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146 4. Relaxation de la volatilité where It is thus easy to obtain the estimate: β ′ (t) = 2λ 2 ln(T/t) ln(T e3/2 /∆t) Varendo [σ2 (t) | ω0] ≤ Eendo[σ 2 (t) | ω0] ( T ∆t )2β(t) 1/2 − 1 6λ2 ln(te3/2 /∆t) Eendo[σ 2 (t) | ω0] We thus conclude, that, for s large enough (i.e., α(s) large enough): Eendo[σ 2 (t) | ω0] − σ 2 (t) Varendo [σ 2 (t) | ω0] 1 6λ 2 ln(te 3/2 /∆t) Over the first decade ∆t ≤ t ≤ 10∆t, the deviation of the conditional mean volatility from the unconditional volatility σ 2 (t) is greater than the conditional variance, which ensures the existence of a strong deterministic component of the conditional response above the stochastic components. Expressions (24,22) are our two main predictions. These equations predict that the conditional response function Eendo[σ2 (t) | ω0] of the volatility decays as a power law ∼ 1/tα of the time since the endogeneous shock, with an exponent α ≈ 2ω0 − ln(σ2 λ2 (t)) C(0) which depends linearly upon the amplitude ω0 of the shock. Note in particular, that α changes sign: it is positive for w0 > 1 2 ln(σ2 (t)) and negative otherwise. Appendix D: Determination of the sources of endogeneous shocks What is the source of endogeneous shocks characterized by the response function (21)? To answer, let us consider the process W (t) ≡ t −∞ dτ η(τ), where η(t) is a standardized Gaussian white noise which captures the information flow impacting on the volatility, as defined in (5). Extending the property (16), we find that t Cov[W (t), ω0] Eendo[W (t) | ω0] = · (ω0 − E[ω0]) ∝ (ω0 − E[ω0]) dτ K(−τ) . (29) Var[ω0] −∞ Expression (29) predicts that the expected path of the continuous information flow prior to the endogeneous shock (i.e., for t < 0) grows like ∆W (t) = η(t)∆t ∼ K(−t)∆t ∼ ∆t/ √ −t for t < 0 upon the approach to the time t = 0 of the large endogeneous shock. In other words, conditioned on the observation of a large endogeneous shock, there is specific set of trajectories of the news flow η(t) that led to it. These conditional news flows have an expectation given by (29). References Adam, M.C. and A. Szafarz, 1992, Speculative Bubbles and Financial Markets, Oxford Economic Papers 44, 626-640. Arneodo, A., J.F. Muzy and D. Sornette, 1998, Direct causal cascade in the stock market, The European physical Journal B 2, 277-282. Bacry, E., J. Delour and J.F. Muzy, 2001, Multifractal ranom walk, Physical Review E 64, 026103. 10 (26) (27) (28)
Baillie, R.T., 1996, Long memory processes and fractional integration in econometrics, Journal of Econometrics 73, 5-59. Barro, R.J., E.F. Fama, D.R. Fischel, A.H. Meltzer, R. Roll and L.G. Telser, 1989, Black monday and the future of financial markets, edited by R.W. Kamphuis, Jr., R.C. Kormendi and J.W.H. Watson (Mid American Institute for Public Policy Research, Inc. and Dow Jones-Irwin, Inc.). Camerer, C., 1989, Bubbles and Fads in Asset Prices, Journal of Economic Surveys 3, 3-41. Cutler, D., J. Poterba and L. Summers, What Moves Stock Prices? 1989, Journal of Portfolio Management, Spring, 4-12. Ding, Z., Granger, C.W.J. and Engle, R., 1993, A long memory property of stock returns and a new model, Journal of Empirical Finance 1, 83-106. Ding, Z., Granger, C.W.J., 1996, Modeling volatility persistence of speculative returns: A new approach, Journal of Econometrics 73, 185-215. Feigenbaum, J.A., 2001, A statistical analysis of log-periodic precursors to financial crashes, Quantitative Finance 1(3), 346-360. Granger, C.W.J. and Ding, Z., 1996, Varieties of long memory models, Journal of Econometrics 73, 61-77. Johansen, A. and Sornette, 1999, Critical Crashes, Risk 12 (1), 91-94. Lillo, F. and R.N. Mantegna, Power-law relaxation in a complex system: the fluctuation decay after a financial market crash, preprint cond-mat/0111257 Mandelbrot, B.B., 1997, Fractals and scaling in finance : discontinuity, concentration, risk : selecta volume E, New York : Springer. Müller, U.A., M.M. Dacorogna, R. Dav, R.B. Olsen, O.V. Pictet and J.E. von Weizs”cker, 1997, Volatilities of Different Time Resolutions - Analyzing the Dynamics of Market Components, Journal of Empirical Finance 4, No. 2-3, 213-240. Muzy, J.F., J. Delour and E. Bacry, 2000, Modelling fluctuations of financial time series: from cascade process to stochastic volatility model, The European physical Journal B 17, 537-548. Muzy, J.-F., D. Sornette, J. Delour and A. Arneodo, 2001, Multifractal returns and Hierarchical Portfolio Theory, Quantitative Finance 1 (1), 131-148. Orléan, A., 1989, Mimetic contagion and speculative bubbles, Theory and Decision 27, 63-92. Orléan, A., 1995, Bayesian interactions and collective dynamics of opinion: Heard behavior and mimetic contagion, journal of Economic Behavior & Organization 28, 257-274. Kim, S., N. Shepard, and S. Chib , 1998, Stochastic volatility: Likelyhood inference and comparison with ARCH models, Review of Economic Studies 65, 361-393. Shiller, R.J., 2000, Irrational exuberance (Princeton University Press, Princeton, NJ). Sornette, D. and A. Johansen, 2001, Significance of log-periodic precursors to financial crashes, Quantitative Finance 1 (4), 452-471. Sornette, D. and J.V. Andersen, 2002, A Nonlinear Super-Exponential Rational Model of Speculative Financial Bubbles, Int. J. Mod. Phys. C 13 (2), 171-188. 11 147
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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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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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196 8. Tests de copule gaussienne
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198 8. Tests de copule gaussienne 2
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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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