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278 9. Mesure de la dépendance extrême entre deux actifs financiers Putting together equations (E.126) and (E.137) we obtain ∞ 1 dx ɛ 1 1 xν Cν which concludes the proof. 1 + 1 ν E.4 Derivation of equation (E.105) From equation (E.111), we can deduce ¯FY (y) = Using equations (E.98) and (E.102), we obtain where x−x0 ɛ 2 ν+1 2 Pɛ(α ˜ Yu + η − αy) = Pɛ(γ ˜ Yu − αy) · − 1 xν = O(ɛ 0 min{ν,1} ) , (E.138) ν−1 ν 2 Cν yν −2 1 + O(y ) . (E.139) γ = α 1 + Putting together these results yields for the leading order ∞ dy ¯ FY (y) · Pɛ(α ˜ Yu + η − αy) = ˜Yu = ν ν−1 1 + O( ˜ Y −2 u ) , (E.140) σ ν1/ν . (E.141) α ∞ dy ˜Yu 2 Cν ν α ˜ Yu ν−1 ν 2 Cν yν · ∞ 1 σ dx 1 · xν where the change of variable x = y has been performed in the last equation. ˜Yu Cν 1 + (γ ˜ Yu−αy) 2 ν σ 2 1 + 1 ν Cν α ˜ Yu σ x− γ α σ α ˜ Yu ν+1 2 2 ν+1 2 , (E.142) We now apply lemma 2 with x0 = γ σ α > 1 and ɛ = α ˜ which goes to zero as u goes to 1. This gives Yu which shows that thus which finally yields ∞ dy ¯ FY (y) · Pɛ(α ˜ Yu + η − αy) ∼u→1 ˜Yu Pr[X > F −1 −1 X (u), Y > FY (u)] ∼u→1 F −1 Y ( ˜ Yu) Pr[X > F −1 X −1 (u)|Y > F (u)] ∼u→1 λ = Y 1 1 + σ α 40 ν ν−1 2 Cν α ˜ Y ν u ν α γ ν α = (1 − u) γ , (E.143) , (E.144) ν α , (E.145) γ ν α , (E.146) γ ν . (E.147)
9.1. Les différentes mesures de dépendances extrêmes 279 References Abramovitz, E. and I.A. Stegun, 1972, Handbook of Mathematical functions (Dover Publications, New York). Andersen, J.V. and D. Sornette, 2001, Have your cake and eat it too: increasing returns while lowering large risks! Journal of Risk Finance 2, 70-82. Ang, A. and G. Bekaert, 2001, International asset allocation with regime shifts, forthcoming Review of Financial Studies. . Ang, A. and J. Chen, 2001, Asymmetric correlations of equity portfolios, forthcoming Journal of Financial Economics. Baig, T. and I. Golgfajn, 1998, Financial market contagion in the Asian crisis, IMF Working Paper. Bhansali, V. and M.B. Wise, 2001, Forecasting portfolio risk in normal and stressed markets, working paper (preprint at http://xxx.lanl.gov/abs/nlin.AO/0108022) Bookstaber, R., 1997, Global risk management: are we missing the point? Journal of Portfolio Management, 23, 102-107. Boyer, B.H., M.S Gibson and M. Lauretan, 1997, Pitfalls in tests for changes in correlations, International Finance Discussion Paper 597, Board of the Governors of the Federal Reserve System. Davis, R.A., T. Mikosch and B. Basrak, 1999, Sample ACF of multivariate stochastic recurrence equations with application to GARCH, Working paper. Calvo, S. and C.M. Reinhart, 1995, Capital flows to Latin Americ: Is there evidence of contagion effects?, in G.A. Calvo, M. Goldstein and E. Haochreiter,eds: Private Captial Flows to Emerging Market After the Mexican Crisis (Institue for Internoational Economics, Washington DC) Cizeau, P., M. Potters and J.P. Bouchaud, 2001, Correlation structure of extreme stock returns, Quantitative Finance 1, 217-222. Claessen, S., R.W. Dornbush and Y.C. Park, 2001, Contagion: Why crises spread and how this can be stopped, in S. Cleassens and K.J. Forbes eds, International Financial Contagion (Kluwer Academic Press). Coles, S., J. Heffernan and J. Tawn, 1999, Dependence measures for extreme value analyses, Extremes 2, 339-365. Embrechs, P., A.J. McNeil and D. Straumann, 1999, Correlation : Pitfalls and Alternatives. Risk, 69-71. Embrechts, P., A.J. McNeil and D. Straumann, 2001, Correlation and Dependency in Risk Management : Properties and Pitfalls, in : Dempster, M., ed., Value at Risk and Beyond (Cambridge University Press). Forbes, K.J. and R. Rigobon, 2002, No contagion, only interdependence: measuring stock market comovements, forthcoming Journal of Finance. Frees, E. and E. Valdez, 1998, Understanding Relationships using copulas, North Americam Actuarial Journal 2, 1-25. 41
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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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328 9. Mesure de la dépendance ext
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Chapitre 10 La mesure du risque Dan
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10.1. La théorie de l’utilité 3
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10.2. Les mesures de risque cohére
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10.3. Les mesures de fluctuations 3
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10.4. Conclusion 361 et de manière
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Chapitre 11 Portefeuilles optimaux
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11.2. Prise en compte des grands ri
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11.3. Equilibre de marché 367 s’
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11.5. Annexe 369 Collective Origin
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11.5. Annexe 371 point λ = 1. Thus
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Chapitre 12 Gestion des risques gra
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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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