极端金融风险

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 Comnlex Svstems

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 Historv 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 R;esults 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 R,eturns 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 R,andom Variables

3.5 Simulation of Dependent Data with a Prescribed Copula

3.5.1 Simulation of Random Variables Characterized

3.5.2 Simulation of Random Variables Characterized

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 R,andom Variables

3.B Sketch of a Proof of a Large Deviation Theorem for Portfolios Made of Weibull R,andom Variables

3.C Relation Between the Objectiveand the Risk-Neutral Copula

4 Measures of Dependences

5 Description of Financial Dependences with Copulas

6 Measuring Extreme Dependences

7 Summary and Outlook

References

In the financial sector， crashes probably represent the most striking eventsamong all possible extreme phenomena， with an impact and frequency thathas been increasing in the last two decades. Consider the worldwidecrash in October 1987 which evaporated more than one thousand billion dol-lars in a few days or the more recent collapse of the internet bubble in whichmore than one-third of the world capitalization of 1999 disappeared afterMarch 2000. Finance and stock markets are based on the fluid convertibilityof stocks into money and vice versa. Thus， to work well， money is requestedto be a reliable standard of value， that is， an effective 8tore of value， hence theconcerns with the negative impacts of inflation.