Advanced
StochasticModels,
RiskAssessment,
andPortfolio
Optimization
TheIdealRisk,Uncertainty,
andPerformanceMeasures
SVETLOZART.RACHEV
STOYANV.STOYANOV
FRANKJ.FABOZZI
JohnWiley&Sons,Inc.共有403页。

Contents
Preface xiii
Acknowledgments xv
AbouttheAuthors xvii
CHAPTER1
ConceptsofProbability1
1.1Introduction 1
1.2BasicConcepts 2
1.3DiscreteProbabilityDistributions 2
1.3.1BernoulliDistribution 3
1.3.2BinomialDistribution 3
1.3.3PoissonDistribution 4
1.4ContinuousProbabilityDistributions 5
1.4.1ProbabilityDistributionFunction,Probability
DensityFunction,andCumulativeDistribution
Function 5
1.4.2TheNormalDistribution 8
1.4.3ExponentialDistribution 10
1.4.4Student!ˉ t-distribution 11
1.4.5ExtremeValueDistribution 12
1.4.6GeneralizedExtremeValueDistribution 12
1.5StatisticalMomentsandQuantiles 13
1.5.1Location 13
1.5.2Dispersion 13
1.5.3Asymmetry 13
1.5.4ConcentrationinTails 14
1.5.5StatisticalMoments 14
1.5.6Quantiles 16
1.5.7SampleMoments 16
1.6JointProbabilityDistributions 17
1.6.1ConditionalProbability 18
1.6.2De?nitionofJointProbabilityDistributions19

1.6.3MarginalDistributions 19
1.6.4DependenceofRandomVariables 20
1.6.5CovarianceandCorrelation 20
1.6.6MultivariateNormalDistribution 21
1.6.7EllipticalDistributions 23
1.6.8CopulaFunctions 25
1.7ProbabilisticInequalities 30
1.7.1Chebyshev!ˉsInequalit 30
!
1.7.2Frechet-HoeffdingInequality 31
1.8Summary 32
CHAPTER2
Optimization35
2.1Introduction 35
2.2UnconstrainedOptimization 36
2.2.1MinimaandMaximaofaDifferentiable
Function 37
2.2.2ConvexFunctions 40
2.2.3QuasiconvexFunctions 46
2.3ConstrainedOptimization 48
2.3.1LagrangeMultipliers 49
2.3.2ConvexProgramming 52
2.3.3LinearProgramming 55
2.3.4QuadraticProgramming 57
2.4Summary 58
CHAPTER3
ProbabilityMetrics61
3.1Introduction 61
3.2MeasuringDistances:TheDiscreteCase 62
3.2.1SetsofCharacteristics 63
3.2.2DistributionFunctions 64
3.2.3JointDistribution 68
3.3Primary,Simple,andCompoundMetrics 72
3.3.1AxiomaticConstruction 73
3.3.2PrimaryMetrics 74
3.3.3SimpleMetrics 75
3.3.4CompoundMetrics 84
3.3.5MinimalandMaximalMetrics 86
3.4Summary 90
3.5TechnicalAppendix 90

3.5.1RemarksontheAxiomaticConstructionof
ProbabilityMetrics
3.5.2ExamplesofProbabilityDistances
3.5.3MinimalandMaximalDistances
obabilityMetrics
Introduction
TheClassicalCentralLimitTheorem
4.2.1TheBinomialApproximationtotheNormal
Distribution
4.2.2TheGeneralCase
4.2.3EstimatingtheDistancefromtheLimit
Distribution
TheGeneralizedCentralLimitTheorem
4.3.1StableDistributions
4.3.2ModelingFinancialAssetswithStable
Distributions
ConstructionofIdealProbabilityMetrics
4.4.1De?nition
4.4.2Examples
Summary
TechnicalAppendix
4.6.1TheCLTConditions
4.6.2RemarksonIdealMetrics
nderUncertainty
Introduction
ExpectedUtilityTheory
5.2.1St.PetersburgParadox
5.2.2ThevonNeumann¨CMorgensternExpecte
UtilityTheory
5.2.3TypesofUtilityFunctions
StochasticDominance
5.3.1First-OrderStochasticDominance
5.3.2Second-OrderStochasticDominance
5.3.3Rothschild-StiglitzStochasticDominance
5.3.4Third-OrderStochasticDominance
5.3.5Ef?cientSetsandthePortfolioChoiceProble
5.3.6ReturnversusPayoff5.4ProbabilityMetricsandStochasticDominance 157
5.5Summary 161
5.6TechnicalAppendix 161
5.6.1TheAxiomsofChoice 161
5.6.2StochasticDominanceRelationsofOrder n 163
5.6.3ReturnversusPayoffandStochasticDominance164
5.6.4OtherStochasticDominanceRelations 166
TER6
skandUncertainty171
6.1Introduction 171
6.2MeasuresofDispersion 174
6.2.1StandardDeviation 174
6.2.2MeanAbsoluteDeviation 176
6.2.3SemistandardDeviation 177
6.2.4AxiomaticDescription 178
6.2.5DeviationMeasures 179
6.3ProbabilityMetricsandDispersionMeasures 180
6.4MeasuresofRisk 181
6.4.1Value-at-Risk 182
6.4.2ComputingPortfolioVaRinPractice 186
6.4.3BacktestingofVaR 192
6.4.4CoherentRiskMeasures 194
6.5RiskMeasuresandDispersionMeasures 198
6.6RiskMeasuresandStochasticOrders 199
6.7Summary 200
6.8TechnicalAppendix 201
6.8.1ConvexRiskMeasures 201
6.8.2ProbabilityMetricsandDeviationMeasures202
TER7
verageValue-at-Risk207
7.1Introduction 207
7.2AverageValue-at-Risk 208
7.3AVaREstimationfromaSample 214
7.4ComputingPortfolioAVaRinPractice 216
7.4.1TheMultivariateNormalAssumption 216
7.4.2TheHistoricalMethod 217
7.4.3TheHybridMethod 217
7.4.4TheMonteCarloMethod 218
7.5BacktestingofAVaR 2207.6SpectralRiskMeasures
7.7RiskMeasuresandProbabilityMetrics
7.8Summary
7.9TechnicalAppendix
7.9.1CharacteristicsofConditionalLoss
Distributions
7.9.2Higher-OrderAVaR
7.9.3TheMinimizationFormulaforAVaR
7.9.4AVaRforStableDistributions
7.9.5ETLversusAVaR
7.9.6RemarksonSpectralRiskMeasures
TER8
timalPortfolios
8.1Introduction
8.2Mean-VarianceAnalysis
8.2.1Mean-VarianceOptimizationProblem
8.2.2TheMean-VarianceEf?cientFrontier
8.2.3Mean-VarianceAnalysisandSSD
8.2.4AddingaRisk-FreeAsset
8.3Mean-RiskAnalysis
8.3.1Mean-RiskOptimizationProblems
8.3.2TheMean-RiskEf?cientFrontier
8.3.3Mean-RiskAnalysisandSSD
8.3.4RiskversusDispersionMeasures
8.4Summary
8.5TechnicalAppendix
8.5.1TypesofConstraints
8.5.2QuadraticApproximationstoUtilityF
8.5.3SolvingMean-VarianceProblemsinPr
8.5.4SolvingMean-RiskProblemsinPractic
8.5.5Reward-RiskAnalysis
TER9
nchmarkTrackingProblems
9.1Introduction
9.2TheTrackingErrorProblem
9.3RelationtoProbabilityMetrics
9.4Examplesofr.d.Metrics
9.5NumericalExample
9.6Summary9.7TechnicalAppendix
9.7.1DeviationMeasuresandr.d.Metrics
9.7.2RemarksontheAxioms
9.7.3Minimalr.d.Metrics
9.7.4LimitCasesof L
(X,Y)and (X,Y)
p
p
9.7.5Computingr.d.MetricsinPractice
TER10
rformanceMeasures
0.1Introduction
0.2Reward-to-RiskRatios
10.2.1RRRatiosandtheEf?cientPortfolios
10.2.2LimitationsintheApplicationof
Reward-to-RiskRatios
10.2.3TheSTARR
10.2.4TheSortinoRatio
10.2.5TheSortino-SatchellRatio
10.2.6AOne-SidedVariabilityRatio
10.2.7TheRachevRatio
0.3Reward-to-VariabilityRatios
10.3.1RVRatiosandtheEf?cientPortfolios
10.3.2TheSharpeRatio
10.3.3TheCapitalMarketLineandtheSharpeRatio
0.4Summary
0.5TechnicalAppendix
10.5.1ExtensionsofSTARR
10.5.2QuasiconcavePerformanceMeasures
10.5.3TheCapitalMarketLineandQuasiconcave
Ratios
10.5.4NonquasiconcavePerformanceMeasures
10.5.5ProbabilityMetricsandPerformanceMeasures