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110 3. Distributions exponentielles étirées contre distributions régulièrement variables (a) Independent data Stretched-Exponential c=0.7 Stretched-Exponential c=0.3 Pareto Distribution b=3 quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1 N/k=4 N/k=4 N/k=4 mean 0.0791 0.0689 0.1172 0.1432 mean 0.3843 0.4340 0.6301 0.7768 mean 0.3460 0.3496 0.3359 0.3334 emp. Std 0.5156 0.3614 0.1652 0.1154 emp. Std 0.5643 0.3925 0.1800 0.1340 emp. Std 0.5384 0.3631 0.1760 0.1175 th. Std 0.5148 0.3635 0.1636 0.1161 th. Std 0.5393 0.3847 0.1786 0.1302 th. Std 0.5358 0.3791 0.1691 0.1195 N/k=10 N/k=10 N/k=10 mean 0.0726 0.0652 0.0880 0.1083 mean 0.3613 0.3819 0.5028 0.5950 mean 0.3134 0.3379 0.3213 0.3382 emp. Std 0.8218 0.5925 0.2602 0.1839 emp. Std 0.8961 0.6141 0.2741 0.1963 emp. Std 0.9003 0.6056 0.2551 0.1933 th. Std 0.8133 0.5745 0.2577 0.1827 th. Std 0.8493 0.6027 0.2755 0.1983 th. Std 0.8425 0.5982 0.2668 0.1892 (b) Dependent data Stretched-Exponential c=0.7 with long memory Stretched-Exponential c=0.3 with long memory Pareto b=3 with long memory quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1 quantile 0.005 0.01 0.05 0.1 N/k=4 N/k=4 N/k=4 mean -0.0439 -0.0736 -0.2134 -0.3395 mean 0.1297 0.1215 0.0410 0.0235 mean 0.1458 0.1261 -0.0023 -0.0790 emp. Std 0.5396 0.3599 0.1643 0.1128 emp. Std 0.5557 0.3756 0.1633 0.1130 emp. Std 0.5572 0.3765 0.1633 0.1125 th. Std 0.5073 0.3576 0.1579 0.1108 th. Std 0.5183 0.3661 0.1620 0.1143 th. Std 0.5194 0.3663 0.1612 0.1130 46 N/k=10 N/k=10 N/k=10 mean -0.0371 -0.0307 -0.0928 -0.1781 mean 0.0961 0.1275 0.0682 0.0521 mean 0.1239 0.1461 0.0584 0.0196 emp. Std 0.8325 0.5754 0.2561 0.1821 emp. Std 0.8314 0.5796 0.2607 0.1801 emp. Std 0.8352 0.5811 0.2608 0.1801 th. Std 0.8028 0.5680 0.2524 0.1771 th. Std 0.8158 0.5793 0.2570 0.1814 th. Std 0.8188 0.5808 0.2567 0.1807 Table 3: Pickands estimates (9) of the parameter ξ for the Generalized Pareto Distribution (7) for thresholds u corresponding to quantiles 90%,95%,99% ans 99.5% and two different values of the ratio N/k respectively equal to 4 and 10. 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.
(a) Dow Jones Positive Tail Negative Tail Maximum Maximum cluster 10 50 100 200 cluster 10 50 100 200 mean 0.2195 0.2150 0.2429 0.2884 mean 0.1346 0.1012 0.1222 0.1700 Emp Std 0.0850 0.1163 0.1342 0.1238 Emp Std 0.1318 0.1437 0.1549 0.1671 GPD GPD quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 mean 0.2479 0.2493 0.1808 0.3458 mean 0.2121 0.0244 0.2496 0.3118 Emp Std 0.0224 0.0436 0.0654 0.1209 Emp Std 0.0294 0.0344 0.0979 0.1452 (b) Nasdaq (Raw data) Positive Tail Negative Tail Maximum Maximum cluster 10 50 100 200 cluster 10 50 100 200 mean 1.2961 1.1992 0.9858 0.7844 mean 1.2105 0.4707 0.4146 0.4200 Emp Std 0.2837 0.2955 0.4377 0.4679 Emp Std 0.2780 0.4625 0.4117 0.3980 GPD GPD quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 mean 0.2123 0.3025 0.5973 0.5973 mean 0.1477 0.2189 0.5973 0.5973 Emp Std 0.0693 0.0712 0 0 Emp Std 0.0468 0.0798 0 0 (c) Nasdaq (Correted data) Positive Tail Negative Tail Maximum Maximum cluster 10 50 100 200 cluster 10 50 100 200 mean 0.2163 0.2652 0.2738 0.2771 mean 0.2530 0.2276 0.3173 0.3681 Emp Std 0.2461 0.2825 0.2364 0.2604 Emp Std 0.3345 0.3273 0.4303 0.4289 GPD GPD quantile 0.9 0.95 0.99 0.995 quantile 0.9 0.95 0.99 0.995 mean 0.2569 0.4945 0.5943 0.5973 mean 0.2560 0.3185 0.5929 0.5847 Emp Std 0.1287 0.1683 0.0304 0.0000 Emp Std 0.1794 0.2137 0.0446 0.0883 Table 4: Mean values and standard deviations of the Maximum Likelihood estimates of the parameter ξ for the distribution of maximum (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), are presented the results for the Dow Jones, in panel (b) for the Nasdaq for raw data and in panel (c) the Nasdaq corrected for the “lunch effect”. 47 111
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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 81 and 82: empirical analog FN(x), estimated f
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Page 85 and 86: 5 Comparison of the descriptive pow
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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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Page 139 and 140: 2 Long-range memory and distinction
Page 141 and 142: Figure 2: Measuring the conditional
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Page 145 and 146: Appendix C: “Conditional response
Page 147 and 148: Baillie, R.T., 1996, Long memory pr
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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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