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140 4. Relaxation de la volatilité ∫ dt E[σ(t) | ω 0 ] − E[σ(t)] 10 −1 10 −2 10 0 Nikkei 250 : Aug 19, 1991 S&P 500 : Aug 19, 1991 FT−SE 100 : Aug 19, 1991 CAC 40 : Sep 11, 2001 S&P 500 : Oct 19, 1987 10 1 Time (trading days) Slope α = 1/2 August 19, 1991 : Putsh against President Gorbachev September 11, 2001: Attack against the WTC Figure 1: Cumulative excess volatility at scale ∆t, that is, integral over time of Eexo[σ 2 (t) | ω0] − σ 2 (t), due to the volatility shock induced by the coup against President Gorbachev observed in three British, Japanese and USA indices and the shock induced by the attack of September 11, 2002 against the World Trade Center. The dashed line is the theoretical prediction obtained by integrating (1), which gives a ∝ √ t time-dependence. The cumulative excess volatility following the crash of October 1987 is also shown with circles. Notice that the slope of the non-constant curve for the October 1987 crash is very different from the value 1/2 expected and observed for exogeneous shocks. This crash and the resulting volatility relaxation can be interpreted as an endogeneous event. 4 10 2
Figure 2: Measuring the conditional volatility response exponent α(s) for S&P 100 intradaily time series as a function of the endogeneous shock amplitude parameterized by s, defined by (19). (a) The original 5 minute intradaily time series from 04/08/1997 to 12/24/2001. The 5 minute de-seasonalized squared returns are aggregated in order to estimate the 40 minutes and daily volatilities. (b) 40 minute log-volatility covariance C40(τ) as a function of the logarithm of the lag τ. The MRW theoretical curve with λ 2 = 0.018 and T = 1 year (dashed line) provides an excellent fit of the data up to lags of one month. (c) Conditional volatility response ln(Eendo[σ 2 (t) | s]) as a function of ln(t) for three shocks with amplitudes given by s = −1, 0, 1. (d) Estimated exponent α(s) for ∆t = 40 minutes (•) as a function of s. The solid line is the prediction corresponding to Eq. (22). The dashed line corresponds to the empirical MRW estimate obtained by averaging over 500 Monte-Carlo trials. It fits more accurately for negative s (volatility lower than normal) due to the fact that the estimations of the variance by aggregation over smaller scales is very noisy for small variance values. The error bars give the 95 % confidence intervals estimated by Monte-Carlo trials of the MRW process. In the inset, α(s) is compared for ∆t = 40 minutes (•) and ∆t = 1 day (×). 5 141
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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
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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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Page 93 and 94: B Minimum Anderson-Darling Estimato
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Page 139: 2 Long-range memory and distinction
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Page 145 and 146: Appendix C: “Conditional response
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Page 159 and 160: Chapitre 6 Comportements mimétique
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190 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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