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108 3. Distributions exponentielles étirées contre distributions régulièrement variables Mean St. Dev. Skewness Ex. Kurtosis Jarque-Bera Nasdaq (5 minutes) † 1.80 ·10 −6 6.61 ·10 −4 0.0326 11.8535 1.30 ·10 5 (0.00) Nasdaq (1 hour) † 2.40 ·10 −5 3.30 ·10 −3 1.3396 23.7946 4.40 ·10 4 (0.00) Nasdaq (5 minutes) ‡ - 6.33 ·10 −9 3.85 ·10 −4 -0.0562 6.9641 4.50 ·10 4 (0.00) Nasdaq (1 hour) ‡ 1.05 ·10 −6 1.90 ·10 −3 -0.0374 4.5250 1.58 ·10 3 (0.00) Dow Jones (1 day) 8.96·10 −5 4.70 ·10 −3 -0.6101 22.5443 6.03 ·10 5 (0.00) Dow jones (1 month) 1.80 ·10 −3 2.54 ·10 −2 -0.6998 5.3619 1.28 ·10 3 (0.00) Table 1: Descriptive statistics for the Dow Jones returns calculated over one day and one month and for the Nasdaq returns calculated over five minutes and one hour. The numbers within parenthesis represent the p-value of Jarque-Bera’s normality test. (†) raw data, (‡) data corrected for the U-shape of the intra-day volatility due to the lunch effect. 44
(a) Independent Data Stretched-Exponential c=0.7 Stretched-Exponential c=0.3 Pareto Distribution b=3 Maximum Maximum Maximum cluster 10 50 100 200 cluster 10 50 100 200 cluster 10 50 100 200 mean 0.1870 0.1734 0.2032 0.2290 mean 0.9313 0.9910 1.0617 1.1011 mean 0.3669 0.4007 0.4455 0.4769 Emp Std 0.0262 0.0340 0.0382 0.0415 Emp Std 0.0380 0.0464 0.0492 0.0508 Emp Std 0.0267 0.0353 0.0384 0.0407 GPD GPD GPD quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 mean 0.1108 0.0758 -0.1374 -0.0509 mean -0.1234 0.1947 -0.2147 -0.2681 mean 0.3336 0.3271 0.3193 0.2585 Emp Std 0.0346 0.1579 0.4648 0.4147 Emp Std 0.8139 0.6197 0.6625 0.597 Emp Std 0.0454 0.0604 0.1271 0.2991 (b) Dependent Data Stretched-Exponential c=0.7 with long memory Stretched-Exponential c=0.3 with long memory Pareto with long memory b=3 Maximum Maximum Maximum cluster 10 50 100 200 cluster 10 50 100 200 cluster 10 50 100 200 mean -0.2169 -0.2226 -0.2027 -0.1918 mean 0.1230 0.1143 0.1359 0.1569 mean 0.0566 0.0606 0.0794 0.0978 Emp Std 0.1553 0.1512 0.1553 0.1718 Emp Std 0.0328 0.0414 0.0450 0.0443 Emp Std 0.0422 0.0495 0.0533 0.0560 45 GPD GPD GPD quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 mean -0.1005 -0.0593 -0.0189 -0.0314 mean 0.0732 0.0740 0.0031 -0.0559 mean 0.2952 0.3203 0.3153 0.2570 Emp Std 0.0386 0.0503 0.1097 0.1700 Emp Std 0.0942 0.1804 0.3744 0.4822 Emp Std 0.0548 0.0694 0.1812 0.3382 Table 2: Mean values and standard deviations of the Maximum Likelihood estimates of the parameter ξ (inverse of the Pareto exponent) for the distribution of maxima (cf. equation 4) when data are clustered in samples of size 10,50,100 and 200 and for the Generalized Pareto Distribution (7) for thresholds u corresponding to quantiles 90%,95%,99% ans 99.5%. In panel (a), we have used iid samples of size 10000 drawn from a Stretched-Exponential distribution with c = 0.7 and c = 0.3 and a Pareto distribution with tail index b = 3, while in panel (b) the samples are drawn from a long memory process with Stretched-Exponential marginals and regularly-varying marginal as explained in the text. 109
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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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Page 85 and 86: 5 Comparison of the descriptive pow
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Page 89 and 90: tail (positive or negative) of the
Page 91 and 92: Equation (46) depends on c only and
Page 93 and 94: B Minimum Anderson-Darling Estimato
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Page 103 and 104: References Andersen, J.V. and D. So
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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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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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