極端金融風險（簡體書）

作者:(法)馬勒沃根

《極端金融風險》由世界圖書出版公司北京公司出版。

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

2.2.3 Empirical Search for Power Law Tails and Possible Alternatives In the early 1960s, Mandelbrot （339） and Fama （157） presented evidence that distributions of returns can be well approximated by a symmetric Lévy stable law with tail index b about 1.7. These estimates of the tail index have recently been supported by Mittnik et al. （362）, and slightly different indices of the stable law （b = 1.4） were suggested by Mantegna and Stanley （345, 346）.
On the other hand, there are numerous evidences of a larger value of the tail index b ≌ 3 （217, 312, 320, 322, 367）. See also the various alternative parameterizations in terms of the Student distribution （62, 275）, or Pearson Type-Ⅶ distributions （368）, which all have an asymptotic power law tail and are regularly varying. Thus, a general conclusion of this group of authors concerning tail fatness can be formulated as follows: the tails of the distribution of returns are heavier than a Gaussian tail and heavier than an exponential tail; they certainly admit the existence of a finite variance （b > 2）, whereas the existence of the third （skewness） and the fourth （kurtosis） moments is questionable.
These two classes of results are contradictory only on the surface, because they actually do not apply to the same quantiles of the distributions of returns. Indeed, Mantegna and Stanley （345） have shown that the distribution of returns of the Standard & Poor's 500 index can be described accurately by a Lévy stable law only within a limited range up to about 5 standard deviations, while a faster decay （approximated by an exponential or a power law with larger exponent） of the distribution is observed beyond. This almostbut-not-quite Lévy stable description could explain （at least, in part） the slow convergence of the distribution of returns to the Gaussian law under time aggregation （72, 451）; and it is precisely outside this range of up to 5 standard deviations, where the Lévy law does not apply anymore that a tail index b ≌ 3 has been estimated. Indeed, most authors who have reported a tail index b ≌ 3 have used some optimality criteria for choosing the sample fractions （i.e., the largest values） for the estimation of the tail index. Thus, unlike the authors supporting stable laws, they have used only a fraction of the largest （positive tail） and smallest （negative tail） sample values.