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32 2. Modèles phénoménologiques de cours 2.1 Bulles rationnelles multi-dimensionnelles et queues épaisses Lux et Sornette (2002) ont démontré que les queues des distributions inconditionnelles des incréments de prix et de rendements associées au modèle de bulles rationnelles de Blanchard et Watson (1982) suivent des lois de puissance (décroissent de manière hyperbolique), dont l’exposant de queue µ est inférieur à un sur un large domaine. Bien que les queues en lois de puissance soient une caractéristique marquante relevée sur les données empiriques, la valeur numérique µ < 1 est en désaccord avec les estimations habituelles qui, selon ce type de modèle, donnent µ 3. Parmi les quatre hypothèses soutenant le modèle de bulles rationnelles de Blanchard et Watson (rationalité des agents, condition de non arbitrage, dynamique multiplicative, bulles indépendantes pour chaque actif), nous démontrons que le même résultat µ < 1 reste valable lorsque que l’on relaxe la dernière de ces hypothèses, i.e en permettant le couplage entre les différentes bulles des divers actifs. En conséquence, des extensions non linéaires de la dynamique des bulles ou une relaxation partielle du principe d’évaluation rationnel des prix sont nécessaires si l’on souhaite rendre compte des observations empiriques. Reprint from : Y. Malevergne and D. Sornette (2001), “Multi-dimensional bubbles and fat tails”, Quantitative Finance 1, 533-541.
2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 33 Q UANTITATIVE F INANCE V OLUME 1 (2001) 533–541 RESEARCH PAPER I NSTITUTE OF P HYSICS P UBLISHING quant.iop.org Multi-dimensional rational bubbles and fat tails Y Malevergne 1,2 and D Sornette 1,3 1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622, Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France 2 Institut de Science Financière et d’Assurances, Université Lyon I, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France 3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science, University of California, Los Angeles, CA 90095, USA E-mail: Yannick.Malevergne@unice.fr and sornette@unice.fr Received 20 July 2001, in final form 20 August 2001 Published 14 September 2001 Online at stacks.iop.org/Quant/1/533 Abstract Lux and Sornette have demonstrated that the tails of the unconditional distributions of price differences and of returns associated with the model of rational bubbles of Blanchard and Watson follow power laws (i.e. exhibit hyperbolic decline), with an asymptotic tail exponent µ
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have been chosen to converge to 1 a
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5 Comparison of the descriptive pow
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ased on the fact that the quantity
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tail (positive or negative) of the
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Equation (46) depends on c only and
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B Minimum Anderson-Darling Estimato
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C Local exponent In this Appendix,
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D Testing non-nested hypotheses wit
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where α = (c ∗ ,d ∗ ). It can
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where E0[·] denotes the expectatio
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References Andersen, J.V. and D. So
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Gouriéroux C. and A. Monfort, 1994
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Rubinstein, M., 1973, The fundament
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(a) Independent Data Stretched-Expo
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(a) Dow Jones Positive Tail Negativ
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Nasdaq Dow Jones Pos. Tail Neg. Tai
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Dow Jones positive tail Dow Jones n
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Variation coefficient Variation coe
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Mean excess function Mean excess fu
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Hill‘s estimate b u Hill‘s esti
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Wilks statistic (doubled log−like
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Tail 1−F(x) and parameter b Tail
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1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4
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Chapitre 4 Relaxation de la volatil
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Volatility Fingerprints of Large Sh
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2 Long-range memory and distinction
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Figure 2: Measuring the conditional
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the system to the accumulation of m
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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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