衍生工具与风险管理第4章课件

D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:1Chapter4:OptionPricingModels:TheBinomialModelYoucanthinkofaderivativeasamixtureofitsconstituentunderliersmuchasacakeisamixtureofeggs,flour,andmilkincarefullyspecifiedproportions.Thederivative’smodelprovidesarecipeforthemixture,onewhoseingredients’quantitiesvarywithtime.EmanuelDermanRisk,July,2001,p.48D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:2ImportantConceptsinChapter4TheconceptofanoptionpricingmodelTheone-andtwo-periodbinomialoptionpricingmodelsExplanationoftheestablishmentandmaintenanceofarisk-freehedgeIllustrationofhowearlyexercisecanbecapturedTheextensionofthebinomialmodeltoanynumberoftimeperiodsAlternativespecificationsofthebinomialmodelD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:3DefinitionofamodelAsimplifiedrepresentationofrealitythatusescertaininputstoproduceanoutputorresultDefinitionofanoptionpricingmodelAmathematicalformulathatusesthefactorsthatdetermineanoption’spriceasinputstoproducethetheoreticalfairvalueofanoption.D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:4TheOne-PeriodBinomialModelConditionsandassumptionsOneperiod,twooutcomes(states)S=currentstockpriceu=1+returnifstockgoesupd=1+returnifstockgoesdownr=risk-freerateValueofEuropeancallatexpirationoneperiodlaterCu=Max(0,Su-X)orCd=Max(0,Sd-X)SeeFigure4.1,p.98D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:5TheOne-PeriodBinomialModel(continued)Importantpoint:d1+rutopreventarbitrageWeconstructahedgeportfolioofhsharesofstockandoneshortcall.Currentvalueofportfolio:V=hS-CAtexpirationthehedgeportfoliowillbeworthVu=hSu-CuVd=hSd-CdIfwearehedged,thesemustbeequal.SettingVu=VdandsolvingforhgivesD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:6TheOne-PeriodBinomialModel(continued)ThesevaluesareallknownsohiseasilycomputedSincetheportfolioisriskless,itshouldearntherisk-freerate.ThusV(1+r)=Vu(orVd)SubstitutingforVandVu(hS-C)(1+r)=hSu-CuAndthetheoreticalvalueoftheoptionisSdSuCChduD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:7TheOne-PeriodBinomialModel(continued)Thisisthetheoreticalvalueofthecallasdeterminedbythestockprice,exerciseprice,risk-freerate,andupanddownfactors.Theprobabilitiesoftheupanddownmoveswereneverspecified.Theyareirrelevanttotheoptionprice.d)-d)/(u-r(1=pwherer1p)C(1pCCduD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:8TheOne-PeriodBinomialModel(continued)AnIllustrativeExampleS=100,X=100,u=1.25,d=0.80,r=.07FirstfindthevaluesofCu,Cd,h,andp:Cu=Max(0,100(1.25)-100)=Max(0,125-100)=25Cd=Max(0,100(.80)-100)=Max(0,80-100)=0h=(25-0)/(125-80)=.556p=(1.07-0.80)/(1.25-0.80)=.6TheninsertintotheformulaforC:14.021.070.0).4((.6)25CD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:9TheOne-PeriodBinomialModel(continued)AHedgedPortfolioShort1,000callsandlong1000h=1000(.556)=556shares.SeeFigure4.2,p.101.Valueofinvestment:V=556($100)-1,000($14.02)=$41,580.(Thisishowmuchmoneyyoumustputup.)Stockgoesto$125Valueofinvestment=556($125)-1,000($25)=$44,500Stockgoesto$80Valueofinvestment=556($80)-1,000($0)=$44,480D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:10TheOne-PeriodBinomialModel(continued)AnOverpricedCallLetthecallbesellingfor$15.00Youramountinvestedis556($100)-1,000($15.00)=$40,600Youwillstillendupwith$44,500,whichisa9.6%return.Everyonewilltakeadvantageofthis,forcingthecallpricetofallto$14.02Youinvested$41,580andgotback$44,500,a7%return,whichistherisk-freerate.D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:11AnUnderpricedCallLetthecallbepricedat$13Sellshort556sharesat$100andbuy1,000callsat$13.Thiswillgenerateacashinflowof$42,600.Atexpiration,youwillenduppayingout$44,500.Thisislikealoaninwhichyouborrowed$42,600andpaidback$44,500,arateof4.46%,whichbeatstherisk-freeborrowingrate.TheOne-PeriodBinomialModel(continued)D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:12TheTwo-PeriodBinomialModelWenowletthestockgoupanotherperiodsothatitendsupSu2,SudorSd2.SeeFigure4.3,p.105.Theoptionexpiresaftertwoperiodswiththreepossiblevalues:X]SdMax[0,CX]SudMax[0,CX]SuMax[0,C2dud2u22D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:13Afteroneperiodthecallwillhaveoneperiodtogobeforeexpiration.Thus,itwillwortheitherofthefollowingtwovaluesThepriceofthecalltodaywillbeTheTwo-PeriodBinomialModel(continued)r1p)C(1pCCor,r1p)C(1pCC22ddududuuD.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:14TheTwo-PeriodBinomialModel(continued)2d2udu2dur)(1Cp)(1p)C2p(1CpCaswrittenbealsocanwhichr1p)C(1pCC22•Thehedgeratiosaredifferentinthedifferentstates:2dudd2uduuduSdSudCCh,SudSuCCh,SdSuCCh22D.M.ChanceAnIntroductiontoDerivativesandRiskManagement,6thed.Ch.4:15TheTwo-PeriodBinomialModel(continued)AnIllustrativeExampleSu2=100(1.25)2=156.25Sud=100(1.25)(.80)=100Sd2=100(.80)2=64Thecalloptionprice