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40 2. Modèles phénoménologiques de cours Y Malevergne and D Sornette Q UANTITATIVE F INANCE The no-arbitrage condition thus implies: EQ[A] = 1 δ Id. We now show that the converse is true, namely that if EQ[A] = 1 δ Id is true, then the no-arbitrage condition is verified. Let us thus assume that EQ[A] = 1 δ Id holds. Then W ′ EQ[Pt+1|Ft] = EQ[W ′ t+1 · Xt+1|Ft] (A.7) = W ′ t+1 · EQ[Xt+1|Ft] (A.8) = W ′ t+1 · EQ[At+1Xt + Bt+1|Ft] (A.9) = W ′ t+1 · EQ[At+1|Ft] · Xt (A.10) = W ′ t+1 · EQ[A] · Xt (A.11) = 1 ′ W t+1 δ · Xt. (A.12) The condition that the portfolio is self-financing is t+1 Xt = W ′ t Xt, which means that the weights can be rebalanced a priori arbitrarily between the assets with the constraint that the total wealth at the same time remains constant. We can thus write EQ[t+1|Ft] = 1 ′ W t+1 · Xt (A.13) δ = 1 δ t. (A.14) Therefore, the discounted process {t} is a Q-martingale. Appendix B. Proof that δ (i) is the discount factor for the ith bubble component in the historical space Here, we express the no-free-lunch condition in the historical space (or real space). The condition we will obtain is the socalled ‘rational expectation condition’, which is a little bit less general than the condition detailed in the previous appendix A. Given the prices {X (i) k }kt of an asset, labelled by i, until the date t, the best estimation of its price at t +1isEP[X (i) t+1 |Ft]. So, the RE condition leads to EP[X (i) t+1 |Ft] − X (i) t X (i) = r t (i) t , (A.15) where r (i) t is the return of the asset i between t and t +1. As previously, we will assume in what follows that r (i) t = r (i) is time independent. Thus, the rational expectation condition for the assets i reads X (i) t = δ (i) · EPX = δ (i) · EP X (i) t+1 |Ft X (i) t+1 |Ft , (A.16) , (A.17) where δ (i) is the discount factor. A priori, each asset has a different return. Thus, introducing the vector ˜Xt whose ith component is X(i) t δ (i) ,we can summarize the rational expectation condition as 540 ˜Xt = EP [Xt+1|Ft] . (A.18) Again, we evaluate the conditional expectation of (16), and using the fact that {At} are i.i.d., we have = EP[A]Xt−1. (A.19) EP Xt|Ft−1 This equation together with (A.18), leads to ˜Xt−1 = EP[A]Xt−1, (A.20) which can be rewritten EP[A] − δ−1 Xt−1 = 0, (A.21) where δ−1 = diag[δ (1)−1 ...δ (d)−1 ] is the matrix whose ith diagonal component is δ (i)−1 and 0 elsewhere. The equation (A.21) must be true for every Xt−1 ∈ Rd , thus EP[A] = δ−1 , (A.22) which is the result announced in section (3.3.2). Appendix C. Proof of proposition 2 on the condition κ1 < 1 First step. Behaviour of the function 1 f(κ)= lim ln EPA ||An ...A1|| n→∞ n κ1 in the interval [0,κ0]. (A.23) In the fourth step of the proof of theorem 3, Kesten (1973) shows that the function f has the following properties: • f is continuous on [0,κ0], • f(0) = 0 and f(κ0) >0, • f ′ (0) 1, it is necessary that f(1)
2.1. Bulles rationnelles multi-dimensionnelles et queues épaisses 41 Q UANTITATIVE F INANCE Multi-dimensional rational bubbles f(k) f(1) 0 Figure A1. Schematic shape of the function f(κ)defined in (A.23). where ρ(EPA [A]) is the spectral radius of A. By definition, || EPA [A] n || | EPA [A] n xmax| =λ n max . (A.29) Then 1 lim n→∞ n ln EPA ||An 1 ...A1|| lim ln ρ(EPA [A])n n→∞ n = ln ρ(EPA [A]) . (A.30) Now, suppose that ρ(EPA [A]) 1. We obtain 1 f(1) = lim n→∞ n ln EPA ||An ...A1|| 0, (A.31) which is in contradiction with the necessary condition (A.24). Thus, References f(1)
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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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206 8. Tests de copule gaussienne c
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228 8. Tests de copule gaussienne f
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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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ln(T/T 0 ) T Figure 2: Logarithm
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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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invested in the risky fund depends
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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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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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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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