4.布朗运动与伊藤公式.ppt
- 文档编号:2211921
- 上传时间:2022-10-27
- 格式:PPT
- 页数:53
- 大小:358KB
4.布朗运动与伊藤公式.ppt
《4.布朗运动与伊藤公式.ppt》由会员分享,可在线阅读,更多相关《4.布朗运动与伊藤公式.ppt(53页珍藏版)》请在冰豆网上搜索。
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
- 配套讲稿:
如PPT文件的首页显示word图标,表示该PPT已包含配套word讲稿。双击word图标可打开word文档。
- 特殊限制:
部分文档作品中含有的国旗、国徽等图片,仅作为作品整体效果示例展示,禁止商用。设计者仅对作品中独创性部分享有著作权。
- 关 键 词:
- 布朗运动 公式