Handbook of Volatility Models and Their Applications

548 Pages·6.948 MB·English

by  Bauwens L., Hafner C., Laurent S. (Eds.)

Handbook of
Volatility Models
and Their
Applications

Handbook of
Volatility Models
and Their
Applications
Edited by
Luc Bauwens
Christian Hafner
Sebastien Laurent
A John Wiley & Sons, Inc., Publication
Copyright©2012byJohnWiley&Sons,Inc.Allrightsreserved
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LibraryofCongressCataloging-in-PublicationData:
Bauwens,Luc,1952–
Handbookofvolatilitymodelsandtheirapplications/LucBauwens,
ChristianHafner,SebastienLaurent.
p.cm.–(Wileyhandbooksinfinancialengineeringandeconometrics;3)
Includesbibliographicalreferencesandindex.
ISBN978-0-470-87251-2(hardback)
1. Banksandbanking–Econometricmodels. 2. Finance–Econometricmodels.
3. GARCHmodel. I. Hafner,Christian. II. Laurent,Sebastien,1974–III.
Title.
HG1601.B382012
332.01(cid:2)5195--dc23
2011044256
PrintedintheUnitedStatesofAmerica
10987654321
Contents
Preface xvii
Contributors xix
1 Volatility Models 1
1.1 Introduction, 1
1.2 GARCH, 1
1.2.1 UnivariateGARCH, 1
1.2.1.1 StructureofGARCHModels, 3
1.2.1.2 EarlyGARCHModels, 5
1.2.1.3 ProbabilityDistributionsforz , 7
t
1.2.1.4 NewGARCHModels, 9
1.2.1.5 ExplanationofVolatility
Clustering, 15
1.2.1.6 LiteratureandSoftware, 16
1.2.1.7 ApplicationsofUnivariate
GARCH, 16
1.2.2 MultivariateGARCH, 18
1.2.2.1 StructureofMGARCHModels, 19
1.2.2.2 ConditionalCorrelations, 19
1.2.2.3 FactorModels, 23
1.3 StochasticVolatility, 25
1.3.1 LeverageEffect, 26
1.3.2 Estimation, 27
1.3.3 MultivariateSVModels, 28
1.3.4 ModelSelection, 30
1.3.5 EmpiricalExample:S&P500, 31
1.3.6 Literature, 32
1.4 RealizedVolatility, 33
1.4.1 RealizedVariance, 33
1.4.1.1 EmpiricalApplication, 40
1.4.2 RealizedCovariance, 44
v
vi Contents
1.4.2.1 RealizedQuadraticCovariation, 44
1.4.2.2 RealizedBipowerCovariation, 44
Acknowledgments, 45
part one
Autoregressive Conditional Heteroskedasticity and
Stochastic Volatility
2 Nonlinear Models for
Autoregressive Conditional
Heteroskedasticity 49
2.1 Introduction, 49
2.2 TheStandardGARCHModel, 50
2.3 PredecessorstoNonlinearGARCHModels, 51
2.4 NonlinearARCHandGARCHModels, 52
2.4.1 Engle’sNonlinearGARCHModel, 52
2.4.2 NonlinearARCHModel, 53
2.4.3 AsymmetricPowerGARCHModel, 53
2.4.4 SmoothTransitionGARCHModel, 54
2.4.5 DoubleThresholdARCHModel, 56
2.4.6 NeuralNetworkARCHandGARCHModels, 57
2.4.7 Time-VaryingGARCH, 58
2.4.8 FamiliesofGARCHModelsandtheir
ProbabilisticProperties, 59
2.5 TestingStandardGARCHAgainstNonlinearGARCH, 60
2.5.1 SizeandSignBiasTests, 60
2.5.2 TestingGARCHAgainstSmoothTransition
GARCH, 61
2.5.3 TestingGARCHAgainstArtificialNeural
NetworkGARCH, 62
2.6 EstimationofParametersinNonlinearGARCH
Models, 63
2.6.1 SmoothTransitionGARCH, 63
2.6.2 NeuralNetworkGARCH, 64
2.7 ForecastingwithNonlinearGARCHModels, 64
2.7.1 SmoothTransitionGARCH, 64
2.7.2 AsymmetricPowerGARCH, 66
2.8 ModelsBasedonMultiplicativeDecompositionofthe
Variance, 67
2.9 Conclusion, 68
Acknowledgments, 69
Contents vii
3 Mixture and Regime-Switching GARCH
Models 71
3.1 Introduction, 71
3.2 Regime-SwitchingGARCHModelsforAssetReturns, 73
3.2.1 TheRegime-SwitchingFramework, 73
3.2.2 ModelingtheMixingWeights, 75
3.2.3 Regime-SwitchingGARCHSpecifications, 78
3.3 StationarityandMomentStructure, 81
3.3.1 Stationarity, 83
3.3.2 MomentStructure, 87
3.4 RegimeInference,LikelihoodFunction,andVolatility
Forecasting, 89
3.4.1 DeterminingtheNumberofRegimes, 92
3.4.2 VolatilityForecasts, 92
3.4.3 ApplicationofMS-GARCHModelstoStock
ReturnIndices, 93
3.5 ApplicationofMixtureGARCHModelstoDensity
PredictionandValue-at-RiskEstimation, 97
3.5.1 Value-at-Risk, 97
3.5.2 DataandModels, 98
3.5.3 EmpiricalResults, 99
3.6 Conclusion, 102
Acknowledgments, 102
4 Forecasting High Dimensional
Covariance Matrices 103
4.1 Introduction, 103
4.2 Notation, 104
4.3 RollingWindowForecasts, 104
4.3.1 SampleCovariance, 105
4.3.2 ObservableFactorCovariance, 105
4.3.3 StatisticalFactorCovariance, 106
4.3.4 Equicorrelation, 107
4.3.5 ShrinkageEstimators, 108
4.4 DynamicModels, 109
4.4.1 CovarianceTargetingScalarVEC, 109
4.4.2 FlexibleMultivariateGARCH, 110
4.4.3 ConditionalCorrelationGARCHModels, 111
4.4.4 OrthogonalGARCH, 113
4.4.5 RiskMetrics, 114
4.4.6 AlternativeEstimatorsforMultivariateGARCH
Models, 116
viii Contents
4.5 HighFrequencyBasedForecasts, 117
4.5.1 RealizedCovariance, 118
4.5.2 Mixed-FrequencyFactorModelCovariance, 119
4.5.3 RegularizationandBlockingCovariance, 119
4.6 ForecastEvaluation, 123
4.6.1 PortfolioConstraints, 124
4.7 Conclusion, 125
Acknowledgments, 125
5 Mean, Volatility, and Skewness
Spillovers in Equity Markets 127
5.1 Introduction, 127
5.2 DataandSummaryStatistics, 129
5.2.1 Data, 129
5.2.2 Time-VaryingSkewness(Univariate
Analysis), 132
5.2.3 SpilloverModels, 135
5.3 EmpiricalResults, 138
5.3.1 ParameterEstimates, 138
5.3.2 SpilloverEffectsinVarianceandSkewness, 139
5.3.2.1 VarianceRatios, 139
5.3.2.2 PatternandSizeofSkewness
Spillovers, 141
5.4 Conclusion, 144
Acknowledgments, 145
6 Relating Stochastic Volatility
Estimation Methods 147
6.1 Introduction, 147
6.2 TheoryandMethodology, 149
6.2.1 Quasi-MaximumLikelihoodEstimation, 150
6.2.2 GaussianMixtureSampling, 151
6.2.3 SimulatedMethodofMoments, 152
6.2.4 MethodsBasedonImportanceSampling, 153
6.2.4.1 ApproximatingintheBasicIS
Approach, 154
6.2.4.2 ImprovingonISwithIIS, 155
6.2.4.3 AlternativeEfficiencyGainswith
EIS, 156
Contents ix
6.2.5 AlternativeSamplingMethods:SSSand
MMS, 158
6.3 ComparisonofMethods, 160
6.3.1 SetupofData-GeneratingProcessandEstimation
Procedures, 160
6.3.2 ParameterEstimatesfortheSimulation, 161
6.3.3 PrecisionofIS, 163
6.3.4 PrecisionofBayesianMethods, 164
6.4 EstimatingVolatilityModelsinPractice, 165
6.4.1 DescribingReturnDataofGoldmanSachsand
IBMStock, 165
6.4.2 EstimatingSVModels, 167
6.4.3 ExtractingUnderlyingVolatility, 168
6.4.4 RelatingtheReturnsinaBivariateModel, 169
6.5 Conclusion, 172
7 Multivariate Stochastic Volatility
Models 175
7.1 Introduction, 175
7.2 MSVModel, 176
7.2.1 Model, 176
7.2.1.1 LikelihoodFunction, 177
7.2.1.2 PriorDistribution, 178
7.2.1.3 PosteriorDistribution, 179
7.2.2 BayesianEstimation, 179
7.2.2.1 Generationofα, 179
7.2.2.2 Generationofφ, 181
7.2.2.3 Generationof(cid:4), 181
7.2.3 Multivariate-t Errors, 181
7.2.3.1 Generationofν, 182
7.2.3.2 Generationofλ, 183
7.3 FactorMSVModel, 183
7.3.1 Model, 183
7.3.1.1 LikelihoodFunction, 184
7.3.1.2 PriorandPosteriorDistributions, 185
7.3.2 BayesianEstimation, 185
7.3.2.1 Generationofα,φ,and(cid:4), 186
7.3.2.2 Generationoff, 187
7.3.2.3 Generationofλ, 187
7.3.2.4 Generationofβ, 188