布朗运动与伊藤公式课件.ppt

《布朗运动与伊藤公式课件.ppt》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、Chapter4BrownianMotion&ItôFormulaStochasticProcessThepricemovementofanunderlyingassetisastochasticprocess.TheFrenchmathematicianLouisBachelierwasthefirstonetodescribethestocksharepricemovementasaBrownianmotioninhis1900doctoralthesis.introductiontotheBrownianm

2、otionderivethecontinuousmodelofoptionpricinggivingthedefinitionandrelevantpropertiesBrownianmotionderivestochasticcalculusbasedontheBrownianmotionincludingtheItointegral&Itoformula.Allofthedescriptionanddiscussionemphasizeclarityratherthanmathematicalrigor.Co

3、in-tossingProblemDefinearandomvariableItiseasytoshowthatithasthefollowingproperties:&areindependentRandomVariableWiththerandomvariable,definearandomvariableandarandomsequenceRandomWalkConsideratimeperiod[0,T],whichcanbedividedintoNequalintervals.LetΔ=TN,t_n=

4、nΔ,(n=0,1,cdots,N),thenArandomwalkisdefinedin[0,T]:iscalledthepathoftherandomwalk.DistributionofthePathLetT=1,N=4,Δ=1/4,FormofPaththepathformedbylinearinterpolationbetweentheaboverandompoints.ForΔ=1/4case,thereare2^4=16paths.tS1PropertiesofthePathCentralLimi

5、tTheoremForanyrandomsequencewheretherandomvariableX~N(0,1),i.e.therandomvariableXobeysthestandardnormaldistribution:E(X)=0,Var(X)=1.ApplicationofCentralLimitThem.ConsiderlimitasΔ→0.DefinitionofWinnerProcess(BrownianMotion)1)Continuityofpath:W(0)=0,W(t)isacon

6、tinuousfunctionoft.2)Normalincrements:Foranyt>0,W(t)~N(0,t),andfor0

7、ediscountedvalueofanunderlyingassetasfollows:intimeinterval[t,t+Δt],theBTMcanbewrittenasLemmaIfud=1,σisthevolatility,lettingthenunderthemartingalemeasureQ,ProofoftheLemmaAccordingtothedefinitionofmartingalemeasureQ,on[t,t+Δt],thusbystraightforwardcomputation,

8、ProofoftheLemmaMoreover,sinceProofoftheLemmacont.bytheassumptionofthelemma,inputthesevaluestotheori.equation.Thiscompletestheproofofthelemma.GeometricBrownianMotionByTaylorexpansionneglec