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132 3. Distributions exponentielles étirées contre distributions régulièrement variables Tail and parameter b 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Sample tail 1−F(x)(tick) and Pareto−b(thin), simulated crossover Simulated 2 Pareto sample b 1 =0.70; b 2 =1.5; Lower threshold u 0 =1; Crossover point u 1 =10; PX>10≅0.1; ( ) 20 40 60 80 100 x 120 140 160 180 200 Figure 11: Local index β(x) for a distribution constructed by joining two Pareto distributions with exponents b1 = 0.70 and b2 = 1.5 at the cross-over point u1 = 10. 68
1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 x 10 −3 0 log−return x 10 1 10 0 10 −1 10 −2 10 −5 10 1 10 0 10 −1 10 −3 y ∼ x 0.54 10 −4 10 −2 log−return x y ∼ x 0.77 Figure 12: Upper panel: Sample tail (continuous line), local index β(x) (dashed line) and local exponent c(x) (dash-dotted line) for the negative tail of the Nasdaq five minutes returns. Lower panel: doubled logarithmic plot of the local index β(x). Over most of the data set, β(x) increases with the log-return x as a power law of index 0.77 while beyond the quantile 99% (see the inset) it behaves like another power law of a smaller index equal to 0.54. The goodness of fit with these two power laws has been checked by a χ 2 test. 69 10 −3 10 −2 133
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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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Chapitre 3 Distributions exponentie
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Empirical Distributions of Log-Retu
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concerning tail fatness can be form
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To fit our two data sets, section 4
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properties, stressing the problems
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Another problem lies in the determi
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In order to obtain a process with S
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and Sornette (1999) for instance. W
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1. The Pareto distribution: 79 Fu(x
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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
Page 95 and 96: C Local exponent In this Appendix,
Page 97 and 98: D Testing non-nested hypotheses wit
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Page 103 and 104: References Andersen, J.V. and D. So
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Page 107 and 108: Rubinstein, M., 1973, The fundament
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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
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Page 149 and 150: Chapitre 5 Approche comportementale
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Page 157 and 158: 5.5. Conclusion 157 ce qui induit u
Page 159 and 160: Chapitre 6 Comportements mimétique
Page 161 and 162: RESEARCH PAPER Q UANTITATIVE F INAN
Page 163 and 164: A Corcos et al Q UANTITATIVE F INAN
Page 179: 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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