Monte_Carlo蒙特卡洛算法模拟

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1、MonteCarloSimulationFE5107RiskAnalysisandManagementInstructor:XianhuaPengDepartmentofMathematicsHKUSTmaxhpeng@ust.hkFall2014XianhuaPeng(HKUST)MonteCarloSimulationFall20141/59Outline1GeneratingMultivariateNormalandtRandomVectors2SimulatingStockPriceUnderBlack-ScholesModel3TheMonteCarlo

2、FrameworkandOutputAnalysis4CalculatingVaRviaSimulation5ImportanceSampling6RelativeErrorsXianhuaPeng(HKUST)MonteCarloSimulationFall20142/59GeneratingMultivariateNormalandtRandomVectorsOutline1GeneratingMultivariateNormalandtRandomVectors2SimulatingStockPriceUnderBlack-ScholesModel3TheM

3、onteCarloFrameworkandOutputAnalysis4CalculatingVaRviaSimulation5ImportanceSampling6RelativeErrorsXianhuaPeng(HKUST)MonteCarloSimulationFall20143/59GeneratingMultivariateNormalandtRandomVectorsGenerateMultivariateNormalRandomVectorsWewanttosimulateX=(X1,...,Xn)T∼dNn(µ,Σ)LetZ=(Z1,...,Zn

4、)T∼dNn(0,In).WeknowhowtosimulateZ.n×ndTLetA∈R.Weknowµ+AZ∼Nn(µ,AA)TdIfwefindAsuchthatAA=Σ,thenweobtainµ+AZ∼Nn(µ,Σ).HowtofindsuchaA?XianhuaPeng(HKUST)MonteCarloSimulationFall20144/59GeneratingMultivariateNormalandtRandomVectorsSpectralDecompositionofΣSpectraldecompositionofreal-valuedsymm

5、etricmatrixΣ:Σ=ΓΛΓT,whereΛ=diag{λ1,...,λn}AcanbechosentobeppA=Γdiag{λ1,...,λn}XianhuaPeng(HKUST)MonteCarloSimulationFall20145/59GeneratingMultivariateNormalandtRandomVectorsCholeskyDecompositionofΣCholeskydecomposition:anypositivesemi-definitesymmetricmatrixΣcanbedecomposedintoΣ=AAT,wh

6、ereAisalowertriangularmatrixwithallelementsabovediagonalbeingzero.The3-dimensionalcase:σ11σ12σ13a1100a11a21a31σ21σ22σ23=a21a2200a22a32σ31σ32σ33a31a32a3300a33MatlabfunctionforCholeskydecomposition:cholBecareful!IfC=chol(Σ),thenΣ=CTC!MatlabfunctionforsimulatingNn(µ,Σ):mvnrnd

7、XianhuaPeng(HKUST)MonteCarloSimulationFall20146/59GeneratingMultivariateNormalandtRandomVectorsGenerateMultivariatetRandomVectorsTherandomvectorXhasamultivariatetdistributionwithdofν(denotedbytn(ν,µ,Σ)),ifHX=µ+p,whereV/νdH∼Nn(0,Σ)d2V∼χνandVisindependentofZXcanbesimulatedbysimulatingHa

8、ndVre