期权及其衍生市场-chapter

2022/10/29FinancialEngineeringChapter10

ModeloftheBehavior

ofStockPrices

2022/10/29FinancialEngineeringStochasticProcessesAstochasticprocessisavariablethatevolvesovertimeinawaythatisatleastinpartrandom.i.e.temperatureandIBMstockpriceAstochasticprocessisdefinedbyaprobabilitylawfortheevolutionxtofavariableovertimet.Forgiventimes,wecancalculatetheprobabilitythatthecorrespondingvaluesx1,x2,x3,etc.lieinsomespecifiedrange.2022/10/29FinancialEngineeringCategorizationofStochasticProcessesDiscretetime;discretevariableRandomwalk:ifcanonlytakeondiscretevaluesDiscretetime;continuousvariable

isanormallydistributedrandomvariablewithzeromean.Continuoustime;discretevariableContinuoustime;continuousvariable2022/10/29FinancialEngineeringModelingStockPricesWecanuseanyofthefourtypesofstochasticprocessestomodelstockpricesThecontinuoustime,continuousvariableprocessprovestobethemostusefulforthepurposesofvaluingderivativesecurities2022/10/29FinancialEngineeringMarkovProcessesInaMarkovprocessfuturemovementsinavariabledependonlyonwhereweare,notthehistoryofhowwegotwhereweareWewillassumethatstockpricesfollowMarkovprocesses2022/10/29FinancialEngineeringWeak-FormMarketEfficiencyTheassertionisthatitisimpossibletoproduceconsistentlysuperiorreturnswithatradingrulebasedonthepasthistoryofstockprices.Inotherwordstechnicalanalysisdoesnotwork.AMarkovprocessforstockpricesisclearlyconsistentwithweak-formmarketefficiency2022/10/29FinancialEngineeringExampleofaDiscreteTimeContinuousVariableModelAstockpriceiscurrentlyat$40Attheendof1yearitisconsideredthatitwillhaveaprobabilitydistributionof f(40,10)wheref(m,s)isanormaldistributionwithmeanmandstandarddeviations.2022/10/29FinancialEngineeringQuestionsWhatistheprobabilitydistributionofthestockpriceattheendof 2years? ½years? ¼years?

Dtyears?Takinglimitswehavedefinedacontinuousvariable,continuoustimeprocess2022/10/29FinancialEngineeringVariances&StandardDeviationsInMarkovprocesseschangesinsuccessiveperiodsoftimeareindependentThismeansthatvariancesareadditiveStandarddeviationsarenotadditive2022/10/29FinancialEngineeringVariances&StandardDeviations(continued)Inourexampleitiscorrecttosaythatthevarianceis100peryear.Itisstrictlynotcorrecttosaythatthestandarddeviationis10peryear.2022/10/29FinancialEngineeringAWienerProcess(BrownianMotion)WeconsideravariablezwhosevaluechangescontinuouslyThechangeinasmallintervaloftimeDtisDz

ThevariablefollowsaWienerprocessif 1. 2.ThevaluesofDzforany2different(non-overlapping)periodsoftimeareindependent2022/10/29FinancialEngineeringPropertiesofaWienerProcessMeanof[z(T)–

z(0)]is0Varianceof[z(T)–

z(0)]isT

Standarddeviationof[z(T)–

z(0)]is2022/10/29FinancialEngineeringTakingLimits...Whatdoesanexpressioninvolvingdzanddtmean?ItshouldbeinterpretedasmeaningthatthecorrespondingexpressioninvolvingDzandDtistrueinthelimitasDttendstozeroInthisrespect,stochasticcalculusisanalogoustoordinarycalculus

2022/10/29FinancialEngineeringGeneralizedWienerProcessesAWienerprocesshasadriftrate(ieaveragechangeperunittime)of0andavariancerateof1InageneralizedWienerprocessthedriftrate&thevarianceratecanbesetequaltoanychosen constants2022/10/29FinancialEngineeringGeneralizedWienerProcesses

(continued)

ThevariablexfollowsageneralizedWienerprocesswithadriftrateofa&avariancerateofb2ifdx=adt+bdz

or:x(t)=x0+at+bz(t)2022/10/29FinancialEngineeringGeneralizedWienerProcesses

(continued)MeanchangeinxintimeTisaTVarianceofchangeinxintimeTisb2TStandarddeviationofchangeinxintimeTis2022/10/29FinancialEngineeringTheExampleRevisitedAstockpricestartsat40&hasaprobabilitydistributionoff(40,10)attheendoftheyearIfweassumethestochasticprocessisMarkovwithnodriftthentheprocessisdS=10dzIfthestockpricewereexpectedtogrowby$8onaverageduringtheyear,sothattheyear-enddistributionisf(48,10),theprocessisdS=8dt+10dz2022/10/29FinancialEngineeringWhy?(1)It’stheonlywaytomakethevarianceof(xT-x0)dependonTandnotonthenumberofsteps.1.DividetimeupintondiscreteperiodsoflengthΔt,n=T/Δt.IneachperiodthevariablexeithermovesupordownbyanamountΔhwiththeprobabilitiesofpandqrespectively.2022/10/29FinancialEngineeringWhy?(2)2.thedistributionforthefuturevaluesofx:E(Δx)=(p-q)ΔhE[(Δx)2]=p(Δh)2+q(-Δh)2So,thevarianceofΔxis:E[(Δx)2]-[E(Δx)]2=[1-(p-q)2](Δh)2=4pq(Δh)23.Sincethesuccessivestepsoftherandomwalkareindependent,thecumulatedchange(xT-x0)isabinomialrandomwalkwithmean:n(p-q)Δh=t(p-q)Δh/Δtandvariance:n[1-(p-q)2](Δh)2=

4pqt(Δh)2/Δt2022/10/29FinancialEngineeringWhy?(3)WhenletΔtgotozero,wewouldlikethemeanandvarianceof(xT-x0)toremainunchanged,andtobeindependentoftheparticularchoiceofp,q,ΔhandΔt.Theonlywaytogetitistoset:

and

then2022/10/29FinancialEngineeringWhy?(4)WhenΔtgoestozero,thebinomialdistributionconvergestoanormaldistribution,withmeanandvariance

2022/10/29FinancialEngineeringSamplepath(a=0.2peryear,b=1.0peryear)Takingatimeintervalofonemonth,thencalculatingatrajectoryforxtusingtheequation:

Atrendof0.2peryearimpliesatrendof0.0167permonth.Astandarddeviationof1.0peryearimpliesavarianceof1.0peryear,andhenceavarianceof0.0833permonth,sothatthestandarddeviationinmonthlytermsis0.2887.SeeInvestmentunderuncertainty,p662022/10/29FinancialEngineeringForecastusinggeneralizedBrownianMotionGiventhevalueofx(t)forDec.1974,X1974,theforecastedvalueofxforatimeTmonthsbeyondDec.1974isgivenby:SeeInvestmentunderuncertainty,p67Inthelongrun,thetrendisthedominantdeterminantofBrownianMotion,wherasintheshortrun,thevolatilityoftheprocessdominates.2022/10/29FinancialEngineeringWhyaGeneralizedWienerProcessisnotAppropriateforStocksForastockpricewecanconjecturethatitsexpectedproportionalchangeinashortperiodoftimeremainsconstantnotitsexpectedabsolutechangeinashortperiodoftimeWecanalsoconjecturethatouruncertaintyastothesizeoffuturestockpricemovementsisproportionaltothelevelofthestockpriceThepriceofastockneverfallbelowzero.2022/10/29FinancialEngineering

ItoProcessInanItoprocessthedriftrateandthevarianceratearefunctionsoftimedx=a(x,t)dt+b(x,t)dz

or:ThediscretetimeequivalentisonlytrueinthelimitasDttendstozero2022/10/29FinancialEngineeringAnItoProcessforStockPrices

wheremistheexpectedreturnsisthevolatility.Thediscretetimeequivalentis2022/10/29FinancialEngineeringMonteCarloSimulationWecansamplerandompathsforthestockpricebysamplingvaluesforeSupposem=0.14,s=0.20,andDt=0.01,then2022/10/29FinancialEngineeringMonteCarloSimulation–OnePath

2022/10/29FinancialEngineeringIto’sLemmaIfweknowthestochasticprocessfollowedbyx,Ito’slemmatellsusthestochasticprocessfollowedbysomefunctionG(x,t)Sinceaderivativesecurityisafunctionofthepriceoftheunderlying&time,Ito’slemmaplaysanimportantpartintheanalysisofderivativesecurities2022/10/29FinancialEngineeringTaylorSeriesExpansionATaylor’sseriesexpansionofG(x,t)gives2022/10/29FinancialEngineeringIgnoringTermsofHigherOrderThanDt2022/10/29FinancialEngineeringSubstitutingforDx2022/10/29FinancialEngineeringThee2DtTerm2022/10/29FinancialEngi