期货期权及其衍生品配套课件(全34章)Ch12PPT文档格式.ppt
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期货期权及其衍生品配套课件(全34章)Ch12PPT文档格式.ppt
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WienerProcessesandItsLemma,Chapter12,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,1,TypesofStochasticProcesses,Discretetime;@#@discretevariableDiscretetime;@#@continuousvariableContinuoustime;@#@discretevariableContinuoustime;@#@continuousvariable,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,2,ModelingStockPrices,WecanuseanyofthefourtypesofstochasticprocessestomodelstockpricesThecontinuoustime,continuousvariableprocessprovestobethemostusefulforthepurposesofvaluingderivatives,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,3,MarkovProcesses(Seepages259-60),InaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegotwhereweareWeassumethatstockpricesfollowMarkovprocesses,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,4,Weak-FormMarketEfficiency,Thisassertsthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.AMarkovprocessforstockpricesisconsistentwithweak-formmarketefficiency,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,5,ExampleofaDiscreteTimeContinuousVariableModel,Astockpriceiscurrentlyat$40Attheendof1yearitisconsideredthatitwillhaveanormalprobabilitydistributionofwithmean$40andstandarddeviation$10,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,6,Questions,Whatistheprobabilitydistributionofthestockpriceattheendof2years?
@#@years?
@#@years?
@#@Dtyears?
@#@Takinglimitswehavedefinedacontinuousvariable,continuoustimeprocess,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,7,Variances&@#@StandardDeviations,InMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthatvariancesareadditiveStandarddeviationsarenotadditive,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,8,Variances&@#@StandardDeviations(continued),Inourexampleitiscorrecttosaythatthevarianceis100peryear.Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,9,AWienerProcess(Seepages261-63),WeconsideravariablezwhosevaluechangescontinuouslyDefinef(m,v)asanormaldistributionwithmeanmandvariancevThechangeinasmallintervaloftimeDtisDzThevariablefollowsaWienerprocessifThevaluesofDzforany2different(non-overlapping)periodsoftimeareindependent,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,10,PropertiesofaWienerProcess,Meanofz(T)z(0)is0Varianceofz(T)z(0)isTStandarddeviationofz(T)z(0)is,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,11,TakingLimits.,Whatdoesanexpressioninvolvingdzanddtmean?
@#@ItshouldbeinterpretedasmeaningthatthecorrespondingexpressioninvolvingDzandDtistrueinthelimitasDttendstozeroInthisrespect,stochasticcalculusisanalogoustoordinarycalculus,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,12,GeneralizedWienerProcesses(Seepage263-65),AWienerprocesshasadriftrate(i.e.averagechangeperunittime)of0andavariancerateof1InageneralizedWienerprocessthedriftrateandthevarianceratecanbesetequaltoanychosenconstants,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,13,GeneralizedWienerProcesses(continued),ThevariablexfollowsageneralizedWienerprocesswithadriftrateofaandavariancerateofb2ifdx=adt+bdz,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,14,GeneralizedWienerProcesses(continued),MeanchangeinxintimeTisaTVarianceofchangeinxintimeTisb2TStandarddeviationofchangeinxintimeTis,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,15,TheExampleRevisited,Astockpricestartsat40andhasaprobabilitydistributionoff(40,100)attheendoftheyearIfweassumethestochasticprocessisMarkovwithnodriftthentheprocessisdS=10dzIfthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionisf(48,100),theprocesswouldbedS=8dt+10dz,Options,Futures,andOtherDerivatives,7thInternationalEdition,CopyrightJohnC.Hull2008,16,ItProcess(Seepages265),InanItprocessthedriftrateandthevarianceratearefunctionsoftimedx=a(x,t)dt+b(x,t)dzThediscretetimeequivalentisonlytrueinthel
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