期权期货及其他衍生产品第9版赫尔-7

StochasticProcesses•Describesthewayinwhichavariablesuchasastockprice,exchangerateorinterestratechangesthroughtime•IncorporatesuncertaintiesOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20142Example1•Eachdayastockprice-increasesby$1withprobability30%-staysthesamewithprobability50%-reducesby$1withprobability20%Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20143Example2•Eachdayastockpricechangeisdrawnfromanormaldistributionwithmean$0.2andstandarddeviation$1Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20144MarkovProcesses(Seepages302-303)•InaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegottowhereweare•IstheprocessfollowedbythetemperatureatacertainplaceMarkov?•WeassumethatstockpricesfollowMarkovprocessesOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20145Weak-FormMarketEfficiency・Thisassertsthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.•AMarkovprocessforstockpricesisconsistentwithweak-formmarketefficiencyOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20146Example•Avariableiscurrently40•ItfollowsaMarkovprocess•Processisstationary(i.e.theparametersoftheprocessdonotchangeaswemovethroughtime)•Attheendof1yearthevariablewillhaveanormalprobabilitydistributionwithmean40andstandarddeviation10Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20147Questions•Whatistheprobabilitydistributionofthestockpriceattheendof2years?•years?•%years?•Dtyears?TakinglimitswehavedefinedacontinuousstochasticprocessOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20148Variances&StandardDeviations•InMarkovprocesseschangesinsuccessiveperiodsoftimeareindependent•Thismeansthatvariancesareadditive•StandarddeviationsarenotadditiveOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull20149Variances&StandardDeviations(continued)•Inourexampleitiscorrecttosaythatthevarianceis100peryear.•Itisstrictlyspeakingnotcorrecttosaythatthestandarddeviationis10peryear.Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201410AWienerProcess(Seepages304-305)•Definef(m,v)asanormaldistributionwithmeanmandvariancev•AvariablezfollowsaWienerprocessif-ThechangeinzinasmallintervaloftimeDtisDz___-A-ThevaluesofDzforany2different(nonoverlapping)periodsoftimeareindependentOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201411PropertiesofaWienerProcess•Meanof[z(T)-z(0)]is0•Varianceof[z(T)-z(0)]isT•Standarddeviationof[z

⑺-z(o)]nsOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201412GeneralizedWienerProcesses(See308)•AWienerprocesshasadriftrate(i.e.averagechangeperunittime)of0andavariancerateof1•InageneralizedWienerprocessthedriftrateandthevarianceratecanbesetequaltoanychosenconstantsOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201413GeneralizedWienerProcesses(continued)Ax=6Z+/)£•Meanchangeinxperunittimeisa•VarianceofchangeinxperunittimeisbOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201414TheExampleRevisited•Astockpricestartsat40andhasaprobabilitydistributionoff(40,100)attheendoftheyear•IfweassumethestochasticprocessisMarkovwithnodriftthentheprocessisdS=10dz•Ifthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionisf(48,100),theprocesswouldbedS=8dt

+10dzOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201416ItoProcess(Seepages308)InanItoprocessthedriftrateandthevarianceratearefunctionsoftimedx=a(x,t)dt+b(x，t)dzThediscretetimeequivalentAx=€Z(X,+b(x,/)8a/aistrueinthelimitasDttendstozeroOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201417WhyaGeneralizedWienerProcessIsNotAppropriateforStocks•Forastockpricewecanconjecturethatitsexpectedpercentagechangeinashortperiodoftimeremainsconstant(notitsexpectedactualchange)•WecanalsoconjecturethatouruncertaintyastothesizeoffuturestockpricemovementsisproportionaltothelevelofthestockpriceOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201418InterestRates•Whatwouldbeareasonablestochasticprocesstoassumefortheshort-terminterestrate?Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201420MonteCarloSimulation•Wecansamplerandompathsforthestockpricebysamplingvaluesfore•Supposem=0.15,s=0.30,andDt=1week(=1/52or0.192years),thenA5=0.15x0.01925+0.30xVO.0192^=0.002885+0.04165^Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201421MonteCarloSimulation-SamplingonePath(SeeTable14.1,page311)WeekStockPriceatStartofPeriodRandomSampleforeChangeinStockPrice,AS0100.000.522.451102.451.446.432108.88-0.86-3.583105.301.466.704112.00-0.69-2.89Options,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201422CorrelatedProcessesSupposedzanddz2areWienerprocesseswithcorrelationrThenwheree1ande2arerandomsamplesfromabivariatestandardnormaldistributionwherecorrelationispOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201423Ito'sLemma(Seepages313-315)•Ifweknowthestochasticprocessfollowedbyx,Ito'slemmatellsusthestochasticprocessfollowedbysomefunctionG(x,t).Whendx=a(x,t)dt+b(x,t)dzthendGdGdt+a2gax1b2dGdxbdz•Sinceaderivativeisafunctionofthepriceoftheunderlyingassetandtime,Ito'slemmaplaysanimportantpartintheanalysisofderivativesOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201424IndicationofWhylto'sLemmaisTrue•ATaylor'sseriesexpansionofG(x,t)givesAG=dGdGd2G2-Ax+-AZ+-Axdxdtdxd2GdZd2G2+Aa?Af-—At+...dxdtdtOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201425gnoringTermsofHigherOrderThanDtInordinarycalculuswehavedGdGAG=-Ax+-AtdxdtInstochasticcalculusthisbecomesAG=dGdGd/d2G2-Ax+-At+/2.-Axdxdtdx2becauseAxhasacomponentwhichisoforderVazOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201426SubstitutingforDxSupposedx=a[x，t}dt+b{x,t)dzsothatAx=aAt+bsThenignoringtermsofhigherorderthanAtAG=dxdtdxOptions,Futures,andOtherDerivatives,9thEdition,Copyright©JohnC.Hull201427TakingLiTakingl