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112 3. Distributions exponentielles étirées contre distributions régulièrement variables (a) Dow Jones Negative Tail Positive Tail quantile 0.995 0.99 0.95 0.90 quantile 0.995 0.99 0.95 0.90 N/k =4 N/k =4 mean 0.2491 -0.0366 0.2721 0.2268 mean 0.6427 -0.3035 0.4006 0.2338 emp. Std 0.3561 0.2382 0.0974 0.0639 emp. Std 0.3913 0.2233 0.0902 0.0695 th. Std 0.5274 0.3590 0.1674 0.1175 th. Std 0.5663 0.3510 0.1710 0.1177 N/k =10 N/k=10 mean 0.4449 -0.2539 0.0279 0.2905 mean -0.2228 0.8057 0.3622 0.3107 emp. Std 0.5354 0.5563 0.1314 0.0943 emp. Std 1.3601 0.5420 0.1301 0.1227 th. Std 0.8619 0.5568 0.2558 0.1877 th. Std 0.7891 0.6552 0.2686 0.1883 (b) Nasdaq (Raw data) Negative Tail Positive Tail quantile 0.995 0.99 0.95 0.90 quantile 0.995 0.99 0.95 0.90 N/k =4 N/k=4 mean 0.3189 0.0006 0.1055 0.0284 mean 1.0709 0.2538 0.1090 0.0386 emp. Std 0.2489 0.1295 0.0649 0.043 emp. Std 0.3461 0.2190 0.0626 0.0384 th. Std 0.5333 0.3605 0.1634 0.1144 th. Std 0.6222 0.3732 0.1634 0.1145 N/k =10 N/k =10 mean 0.2034 -0.1098 0.1634 0.2445 mean -0.8666 1.1228 0.1601 0.2846 emp. Std 0.6173 0.4596 0.0871 0.0571 emp. Std 0.6405 0.3499 0.1157 0.0762 th. Std 0.8281 0.5634 0.2604 0.1863 th. Std 0.7835 0.7042 0.2602 0.1875 (c) Nasdaq (Corrected data) Negative Tail Positive Tail quantile 0.995 0.99 0.95 0.90 quantile 0.995 0.99 0.95 0.90 N/k =4 N/k=4 mean 0.1910 0.3435 0.0616 0.2070 mean 1.4051 -0.1313 0.0255 0.2259 emp. Std 0.2238 0.1441 0.0487 0.0333 emp. Std 0.2299 0.1424 0.0739 0.0390 th. Std 0.5228 0.3787 0.1624 0.1172 th. Std 0.6742 0.3556 0.1617 0.1175 N/k =10 N/k =10 mean 0.0630 0.0543 0.4537 -0.0699 mean -0.3357 1.3961 0.4034 0.0770 emp. Std 0.4899 0.1004 0.0780 0.0614 emp. Std 0.6830 0.2481 0.0891 0.0643 th. Std 0.5137 0.3629 0.1727 0.1150 th. Std 0.7835 0.7521 0.2705 0.1820 Table 5: 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 respectiveley equal to 4 and 10. 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”. 48
Nasdaq Dow Jones Pos. Tail Neg. Tail Pos. Tail Neg. Tail q 10 3 u nu 10 3 u nu 10 2 u nu 10 2 u nu q1=0 0.0053 11241 0.0053 10751 0.0032 14949 0.0028 13464 q2=0.1 0.0573 10117 0.0571 9676 0.0976 13454 0.0862 12118 q3=0.2 0.1124 8993 0.1129 8601 0.1833 11959 0.1739 10771 q4=0.3 0.1729 7869 0.1723 7526 0.2783 10464 0.263 9425 q5=0.4 0.238 6745 0.2365 6451 0.3872 8969 0.3697 8078 q6=0.5 0.3157 5620 0.3147 5376 0.5055 7475 0.4963 6732 q7=0.6 0.406 4496 0.412 4300 0.6426 5980 0.6492 5386 q8=0.7 0.5211 3372 0.5374 3225 0.8225 4485 0.8376 4039 q9=0.8 0.6901 2248 0.7188 2150 1.0545 2990 1.1057 2693 q10=0.9 0.973 1124 1.0494 1075 1.4919 1495 1.6223 1346 q11=0.925 1.1016 843 1.1833 806 1.6956 1121 1.8637 1010 q12=0.95 1.2926 562 1.3888 538 1.9846 747 2.2285 673 q13=0.96 1.3859 450 1.4955 430 2.1734 598 2.4197 539 q14=0.97 1.53 337 1.639 323 2.413 448 2.7218 404 q15=0.98 1.713 225 1.8557 215 2.7949 299 3.1647 269 q16=0.99 2.1188 112 1.8855 108 3.5704 149 4.1025 135 q17=0.9925 2.3176 84 2.4451 81 3.9701 112 4.3781 101 q18=0.995 3.0508 56 2.7623 54 4.5746 75 5.0944 67 Table 6: Significance levels qk and their corresponding lower thresholds uk for the four different samples. The number nu provides the size of the sub-sample beyond the threshold uk. 49 113
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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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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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