4.布朗运动与伊藤公式.ppt

4.布朗运动与伊藤公式.ppt

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Chapter4BrownianMotion&ItFormulaStochasticProcessnThepricemovementofanunderlyingassetisastochasticprocess.nTheFrenchmathematicianLouisBachelierwasthefirstonetodescribethestocksharepricemovementasaBrownianmotioninhis1900doctoralthesis.nintroductiontotheBrownianmotionnderivethecontinuousmodelofoptionpricingngivingthedefinitionandrelevantpropertiesBrownianmotionnderivestochasticcalculusbasedontheBrownianmotionincludingtheItointegral&Itoformula.nAllofthedescriptionanddiscussionemphasizeclarityratherthanmathematicalrigor.Coin-tossingProblemnDefinearandomvariablenItiseasytoshowthatithasthefollowingproperties:

n&areindependentRandomVariablenWiththerandomvariable,definearandomvariableandarandomsequencenRandomWalknConsideratimeperiod0,T,whichcanbedividedintoNequalintervals.Let=TN,t_n=n,（n=0,1,cdots,N）,thennArandomwalkisdefinedin0,T:

niscalledthepathoftherandomwalk.DistributionofthePathnLetT=1,N=4,=1/4,FormofPathnthepathformedbylinearinterpolationbetweentheaboverandompoints.For=1/4case,thereare24=16paths.tS1PropertiesofthePathCentralLimitTheoremnForanyrandomsequencewheretherandomvariableXN（0,1）,i.e.therandomvariableXobeysthestandardnormaldistribution:

E（X）=0,Var（X）=1.ApplicationofCentralLimitThem.nConsiderlimitas0.DefinitionofWinnerProcess（BrownianMotion）n1）Continuityofpath:

W（0）=0,W（t）isacontinuousfunctionoft.n2）Normalincrements:

Foranyt0,W（t）N（0,t）,andfor0s0（0）denotingthenumberofsharesbought（sold）attimet.Forachoseninvestmentstrategy,whatisthetotalprofitatt=T?

AnExamplecont.nPartition0,Tby:

nIfthetransactionsareexecutedattimeonly,thentheinvestmentstrategycanonlybeadjustedontradingdays,andthegain（loss）atthetimeintervalisnThereforethetotalprofitin0,TisDefinitionofItIntegralnIff（t）isanon-anticipatingstochasticprocess,suchthatthelimitexists,andisindependentofthepartition,thenthelimitiscalledtheItIntegraloff（t）,denotedasRemarkofItIntegralnDef.oftheItoIntegraloneoftheRiemannintegral.n-theRiemannsumunderaparticularpartition.nHowever,f（t）-non-anticipating,nHenceinthevalueoffmustbetakenattheleftendpointoftheinterval,notatanarbitrarypointin.nBasedonthequadraticvarianceThem.4.1thatthevalueofthelimitoftheRiemannsumofaWienerprocessdependsonthechoiceoftheinterpoints.nSo,foraWienerprocess,iftheRiemannsumiscalculatedoverarbitrarilypointin,theRiemannsumhasnolimit.RemarkofItIntegral2nIntheaboveproofprocess:

sincethequadraticvariationofaBrownianmotionisnonzero,theresultofanItointegralisnotthesameastheresultofanormalintegral.ItoDifferentialFormulannThisindicatesacorrespondingchangeinthedifferentiationruleforthecompositefunction.ItFormulanLet,whereisastochasticprocess.WewanttoknownThisistheItoformulatobediscussedinthissection.TheItoformulaistheChainRuleinstochasticcalculus.CompositeFunctionofaStochasticProcessnThedifferentialofafunctionisthelinearprincipalpartofitsincrement.DuetothequadraticvariationtheoremoftheBrownianmotion,acompositefunctionofastochasticprocesswillhavenewcomponentsinitslinearprincipalpart.Letusbeginwithafewexamples.ExpansionnBytheTaylorexpansion,nThenneglectingthehigherorderterms,Examplen1DifferentialofRiskyAssetnInarisk-neutralworld,thepricemovementofariskyassetcanbeexpressedby,nWewanttofinddS（t）=?

DifferentialofRiskyAssetcont.nStochasticDifferentialEquationnInarisk-neutralworld,theunderlyingassetsatisfiesthestochasticdifferentialequationwhereisthereturnofoveratimeintervaldt,rdtistheexpectedgrowthofthereturnof,andisthestochasticcomponentofthereturn,withvariance.iscalledvolatility.Theorem4.2（ItoFormula）nVisdifferentiablebothvariables.IfsatisfiesSDEthenProofofTheorem4.2nBytheTaylorexpansionnButProofofTheorem4.2cont.nSubstitutingitintoori.Equ.,wegetnnThusItoformulaistrue.Theorem4.3nIfarestochasticprocessessatisfyingrespectivelythefollowingSDEnthenProofofTheorem4.3nnBytheItoformula,ProofofTheorem4.3cont.nSubstitutingthemintoaboveformulanThustheTheorem4.3isproved.Theorem4.4nIfarestochasticprocessessatisfyingtheaboveSDE,thennProofofTheorem4.4nByItoformulanProofofTheorem4.4cont.nThusbyTheorem4.3,wehavennTheoremisproved.RemarknTheorems4.3-4.4tellus:

nDuetothechangeintheChainRulefordifferentiatingcompositefunctionoftheWienerprocess,theproductruleandquotientrulefordifferentiatingfunctionsoftheWienerprocessarealsochanged.nAlltheseresultsremindusthatstochasticcalculusoperationsaredifferentfromthenormalcalculusoperations!

MultidimensionalItformul