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114 3. Distributions exponentielles étirées contre distributions régulièrement variables Mean AD-statistic for u1 – u9 N-pos N-neg DJ-pos DJ-neg Weibull 1.86 (89.34%) 1.34 (79.18%) 5.43 (99.82%) 3.85 (98.91%) Gen. Pareto 4.45 (99.45%) 2.68 (96.01%) 12.47 (99.996%) 6.44 (99,99%) Gamma 3.59 (98.59%) 2.82 (96.62%) 8.76 (99.996%) 7.23 (99.996%) Exponential 3.64 (98.66%) 2.76 (96.36%) 13.96 (99.996%) 10.10 (99.996%) Pareto 475.2 (99.996%) 441.7 (99.996%) 691.7 (99.996%) 607.9 (99.996%) Mean AD-statistic for u10 – u18 Weibull 1.21 (74.29%) .988 (63.39%) .835 (53,52%) .849 (54.54%) Gen. Pareto 2.29 (93.57%) 1.88 (89.52%) 1.95 (90.28%) 1.36 (79.67%) Gamma 2.49 (95.00%) 1.90 (89.74%) 2.12 (92.01%) 1.63 (86.02%) Exponential 3.26 (97.97%) 1.93 (90.02%) 4.52 (99.10%) 2.70 (96.11%) Pareto 1.80 (88.60%) 1.77 (88.23%) 1.18 (73.01%) 1.65 (86.40%) Table 7: Mean Anderson-Darling distances in the range of thresholds u1-u9 and in the range u10-u18. The figures within parenthesis characterize the goodness of fit: they represent the significance levels with which the considered model can be rejected. 50
Nasdaq Dow Jones Pos. Tail Neg. Tail Pos. Tail Neg. Tail MLE ADE MLE ADE MLE ADE MLE ADE 1 0.256 (0.002) 0.192 0.254 (0.002) 0.191 0.204 (0.002) 0.150 0.199 (0.002) 0.147 2 0.555 (0.006) 0.443 0.548 (0.006) 0.439 0.576 (0.005) 0.461 0.538 (0.005) 0.431 3 0.765 (0.008) 0.630 0.755 (0.008) 0.625 0.782 (0.007) 0.644 0.745 (0.007) 0.617 4 0.970 (0.011) 0.819 0.945 (0.011) 0.800 0.989 (0.010) 0.833 0.920 (0.009) 0.777 5 1.169 (0.014) 1.004 1.122 (0.014) 0.965 1.219 (0.013) 1.053 1.114 (0.012) 0.960 6 1.400 (0.019) 1.227 1.325 (0.018) 1.157 1.447 (0.017) 1.279 1.327 (0.016) 1.169 7 1.639 (0.024) 1.460 1.562 (0.024) 1.386 1.685 (0.022) 1.519 1.563 (0.021) 1.408 8 1.916 (0.033) 1.733 1.838 (0.032) 1.655 1.984 (0.030) 1.840 1.804 (0.028) 1.659 9 2.308 (0.049) 2.145 2.195 (0.047) 1.999 2.240 (0.041) 2.115 2.060 (0.040) 1.921 10 2.759 (0.082) 2.613 2.824 (0.086) 2.651 2.575 (0.067) 2.474 2.436 (0.066) 2.315 11 2.955 (0.102) 2.839 3.008 (0.106) 2.836 2.715 (0.081) 2.648 2.581 (0.081) 2.467 12 3.232 (0.136) 3.210 3.352 (0.145) 3.259 2.787 (0.102) 2.707 2.765 (0.107) 2.655 13 3.231 (0.152) 3.193 3.441 (0.166) 3.352 2.877 (0.118) 2.808 2.782 (0.120) 2.642 14 3.358 (0.183) 3.390 3.551 (0.198) 3.479 2.920 (0.138) 2.841 2.903 (0.144) 2.740 15 3.281 (0.219) 3.306 3.728 (0.254) 3.730 2.989 (0.173) 2.871 3.059 (0.186) 2.870 16 3.327 (0.313) 3.472 3.990 (0.384) 3.983 3.226 (0.263) 3.114 3.690 (0.318) 3.668 17 3.372 (0.366) 3.636 3.917 (0.435) 3.860 3.427 (0.322) 3.351 3.518 (0.350) 3.397 18 3.136 (0.415) 3.326 4.251 (0.578) 4.302 3.818 (0.441) 3.989 4.168 (0.506) 4.395 Table 8: Maximum Likelihood and Anderson-Darling estimates of the Pareto parameter b. Figures within parentheses give the standard deviation of the Maximum Likelihood estimator. 51 115
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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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A Corcos et al Q UANTITATIVE F INAN
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Deuxième partie Etude des proprié
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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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