目錄
1 On the Origin of Risks and Extremes
1.1 The Multidimensional Nature of Risk and Dependence
1.2 How to Rank Risks Coherently？
1.2.1 Coherent Measures of Risks
1.2.2 Consistent Measures of Risks and Deviation Measures.
1.2.3 Examples of Consistent Measures of Risk
1.3 Origin of Risk and Dependence
1.3.1 The CAPM View
1.3.2 The Arbitrage Pricing Theory （APT） and the Fama-French Factor Model
1.3.3 The Efficient Market Hypothesis
1.3.4 Emergence of Dependence Structures in the Stock Markets
1.3.5 Large Risks in Complex Systems
Appendix
1.A Why Do Higher Moments Allow us to Assess Larger Risks？
2 Marginal Distributions of Returns
2.1 Motivations
2.2 A Brief History of Return Distributions
2.2.1 The Gaussian Paradigm
2.2.2 Mechanisms for Power Laws in Finance
2.2.3 Empirical Search for Power Law Tails and Possible Alternatives
2.3 Constraints from Extreme Value Theory
2.3.1 Main Theoretical Results on Extreme Value Theory
2.3.2 Estimation of the Form Parameter and Slow Convergence to Limit Generalized Extreme Value （GEV） and Generalized Pareto （GPD） Distributions
2.3.3 Can Long Memory Processes Lead to Misleading Measures of Extreme Properties？
2.3.4 GEV and GPD Estimators of the Distributions of Returns of the Dow Jones and Nasdaq Indices
2.4 Fitting Distributions of Returns with Parametric Densities
2.4.1 Definition of Two Parametric Families
2.4.2 Parameter Estimation Using Maximum Likelihood and Anderson-Darling Distance
2.4.3 Empirical Results on the Goodness-of-Fits
2.4.4 Comparison of the Descriptive Power of the Different Families
2.5 Discussion and Conclusions
2.5.1 Summary
2.5.2 Is There a Best Model of Tails？
2.5.3 Implications for Risk Assessment
Appendix
2.A Definition and Main Properties of Multifractal Processes
2.B A Survey of the Properties of Maximum Likelihood Estimators
2.C Asymptotic Variance-Covariance of Maximum Likelihood Estimators of the SE Parameters
2.D Testing the Pareto Model versus the Stretched-Exponential Model
3 Notions of Copulas
3.1 What is Dependence？
3.2 Definition and Main Properties of Copulas
3.3 A Few Copula Families
3.3.1 Elliptical Copulas
3.3.2 Archimedean Copulas
3.3.3 Extreme Value Copulas
3.4 Universal Bounds for Functionals of Dependent Random Variables
3.5 Simulation of Dependent Data with a Prescribed Copula
3.5.1 Simulation of Random Variables Characterized by Elliptical Copulas
3.5.2 Simulation of Random Variables Characterized by Smooth Copulas
3.6 Application of Copulas
3.6.1 Assessing Tail Risk
3.6.2 Asymptotic Expression of the Value-at-Risk
3.6.3 Options on a Basket of Assets
3.6.4 Basic Modeling of Dependent Default Risks
Appendix
3.A Simple Proof of a Theorem on Universal Bounds for Functionals of Dependent Random Variables
3.B Sketch of a Proof of a Large Deviation Theorem for Portfolios Made of Weibull Random Variables
3.C Relation Between the Objective and the Risk-Neutral Copula
4 Measures of Dependences
4.1 Linear Correlations
4.1.1 Correlation Between Two Random Variables
4.1.2 Local Correlation
4.1.3 Generalized Correlations Between N > 2 Random Variables
4.2 Concordance Measures
4.2.1 Kendall's Tau
4.2.2 Measures of Similarity Between Two Copulas
4.2.3 Common Properties of Kendall's Tau, Spearman's Rho and Gini's Gamma
4.3 Dependence Metric
4.4 Quadrant and Orthant Dependence
4.5 Tail Dependence
4.5.1 Definition
4.5.2 Meaning and Refinement of Asymptotic Independence
4.5.3 Tail Dependence for Several Usual Models
4.5.4 Practical Implications
Appendix
4.A Tail Dependence Generated by Student's Factor Model.
5 Description of Financial Dependences with Copulas
5.1 Estimation of Copulas
5.1.1Nonparametric Estimation
5.1.2 Semiparametric Estimation
5.1.3 Parametric Estimation
5.1.4 Goodness-of-Fit Tests
5.2 Description of Financial Data in Terms of Gaussian Copulas
5.2.1 Test Statistics and Testing Procedure
5.2.2 Empirical Results
5.3 Limits of the Description in Terms of the Gaussian Copula
5.3.1 Limits of the Tests
5.3.2 Sensitivity of the Method
5.3.3 The Student Copula: An Alternative？
5.3.4 Accounting for Heteroscedasticity
5.4 Summary
Appendix
5.A Proof of the Existence of a X2-Statistic for Testing Gaussian Copulas
5.B Hypothesis Testing with Pseudo Likelihood
6 Measuring Extreme Dependences
6.1 Motivations
6.1.1 Suggestive Historical Examples
6.1.2 Review of Different Perspectives
6.2 Conditional Correlation Coefficient
6.2.1 Definition
6.2.2 Influence of the Conditioning Set
6.2.3 Influence of the Underlying Distribution for a Given Conditioning Set
6.2.4 Conditional Correlation Coefficient on Both Variables
6.2.5 An Example of Empirical Implementation
6.2.6 Summary
6.3 Conditional Concordance Measures
6.3.1 Definition
6.3.2 Example
6.3.3 Empirical Evidence
6.4 Extreme Co-movements
6.5 Synthesis and Consequences
Appendix
6.A Correlation Coefficient for Gaussian Variables Conditioned on Both X and Y Larger Than u
6.B Conditional Correlation Coefficient for Student's Variables
6.C Conditional Spearman's Rho
7 Summary and Outlook
7.1 Synthesis
7.2 Outlook and Future Directions
7.2.1 Robust and Adaptive Estimation of Dependences
7.2.2 Outliers, Kings, Black Swans and Their Dependence
7.2.3 Endogeneity Versus Exogeneity
7.2.4 Nonstationarity and Regime Switching in Dependence
7.2.5 Time-Varying Lagged Dependence
7.2.6 Toward a Dynamical Microfoundation of Dependences
References
Index
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