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136 4. Relaxation de la volatilité
Volatility Fingerprints of Large Shocks: Endogeneous Versus Exogeneous ∗ D. Sornette 1,2 , Y. Malevergne 1,3 and J.-F. Muzy 4 1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622 Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France 2 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science University of California, Los Angeles, California 90095, USA 3 Institut de Science Financière et d’Assurances - Université Lyon I 43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France 4 Laboratoire Systèmes Physiques de l’Environemment, CNRS UMR 6134 Université de Corse, Quartier Grossetti, 20250 Corte, France email: sornette@unice.fr, Yannick.Malevergne@unice.fr and muzy@univ-corse.fr fax: (310) 206 30 51 Forthcoming Risk Abstract Finance is about how the continuous stream of news gets incorporated into prices. But not all news have the same impact. Can one distinguish the effects of the Sept. 11, 2001 attack or of the coup against Gorbachev on Aug., 19, 1991 from financial crashes such as Oct. 1987 as well as smaller volatility bursts? Using a parsimonious autoregressive process with long-range memory defined on the logarithm of the volatility, we predict strikingly different response functions of the price volatility to great external shocks compared to what we term endogeneous shocks, i.e., which result from the cooperative accumulation of many small shocks. These predictions are remarkably well-confirmed empirically on a hierarchy of volatility shocks. Our theory allows us to classify two classes of events (endogeneous and exogeneous) with specific signatures and characteristic precursors for the endogeneous class. It also explains the origin of endogeneous shocks as the coherent accumulations of tiny bad news, and thus unify all previous explanations of large crashes including Oct. 1987. 1 Introduction A market crash occurring simultaneously on most of the stock markets of the world as witnessed in Oct. 1987 would amount to the quasi-instantaneous evaporation of trillions of dollars. Market crashes are the extreme end members of a hierarchy of market shocks, which shake stock markets repeatedly. Among ∗ We acknowledge helpful discussions and exchanges with E. Bacry and V. Pisarenko. This work was partially supported by the James S. Mc Donnell Foundation 21st century scientist award/studying complex system. 1 137
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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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empirical analog FN(x), estimated f
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have been chosen to converge to 1 a
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 101 and 102: where E0[·] denotes the expectatio
Page 103 and 104: References Andersen, J.V. and D. So
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Page 107 and 108: Rubinstein, M., 1973, The fundament
Page 109 and 110: (a) Independent Data Stretched-Expo
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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 131 and 132: Tail 1−F(x) and parameter b Tail
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Page 135: Chapitre 4 Relaxation de la volatil
Page 139 and 140: 2 Long-range memory and distinction
Page 141 and 142: Figure 2: Measuring the conditional
Page 143 and 144: the system to the accumulation of m
Page 145 and 146: Appendix C: “Conditional response
Page 147 and 148: Baillie, R.T., 1996, Long memory pr
Page 149 and 150: Chapitre 5 Approche comportementale
Page 151 and 152: 5.1. Prix d’un actif et excès de
Page 153 and 154: 5.2. Modèles d’opinion contre mo
Page 155 and 156: 5.4. Conséquences des phénomènes
Page 157 and 158: 5.5. Conclusion 157 ce qui induit u
Page 159 and 160: Chapitre 6 Comportements mimétique
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Page 163 and 164: A Corcos et al Q UANTITATIVE F INAN
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186 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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