发帖
雷达卡
Contents
1 On the Origin of Risks and Extremes . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Multidimensional Nature of Risk
and Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 How to Rank Risks Coherently? . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Coherent Measures of Risks . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Consistent Measures of Risks and Deviation Measures . . 7
1.2.3 Examples of Consistent Measures of Risk . . . . . . . . . . . . . 10
1.3 Origin of Risk and Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 The CAPM View. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 The Arbitrage Pricing Theory (APT)
and the Fama–French Factor Model. . .Arbitrage Pricing Theory (APT)
and the Fama–French Factor Model. . . . . . . . . . . . . . . . . . 18
1.3.3 The Efficient Market Hypothesis . . . . . . . . . . . . . . . . . . . . 20
1.3.4 Emergence of Dependence Structures
in the Stock Markets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.3.5 Large Risks in Complex Systems . . . . . . . . . . . . . . . . . . . . 29
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.A Why Do Higher Moments Allow
us to Assess Larger Risks? . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Marginal Distributions of Returns . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2 A Brief History of Return Distributions . . . . . . . . . . . . . . . . . . . . 37
2.2.1 The Gaussian Paradigm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.2 Mechanisms for Power Laws in Finance . . . . . . . . . . . . . . 39
2.2.3 Empirical Search for Power Law Tails
and Possible Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Constraints from Extreme Value Theory. . . . . . . . . . . . . . . . . . . . 43
2.3.1 Main Theoretical Results on Extreme Value Theory . . . 45
2.3.2 Estimation of the Form Parameter and Slow
Convergence to Limit Generalized Extreme Value
(GEV) and Generalized Pareto (GPD) Distributions . . . 47
XIV Contents
2.3.3 Can Long Memory Processes Lead to Misleading
Measures of Extreme Properties? . . . . . . . . . . . . . . . . . . . . 51
2.3.4 GEV and GPD Estimators of the Distributions
of Returns of the Dow Jones and Nasdaq Indices . . . . . . 52
2.4 Fitting Distributions of Returns with Parametric Densities . . . . 54
2.4.1 Definition of Two Parametric Families . . . . . . . . . . . . . . . 54
2.4.2 Parameter Estimation Using Maximum Likelihood
and Anderson-Darling Distance . . . . . . . . . . . . . . . . . . . . . 60
2.4.3 Empirical Results on the Goodness-of-Fits . . . . . . . . . . . 62
2.4.4 Comparison of the Descriptive Power
of the Different Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5.1 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2.5.2 Is There a Best Model of Tails? . . . . . . . . . . . . . . . . . . . . . 76
2.5.3 Implications for Risk Assessment . . . . . . . . . . . . . . . . . . . . 78
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.A Definition and Main Properties of Multifractal Processes 80
2.B A Survey of the Properties
of Maximum Likelihood Estimators . . . . . . . . . . . . . . . . . . 87
2.C Asymptotic Variance–Covariance of Maximum
Likelihood Estimators of the SE Parameters . . . . . . . . . . 91
2.D Testing the Pareto Model versus
the Stretched-Exponential Model . . . . . . . . . . . . . . . . . . . . 93
3 Notions of Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1 What is Dependence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2 Definition and Main Properties of Copulas . . . . . . . . . . . . . . . . . . 103
3.3 A Few Copula Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.3.1 Elliptical Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.3.2 Archimedean Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.3.3 Extreme Value Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.4 Universal Bounds for Functionals
of Dependent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.5 Simulation of Dependent Data with a Prescribed Copula . . . . . 120
3.5.1 Simulation of Random Variables Characterized
by Elliptical Copulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.5.2 Simulation of Random Variables Characterized
by Smooth Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.6 Application of Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.6.1 Assessing Tail Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.6.2 Asymptotic Expression of the Value-at-Risk . . . . . . . . . . 128
3.6.3 Options on a Basket of Assets . . . . . . . . . . . . . . . . . . . . . . . 131
3.6.4 Basic Modeling of Dependent Default Risks . . . . . . . . . . . 137
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Contents XV
3.A Simple Proof of a Theorem on Universal Bounds
for Functionals of Dependent Random Variables . . . . . . . 138
3.B Sketch of a Proof of a Large Deviation Theorem
for Portfolios Made of Weibull Random Variables . . . . . . 140
3.C Relation Between the Objective
and the Risk-Neutral Copula. . . . . . . . . . . . . . . . . . . . . . . . 143
4 Measures of Dependences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.1 Linear Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
4.1.1 Correlation Between Two Random Variables . . . . . . . . . . 147
4.1.2 Local Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
4.1.3 Generalized Correlations Between N > 2 Random
Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.2 Concordance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.2.1 Kendall’s Tau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.2.2 Measures of Similarity Between Two Copulas . . . . . . . . . 158
4.2.3 Common Properties of Kendall’s Tau,
Spearman’s Rho and Gini’s Gamma . . . . . . . . . . . . . . . . . 161
4.3 Dependence Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.4 Quadrant and Orthant Dependence . . . . . . . . . . . . . . . . . . . . . . . . 164
4.5 Tail Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
4.5.2 Meaning and Refinement of Asymptotic Independence . 168
4.5.3 Tail Dependence for Several Usual Models . . . . . . . . . . . . 170
4.5.4 Practical Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.A Tail Dependence Generated by Student’s Factor Model . 182
5 Description of Financial Dependences with Copulas . . . . . . . 189
5.1 Estimation of Copulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.1.1 Nonparametric Estimation. . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.1.2 Semiparametric Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.1.3 Parametric Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.1.4 Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2 Description of Financial Data in Terms of Gaussian Copulas . . 204
5.2.1 Test Statistics and Testing Procedure . . . . . . . . . . . . . . . . 204
5.2.2 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.3 Limits of the Description in Terms of the Gaussian Copula . . . 212
5.3.1 Limits of the Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
5.3.2 Sensitivity of the Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.3.3 The Student Copula: An Alternative? . . . . . . . . . . . . . . . . 215
5.3.4 Accounting for Heteroscedasticity . . . . . . . . . . . . . . . . . . . 217
5.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
XVI Contents
5.A Proof of the Existence of a χ2-Statistic
for Testing Gaussian Copulas . . . . . . . . . . . . . . . . . . . . . . . 221
5.B Hypothesis Testing with Pseudo Likelihood . . . . . . . . . . . 222
6 Measuring Extreme Dependences . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
京公网安备 11010802022788号
