极端金融风险

章节摘录

　　1.3.5 Large Risks in Complex Systems　　These calculations show that an endogenous small positive correlation betweenall stock-pairs gives rise to large eigenvalues which can then be associated with "market factors." It seems that earlier researches have promoted the other wayaround: existing market factors （stock indices， news agencies， etc.） introduceexogenous market impact which affect different stocks similarly， thereby in-troducing positive correlation and thus large eigenvalues. This is clear fromthe general formulation of （linear） factor models such as the CAPM， APT，and Fama-French approaches in which the returns of all stocks are regressedagainst the same set of factors. Actually， we propose that the two chains ofcause and result may be intrinsically coupled: the correlation structure be-tween stocks is a stable attractor of a self-organized dynamics with positiveand negative feedbacks in which factors exist because correlations exist， andcorrelations exist because factors exist. It would suggest the development ofdynamical factor models， in wluch agents form anticipations on correlationsbased on their calibration of the past behavior of the regression to factors，in order to study the possible types of attractors （single or multiple equilib-ria） in the correlation structure of stocks. This may cast new light on themajor unsolved problem stated in the introduction of this chapter concerningthe relationship between return and risks: perhaps， the concept of return asthe remuneration of risk which is so fundamental in financial theory shouldbe replaced by the concept of the emergence of the risk-return duality， inwhich their relationship can be negative or positive， depending upon circum-stances that remain to be worked out. Moreover， simulations of complex self-organizing systems show that large fluctuations and extreme variations arethe rule rather than the exception.　　The complex system approach， which involves seeing interconnections andrelationships， /.e.， the whole picture as well as the component parts， is nowa-days pervasive in modern control of engineering devices and business manage-ment. A central property of a complex system is the possible occurrence ofcoherent large-scale collective behaviors with a very rich structure， resultingfrom the repeated non-linear interactions among its constituents: the wholeturns out to be much more than the sum of its parts. Most complex systemsaround us do exhibit rare and sudden transitions that occur over time in-tervals that are short compared with the characteristic time scales of theirposterior evolution. Such extreme events express more than anything else theunderlying forces usually hidden by almost perfect balance and thus pro-vide the potential for a better scientific understanding of complex systems.These crises have fundamental societal impacts and range from large nat-ural catastrophes， catastrophic events of environmental degradation， to thefailure of engineering structures， crashes in the stock market， social unrestleading to large-scale strikes and upheaval， economic drawdowns on nationaland global scales， regional power blackouts， traffic gridlocks， diseases and epi-demics， etc. An outstanding scientific question is how such large-scale patternsof catastrophic nature might evolve from a series ofinteractions on the small-est and increasingly larger scales. In complex systems， it has been found thatthe organization of spatial and temporal correlations do not stem， in general，from a nucleation phase diffusing across the system. It results rather from aprogressive and more global cooperative process occurring over the whole sys-tem by repetitive interactions， which is partially described by the distributedcorrelations at the origin of a large eigenvalue as described above. An instancewould be the many occurrences of simultaneous scientific and technical discov-eries signaling the global nature of the maturing process. Recent developmentssuggest that non-traditional approaches， based on the concepts and methodsof statistical and nonlinear physics coupled with ideas and tools from com-putation intelligence could provide novel methods in complexity to direct thenumerical resolution of more realistic models and the identification of rele-vant signatures of large and extreme risks. To address the challenge posed bythe identification and modeling of such outliers， the available theoretical toolscomprise in particular bifurcation and catastrophe theories， dynamical criticalphenomena and the renormalization group， nonlinear dynamical systems， andthe theory of partially （spontaneously or not） broken symmetries. This fieldof research is presently very active and is expected to advance significantlyour understanding， quantification， and control of risks.　　……

书籍目录

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

作者简介

《极端金融风险》内容全面系统，既有很高的实用价值，又有很强的资料收藏价值，涵盖了The Multidimensional Nature of Risk and Dependence、Emergence of Dependence Structures in the Stock Markets、Discussion and Conclusions等内容，各章结构合理，层次清楚、叙述详细、文字流畅，非常适于阅读。

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