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326 9. Mesure de la dépendance extrême entre deux actifs financiers References Bigham, N.H., C.M. Goldie, and J.L. Teugel, 1987, Regular Variation. (Cambridge university press Cambridge). Bouchaus, J.P., D. Sornette, C. Walter and J.P. Aguilar, 1998, Taming large events: Optimal portfolio theory for strongly fluctuating assets, International Journal of Theoretical and Applied Finance, 1, 25-41. Boyer, B.H., M.S. Gibson, and M. Lauretan, 1997, Pitfalls in tests for changes in correlations, Working paper, International Finance Discussion Paper 597, Board of the Governors of the Federal Reserve System. Brennan, M.J., and E.J. Schwarz, 1978, A continuous time approach to the pricing of bonds, Journal of Banking and Finance 3, 133–155. Campbell, J.Y., M. Lettau, B. G. Malkiel, and Y. Xu, 2001, Have individual stocks become more volatile? An empirical exploration of idiosyncratic risk, Journal of Finance 56, 1–43. Carey, M., 1998, Credit risk in private debt portfolio, Journal of Finance 53, 56–61. Cho, Y.-H., and R.F. Engle, 2000, Time-varying betas and asymmetric effects of news : empirical analysis of blue chip stocks, Working paper, University of California, San Diego. Coles, S., J. Heffernan, and J. Tawn, 1999, Dependence measures for extreme value analyses, Extremes 2, 339–365. Cox, J.C, J.E. Ingersoll, and S.A. Ross, 1985, A theory of the term structure of interest rates, Econometrica 51, 385–408. Danielsson, J., L. de Haan, L. Peng, and C.G. de Vries, 2001, Using a bootstrap method to choose the optimal sample fraction in tail index estimation, Journal of Multivariate analysis 76, 226–248. Danielsson, J., and C.G. de Vries, 1997, Tail index and quantile estimation with very high frequency data, Journal of Empirical Finance 4, 241–257. de Haan, L., S.I. Resnick, H. Rootzen, and C.G. de Vries, 1989, Extremal behaviour of solutions to a stochastic difference equation with application to ARCH processes, Stochastic Processes and their Applications 32, 213–224. Embrechts, P., C.P. Kluppelberg, and T. Mikosh, 1997, Modelling Extremal Events. (Springer- Verlag Berlin). Embrechts, P., A.J. McNeil, and D. Straumann, 1999, Correlation: Pitfalls and Alternatives, Risk pp. 69–71. Embrechts, P., A.J. McNeil, and D. Straumann, 2001, Correlation and Dependency in Risk Management: Properties and Pitfalls, in M. Dempster, eds.: Value at Risk and Beyond (Cambridge University Press, ). Engle, R.F., and S. Manganelli, 1999, CAViaR: Conditional autoregressive Value-at-Risk by regression quantiles, Working paper, University of California, San Diego. Fama, E., and J. Mc Beth, 1973, Risk, return and equilibrium: empirical tests, Journal of Political Economy 81, 607–636. 27
9.2. Estimation du coefficient de dépendance de queue 327 Feller, W., 1971, An introduction to probability theory and its applications II. (Wiley New-York). Frees, E., and E. Valdez, 1998, Understanding relationships using copulas, North Americam Actuarial Journal 2, 1–25. Gopikrishnan, P., M. Meyer, L.A.N. Amaral, and H.E. Stanley, 1998, Inverse Cubic Law for the Distribution of Stock Price Variations, European Physical Journal B 3, 139 –140. Gorby, M.B., 2000, A comparative anatomy of credit risk models, Journal of Banking and Finance 24, 119–149. Hall, P., 1990, Using the bootstrap method to estimate mean squared error and select smoothing parameter in non parametric problems, Journal of Multivariate Analysis 32, 177–203. Joe, H., 1997, Multivariate models and dependence concepts. (Chapman & Hall London). Johansen, A., and D. Sornette, 2002, Large stock market price drawdowns are outliers, Journal of Risk 4, 69–110. Jones, C.P., and J.W.Wilson, 1989, Is stock price volatility increasing?, Financial Analysts Journal 45, 20–26. Kandel, S., and R. Staumbaugh, 1987, On correlations and the sensitivity of inference about meanvariance efficiency, Journal of Financial Economics 18, 61–90. Kearns, P., and A.R. Pagan, 1997, Estimating the density tail index for financial time series, Review of Economics and Statistics 79, 171–175. Lindskog, F., 2000, Modelling Dependence with Copulas, Working paper, RiskLab, http : //www.risklab.ch/P apers.html#MT Lindskog. Lintner, J., 1965, The valuation of risk assets and the selection of risky investments in stock portfolios and the capital bugets, Review of Economics and Statitics 41, 13–37. Longin, F.M., 1996, The asymptotic distribution of extrem stock market returns, Journal of Business 96, 383–408. Longin, F.M., and B. Solnik, 2001, Extreme Correlation of International Equity Markets, Journal of Finance 56, 649–676. Loretan, M., and P.C.B. Phillips, 1994, Testing the covariance stationarity of heavy-tailed times series, Journal of Empirical Finance 1, 211–248. Lucas, A., P. Klaassen, P. Spreij, and S. Straetmans, 2001, An analytic approach to credit risk of large corporate bond and loan portfolios, Journal of Banking and Finance 25, 1635–1664. Lux, T., 1996, The stable Paretian hypothesis and the frequency of large returns: an examination of major German stocks, Applied Financial Economics 6, 463–475. Malevergne, Y., and D. Sornette, 2001, Testing the Gaussian copula hypothesis for financial assets dependences, Working paper, (e-print available at http://papers.ssrn.com/abstract=291140). Malevergne, Y., and D. Sornette, 2002, Hedging Extreme Co-Movements, submitted to RISK, preprint at http://arXiv.org/abs/cond-mat/0205636. Mossin, J., 1966, Equilibrium in capital asset market, Econometrica 35, 768–783. 28
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UNIVERSITE DE NICE-SOPHIA ANTIPOLIS
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4 Table des matières 2.2.2 Bulles
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6 Table des matières 7.3.2 Tests n
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8 Table des matières 12.1.1 Distri
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10 Table des matières Références
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12 m’ont souvent été d’un gra
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14 Introduction pour espérer se co
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16 Introduction les distributions e
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18 Introduction
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Chapitre 1 Faits stylisés des rent
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1.1. Rappel des faits stylisés 23
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1.1. Rappel des faits stylisés 25
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1.2. De la difficulté de représen
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1.3. Modélisation des propriétés
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Chapitre 2 Modèles phénoménologi
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2.1. Bulles rationnelles multi-dime
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2.2. Des bulles rationnelles aux kr
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Chapitre 3 Distributions exponentie
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Empirical Distributions of Log-Retu
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concerning tail fatness can be form
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To fit our two data sets, section 4
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properties, stressing the problems
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Another problem lies in the determi
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In order to obtain a process with S
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and Sornette (1999) for instance. W
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1. The Pareto distribution: 79 Fu(x
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empirical analog FN(x), estimated f
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have been chosen to converge to 1 a
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5 Comparison of the descriptive pow
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ased on the fact that the quantity
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tail (positive or negative) of the
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Equation (46) depends on c only and
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B Minimum Anderson-Darling Estimato
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C Local exponent In this Appendix,
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D Testing non-nested hypotheses wit
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where α = (c ∗ ,d ∗ ). It can
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where E0[·] denotes the expectatio
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References Andersen, J.V. and D. So
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Gouriéroux C. and A. Monfort, 1994
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Rubinstein, M., 1973, The fundament
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(a) Independent Data Stretched-Expo
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(a) Dow Jones Positive Tail Negativ
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Nasdaq Dow Jones Pos. Tail Neg. Tai
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Dow Jones positive tail Dow Jones n
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Variation coefficient Variation coe
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Mean excess function Mean excess fu
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Hill‘s estimate b u Hill‘s esti
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Wilks statistic (doubled log−like
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Tail 1−F(x) and parameter b Tail
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1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4
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Chapitre 4 Relaxation de la volatil
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Volatility Fingerprints of Large Sh
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2 Long-range memory and distinction
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Figure 2: Measuring the conditional
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the system to the accumulation of m
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Appendix C: “Conditional response
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Baillie, R.T., 1996, Long memory pr
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Chapitre 5 Approche comportementale
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5.1. Prix d’un actif et excès de
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5.2. Modèles d’opinion contre mo
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5.4. Conséquences des phénomènes
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5.5. Conclusion 157 ce qui induit u
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Chapitre 6 Comportements mimétique
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RESEARCH PAPER Q UANTITATIVE F INAN
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A Corcos et al Q UANTITATIVE F INAN
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Deuxième partie Etude des proprié
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182 7. Etude de la dépendance à l
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192 8. Tests de copule gaussienne
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194 8. Tests de copule gaussienne 1
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200 8. Tests de copule gaussienne t
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202 8. Tests de copule gaussienne T
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204 8. Tests de copule gaussienne a
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206 8. Tests de copule gaussienne c
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208 8. Tests de copule gaussienne t
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210 8. Tests de copule gaussienne (
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212 8. Tests de copule gaussienne R
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228 8. Tests de copule gaussienne f
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236 8. Tests de copule gaussienne T
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238 9. Mesure de la dépendance ext
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Page 363 and 364: Chapitre 11 Portefeuilles optimaux
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Page 369 and 370: 11.5. Annexe 369 Collective Origin
Page 371 and 372: 11.5. Annexe 371 point λ = 1. Thus
Page 373 and 374: Chapitre 12 Gestion des risques gra
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12.1. Comprendre et gérer les risq
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12.1. Comprendre et gérer les risq
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12.2. Minimiser l’impact des gran
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Chapitre 13 Gestion de portefeuille
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VaR-Efficient Portfolios for a Clas
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dependence: from independence to co
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given a series of return {rt}t foll
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eads cV (u1, · · · , un) = 1
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THEOREM 4 (TAIL EQUIVALENCE FOR A S
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Independent Assets Comonotonic Asse
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3.2 Typical recurrence time of larg
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and finally w ∗ i = χ−1 i j
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where the ˆwi’s are solution of
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Choosing x > y > Aɛ, and applying
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Expanding fi(xi) around x ∗ i yie
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for all h ∈ AC. Thus, integrating
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B Asymptotic distribution of the su
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This yields × h+ Sc−2 − 2 dh
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References Acerbi, A. and D. Tasche
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Malevergne, Y. and D. Sornette, 200
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ln(T/T 0 ) T Figure 2: Logarithm
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Chapitre 14 Gestion de Portefeuille
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Multi-Moments Method for Portfolio
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choosen risk measure. Section 5 pre
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The variance of the return X of an
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This property is verified for all c
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µn=α of order n = 2, 4, 6 and 8.
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these two assets or portfolios. The
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Due to the homogeneity property of
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invested in the risky fund depends
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C. Then, if g1(X1), · · · , gn(X
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Differentiating with respect to x1,
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7.2 Transformation of the modified
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In the sequel, we will first evalua
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8.3 Non-symmetric assets In the cas
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y the variance will then increase).
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A Description of the data set We ha
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thus and finally 1 λ1 n−1 = w
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C Composition of the market portfol
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D Generalized capital asset princin
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This brings us back to the problem
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F.2 General case We now consider a
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Harvey, C.R. and A. Siddique, 2000,
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µ µ2 1/2 µ4 1/4 µ6 1/6 µ8 1/8
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Mean (10 −3 ) Variance (10 −3 )
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μ (daily return) x 10−3 2.5 2 1.
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w i w i 0.2 0.15 0.1 0.05 Mean−μ
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Figure 6: Schematic representation
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Empirical Cumulative Distribution 1
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Z 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0
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Z 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0
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10 0 10 −1 10 0 10 −1 10 −1 1
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10 0 10 −1 10 0 10 −1 10 −1 1
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κ 22 20 18 16 14 12 10 8 6 4 2 Exc
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Return μ 0.12 0.11 0.1 0.09 0.08 0
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Conclusions et Perspectives L’obj
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Conclusion 493 en univers gaussien
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Annexe A Evaluation de la conduite
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A.2. Eléments de contexte 497 bora
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A.4. Compétences acquises et ensei
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A.5. Conclusion 501 de la recherche
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Bibliographie ACERBI, C. (2002) :
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Bibliographie 505 BOUCHAUD, J. P. E
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Bibliographie 507 DE FINETTI, B. (1
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Bibliographie 509 GRANGER, C. W. ET
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Bibliographie 511 LILLO, F. ET R. N
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Bibliographie 513 PATTON, A. (2001)
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Bibliographie 515 SUSMEL, R. (1996)
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