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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)TdIfwe?ndAsuchthatAA=Σ,thenweobtainμ+AZ～Nn(μ,Σ).Howto?ndsuchaA?XianhuaPeng(HKUST)MonteCarloSimulationFall20144/59GeneratingMultivariateNormalandtRandomVectorsSpectralDecompositionofΣSpectraldecompositionofreal-valuedsymm
5、etricmatrixΣ:Σ=ΓΛΓT,whereΛ=diag{λ1,...,λn}AcanbechosentobeppA=Γdiag{λ1,...,λn}XianhuaPeng(HKUST)MonteCarloSimulationFall20145/59GeneratingMultivariateNormalandtRandomVectorsCholeskyDecompositionofΣCholeskydecomposition:anypositivesemi-de?nitesymmetricmatrixΣcanbedecomposedintoΣ=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
