanddiscounting,weobtainthevalueoftheoptionas2012612[24206523034779603477]2118e-.?/.??.?.+.?.=.ThevalueoftheEuropeanoptionis2.118.ThiscanalsobecalculatedbyworkingbackthroughthetreeasshowninFigureS13.9.ThesecondnumberateachnodeisthevalueoftheEuropeanoption.b.ThevalueoftheAmericanoptionisshownasthethirdnumberateachnodeonthetree.Itis2.537.ThisisgreaterthanthevalueoftheEuropeanoptionbecauseitisoptimaltoexerciseearlyatnodeC.40.0002.1182.53744.0000.8100.81036.0004.7596.00048.4000.0000.00039.6002.4002.40032.4009.6009.600ABCFigureS13.9:TreetoevaluateEuropeanandAmericanputoptionsinProblem13.22.Ateachnode,uppernumberisthestockprice,thenextnumberistheEuropeanputprice,andthefinalnumberistheAmericanputpriceProblem13.23.Usinga“trial-and-error”approach,estimatehowhighthestrikepricehastobeinProblem13.17forittobeoptimaltoexercisetheoptionimmediately.Trialanderrorshowsthatimmediateearlyexerciseisoptimalwhenthestrikepriceisabove43.2.Thiscanbealsoshowntobetruealgebraically.Supposethestrikepriceincreasesbyarelativelysmallamountq.ThisincreasesthevalueofbeingatnodeCbyqandthevalueofbeingatnodeBby0030347703374eqq-..=..ItthereforeincreasesthevalueofbeingatnodeAby003(065230337403477)0551qqeq-..?.+.=.ForearlyexerciseatnodeAwerequire253705512qq.+.<+or1196q>..Thiscorrespondstothestrikepricebeinggreaterthan43.196.Problem13.24.Astockpriceiscurrently$30.Duringeachtwo-monthperiodforthenextfourmonthsitisexpectedtoincreaseby8%orreduceby10%.Therisk-freeinterestrateis5%.Useatwo-steptreetocalculatethevalueofaderivativethatpaysoff2[max(300)]TS-,whereTSisthestockpriceinfourmonths?IfthederivativeisAmerican-style,shoulditbeexercisedearly?Thistypeofoptionisknownasapoweroption.AtreedescribingthebehaviorofthestockpriceisshowninFigureS13.10.Therisk-neutralprobabilityofanupmove,p,isgivenby005212090602010809ep.?/-.==..-.Calculatingtheexpectedpayoffanddiscounting,weobtainthevalueoftheoptionas393.5]3980.049.323980.06020.027056.0[12/405.02=?+????-eThevalueoftheEuropeanoptionis5.393.ThiscanalsobecalculatedbyworkingbackthroughthetreeasshowninFigureS13.10.ThesecondnumberateachnodeisthevalueoftheEuropeanoption.EarlyexerciseatnodeCwouldgive9.0whichislessthan13.2435.TheoptionshouldthereforenotbeexercisedearlyifitisAmerican.FigureS13.10:TreetoevaluateEuropeanpoweroptioninProblem13.24.Ateachnode,uppernumberisthestockpriceandthenextnumberistheoptionpriceProblem13.25.ConsideraEuropeancalloptiononanon-dividend-payingstockwherethestockpriceis$40,thestrikepriceis$40,therisk-freerateis4%perannum,thevolatilityis30%perannum,andthetimetomaturityissixmonths.a.Calculateu,d,andpforatwosteptreeb.Valuetheoptionusingatwosteptree.c.VerifythatDerivaGemgivesthesameanswerd.UseDerivaGemtovaluetheoptionwith5,50,100,and500timesteps.0.000029.16000.705624.300032.4900F(a)ThisproblemisbasedonthematerialinSection13.8.Inthiscase025t?=.sothat03011618ue.==.,108607du=/=.,and00402508607049591161808607ep.?.-.==..-.(b)and(c)Thevalueoftheoptionusingatwo-steptreeasgivenbyDerivaGemisshowninFigureS13.11tobe3.3739.TouseDerivaGemchoosethefirstworksheet,selectEquityastheunderlyingtype,andselectBinomialEuropeanastheOptionType.AftercarryingoutthecalculationsselectDisplayTree.(d)With5,50,100,and500timestepsthevalueoftheoptionis3.9229,3.7394,3.7478,and3.7545,respectively.FigureS13.11:TreeproducedbyDerivaGemtoevaluateEuropeanoptioninProblem13.25Problem13.26.RepeatProblem13.25foranAmericanputoptiononafuturescontract.Thestrikepriceandthefuturespriceare$50,therisk-freerateis10%,thetimetomaturityissixmonths,andthevolatilityis40%perannum.(a)Inthiscase025t?=.and04012214ue.==.,108187du=/=.,and0102508187045021221408187ep.?.-.==..-.(b)and(c)Thevalueoftheoptionusingatwo-steptreeis4.8604.(d)With5,50,100,and500timestepsthevalueoftheoptionis5.6858,5.3869,5.3981,and5.4072,respectively.Problem13.27.Footnote1showsthatthecorrectdiscountratetousefortherealworldexpectedpayoffinAteachnode:Uppervalue=UnderlyingAssetPriceLowervalue=OptionPriceValuesinredarearesultofearlyexercise.Strikeprice=40Discountfactorperstep=0.9900Timestep,dt=0.2500years,91.25daysGrowthfactorperstep,a=1.0101Probabilityo