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94 3. Distributions exponentielles étirées contre distributions régulièrement variables B.3 Exponential distribution In the exponential case, equation(63) becomes with N wk ∑ k=1 Fk Here again, we can show that this equation admits a unique solution. − 1 (u − xk) = 0, (70) u − xk Fk = 1 − exp − . (71) d 30
C Local exponent In this Appendix, we come back to the notion of a local Pareto exponent discussed above in section 4.3.1, with figure 7 and expression (33). We call the local exponent β. Generally speaking, any positive smooth function g(x) can be represented in the form g(x) = 1/x β(x) , x 1, (72) by defining β(x) = −ln(g(x))/lnx. Thus, the local index β(x) of a distribution F(x) will be defined as β(x) = − 95 ln(1 − F(x)) . (73) ln(x) For example, the log-normal distribution can be mistaken for a power law over several decades with very slowly (logarithmically) varying exponents if its variance is large (see figures 4.2 and 4.3 in section 4.1.3 of (Sornette 2000)). When we approximate a sample distribution by some parametric family with moving index β(x), it is important that β(x) should have as small a variation as possible, i.e., that the representation (72) be parsimonious. Given a sample x1,x2,··· ,xN, drawn from a distribution function F(x), x ≥ 1, the tail index β(x) is consistently estimated by ˆβ(x) = lnN − ln ∑i 1 {xi>x} , (74) ln(x) where 1 {·} is the indicator function which equals one if its argument is true and zero otherwhise. The asymptotic distribution of the estimator is easily derived and reads: with N 1/2 · e ln(x)·ˆ β(x) − e ln(x)·β(x) d −→ N (0,σ 2 ), (75) σ 2 = 1 . (76) F(x) · (1 − F(x)) As an example, let us illustrate the properties of the local index β(x) for regularly varying distributions. The general representation of any regularly varying distribution is given by ¯F(x) = L(x) · x −α , where L(·) is a slowly varying function. In such a case, the local power index can be written as β(x) = α − lnL(x) , (77) ln(x) which goes to α as x goes to infinity, as expected from the build-in slow variation of L(x). The upper panel of figure 10 shows the local index β(x) for a simulated Pareto distribution with power index b = 1.2 (X > 1). The estimate β(x) oscillates very closely to the true value b = 1.2. Let us now assume that we observe a regularly varying local exponent β(x) = L(x) · x c , with c > 0, (78) it would be clearly the charaterization of a Stretched-Exponential distribution ¯F(x) = exp −L ′ (x) · x c , (79) 31
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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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Page 85 and 86: 5 Comparison of the descriptive pow
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Page 113 and 114: Nasdaq Dow Jones Pos. Tail Neg. Tai
Page 121 and 122: Dow Jones positive tail Dow Jones n
Page 123 and 124: Variation coefficient Variation coe
Page 125 and 126: Mean excess function Mean excess fu
Page 127 and 128: Hill‘s estimate b u Hill‘s esti
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Page 135 and 136: Chapitre 4 Relaxation de la volatil
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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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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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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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