statistique, théorie et gestion de portefeuille - Docs at ISFA

More documents

Recommendations

Info

324 9. Mesure de la dépendance extrême entre deux actifs financiers which give lim ¯Fε1,ε2 u→1 [F1 −1 (u) − β1F −1 Y (u) x, F2 −1 (u) − β2F −1 Y (u) x] = 1x>maxl 1 , β1 l2 and following the same calculations as in part A.1, it concludes the proof. β2 , (77) We can now apply Lebesgue’s theorem of dominated convergence (see part A.1 for the justification), which allows us to assert that lim dx fu(x) = u→1 Since lim u→1 the proof of theorem 2 is concluded. dx 1x>maxl 1 , β1 l β2 2 · f(x). (78) dx fu(x) = lim u→1 Pr X1 > F −1 1 (u)|X2 > F −1 2 (u) , (79) = λ, (80) 25
9.2. Estimation du coefficient de dépendance de queue 325 B Proofs of the corollaries B.1 First corollary Corollary 1 If the random variable Y has a rapidly varying distribution function, then λ = 0. Proof : Let us write For a rapidly varying function ¯ FY , we have t PY (tx) FY ¯ (t) = t PY (tx) ¯ FY (tx) ∀x > 1, lim t→∞ FY (tx) FY (t) ¯ · ¯ ¯FY (tx) ¯FY (t) . (81) = 0, (82) while the leftmost factor of the right-hand-side of equation (81) remains bounded as t goes to infinity, so that t PY (tx) FY ¯ (tx) lim t→∞ FY ¯ · (tx) FY ¯ = f(x) = 0 . (83) (t) Since f(x) = 0, we can apply lemma 2 without the hypothesis H3, which concludes the proof. B.2 Second corollary Corollary 2 Let Y be regularly varying with index (−α), and assume that hypothesis H3 is satisfied. Then, the coefficient of (upper) tail dependence is λ = 1 max 1, l β where l denotes the limit, when u → 1, of the ratio FX −1 (u)/FY −1 (u). α , (84) Proof : Karamata’s theorem (see Embrechts, Kluppelberg, and Mikosh (1997, p 567)) ensures that H1 is satisfied with f(x) = α xα+1 , which is sufficient to prove the corollary. To go one step further, let us define where L1(·) and L2(·) are slowly varying functions. ¯Fy(y) = y −α · L1(y), (85) ¯Fε(ε) = ε −α · L2(ε), (86) Using the proposition stated in Feller (1971, p 278), we obtain, for the distribution of the variable X ¯FX(x) ∼ x −α β α for large x. x · L1 + L2(x) , β (87) Assuming now, for simplicity, that L1 (resp. L2) goes to a constant C1 (resp. C2), this implies that H3 is satistified, since FX l = lim u→1 −1 (u) FY −1 = β 1 + (u) C2 βα 1 α C1 This allows us to obtain the equations (10) and (13). 26 . (88)
Page 1:
UNIVERSITE DE NICE-SOPHIA ANTIPOLIS
Page 4 and 5:
4 Table des matières 2.2.2 Bulles
Page 6 and 7:
6 Table des matières 7.3.2 Tests n
Page 8 and 9:
8 Table des matières 12.1.1 Distri
Page 10 and 11:
10 Table des matières Références
Page 12 and 13:
12 m’ont souvent été d’un gra
Page 14 and 15:
14 Introduction pour espérer se co
Page 16 and 17:
16 Introduction les distributions e
Page 18 and 19:
18 Introduction
Page 21 and 22:
Chapitre 1 Faits stylisés des rent
Page 23 and 24:
1.1. Rappel des faits stylisés 23
Page 25 and 26:
1.1. Rappel des faits stylisés 25
Page 27 and 28:
1.2. De la difficulté de représen
Page 29 and 30:
1.3. Modélisation des propriétés
Page 31 and 32:
Chapitre 2 Modèles phénoménologi
Page 33 and 34:
2.1. Bulles rationnelles multi-dime
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
2.2. Des bulles rationnelles aux kr
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Chapitre 3 Distributions exponentie
Page 65 and 66:
Empirical Distributions of Log-Retu
Page 67 and 68:
concerning tail fatness can be form
Page 69 and 70:
To fit our two data sets, section 4
Page 71 and 72:
properties, stressing the problems
Page 73 and 74:
Another problem lies in the determi
Page 75 and 76:
In order to obtain a process with S
Page 77 and 78:
and Sornette (1999) for instance. W
Page 79 and 80:
1. The Pareto distribution: 79 Fu(x
Page 81 and 82:
empirical analog FN(x), estimated f
Page 83 and 84:
have been chosen to converge to 1 a
Page 85 and 86:
5 Comparison of the descriptive pow
Page 87 and 88:
ased on the fact that the quantity
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
Page 99 and 100:
where α = (c ∗ ,d ∗ ). It can
Page 101 and 102:
where E0[·] denotes the expectatio
Page 103 and 104:
References Andersen, J.V. and D. So
Page 105 and 106:
Gouriéroux C. and A. Monfort, 1994
Page 107 and 108:
Rubinstein, M., 1973, The fundament
Page 109 and 110:
(a) Independent Data Stretched-Expo
Page 111 and 112:
(a) Dow Jones Positive Tail Negativ
Page 113 and 114:
Nasdaq Dow Jones Pos. Tail Neg. Tai
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
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
Page 129 and 130:
Wilks statistic (doubled log−like
Page 131 and 132:
Tail 1−F(x) and parameter b Tail
Page 133 and 134:
1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4
Page 135 and 136:
Chapitre 4 Relaxation de la volatil
Page 137 and 138:
Volatility Fingerprints of Large Sh
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
Page 161 and 162:
RESEARCH PAPER Q UANTITATIVE F INAN
Page 163 and 164:
A Corcos et al Q UANTITATIVE F INAN
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179:
Deuxième partie Etude des proprié
Page 182 and 183:
182 7. Etude de la dépendance à l
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
192 8. Tests de copule gaussienne
Page 194 and 195:
194 8. Tests de copule gaussienne 1
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
200 8. Tests de copule gaussienne t
Page 202 and 203:
202 8. Tests de copule gaussienne T
Page 204 and 205:
204 8. Tests de copule gaussienne a
Page 206 and 207:
206 8. Tests de copule gaussienne c
Page 208 and 209:
208 8. Tests de copule gaussienne t
Page 210 and 211:
210 8. Tests de copule gaussienne (
Page 212 and 213:
212 8. Tests de copule gaussienne R
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
228 8. Tests de copule gaussienne f
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
236 8. Tests de copule gaussienne T
Page 238 and 239:
238 9. Mesure de la dépendance ext
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
Page 272 and 273:
Page 274 and 275: 274 9. Mesure de la dépendance ext
Page 345 and 346: Chapitre 10 La mesure du risque Dan
Page 347 and 348: 10.1. La théorie de l’utilité 3
Page 353 and 354: 10.2. Les mesures de risque cohére
Page 359 and 360: 10.3. Les mesures de fluctuations 3
Page 361 and 362: 10.4. Conclusion 361 et de manière
Page 363 and 364: Chapitre 11 Portefeuilles optimaux
Page 365 and 366: 11.2. Prise en compte des grands ri
Page 367 and 368: 11.3. Equilibre de marché 367 s’
Page 369 and 370: 11.5. Annexe 369 Collective Origin
Page 371 and 372: 11.5. Annexe 371 point λ = 1. Thus
Page 373 and 374: Chapitre 12 Gestion des risques gra
Page 375 and 376:
12.1. Comprendre et gérer les risq
Page 377 and 378:
Page 379 and 380:
Page 381 and 382:
12.2. Minimiser l’impact des gran
Page 383 and 384:
Page 385 and 386:
Page 387 and 388:
Chapitre 13 Gestion de portefeuille
Page 389 and 390:
VaR-Efficient Portfolios for a Clas
Page 391 and 392:
dependence: from independence to co
Page 393 and 394:
given a series of return {rt}t foll
Page 395 and 396:
eads cV (u1, · · · , un) = 1
Page 397 and 398:
THEOREM 4 (TAIL EQUIVALENCE FOR A S
Page 399 and 400:
Independent Assets Comonotonic Asse
Page 401 and 402:
3.2 Typical recurrence time of larg
Page 403 and 404:
and finally w ∗ i = χ−1 i j
Page 405 and 406:
where the ˆwi’s are solution of
Page 407 and 408:
Choosing x > y > Aɛ, and applying
Page 409 and 410:
Expanding fi(xi) around x ∗ i yie
Page 411 and 412:
for all h ∈ AC. Thus, integrating
Page 413 and 414:
B Asymptotic distribution of the su
Page 415 and 416:
This yields × h+ Sc−2 − 2 dh
Page 417 and 418:
References Acerbi, A. and D. Tasche
Page 419 and 420:
Malevergne, Y. and D. Sornette, 200
Page 421 and 422:
ln(T/T 0 ) T Figure 2: Logarithm
Page 423 and 424:
Chapitre 14 Gestion de Portefeuille
Page 425 and 426:
Multi-Moments Method for Portfolio
Page 427 and 428:
choosen risk measure. Section 5 pre
Page 429 and 430:
The variance of the return X of an
Page 431 and 432:
This property is verified for all c
Page 433 and 434:
µn=α of order n = 2, 4, 6 and 8.
Page 435 and 436:
these two assets or portfolios. The
Page 437 and 438:
Due to the homogeneity property of
Page 439 and 440:
invested in the risky fund depends
Page 441 and 442:
C. Then, if g1(X1), · · · , gn(X
Page 443 and 444:
Differentiating with respect to x1,
Page 445 and 446:
7.2 Transformation of the modified
Page 447 and 448:
In the sequel, we will first evalua
Page 449 and 450:
8.3 Non-symmetric assets In the cas
Page 451 and 452:
y the variance will then increase).
Page 453 and 454:
A Description of the data set We ha
Page 455 and 456:
thus and finally 1 λ1 n−1 = w
Page 457 and 458:
C Composition of the market portfol
Page 459 and 460:
D Generalized capital asset princin
Page 461 and 462:
This brings us back to the problem
Page 463 and 464:
F.2 General case We now consider a
Page 465 and 466:
Harvey, C.R. and A. Siddique, 2000,
Page 467 and 468:
µ µ2 1/2 µ4 1/4 µ6 1/6 µ8 1/8
Page 469 and 470:
Mean (10 −3 ) Variance (10 −3 )
Page 471 and 472:
μ (daily return) x 10−3 2.5 2 1.
Page 473 and 474:
w i w i 0.2 0.15 0.1 0.05 Mean−μ
Page 475 and 476:
Figure 6: Schematic representation
Page 477 and 478:
Empirical Cumulative Distribution 1
Page 479 and 480:
Z 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0
Page 481 and 482:
Z 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0
Page 483 and 484:
10 0 10 −1 10 0 10 −1 10 −1 1
Page 485 and 486:
10 0 10 −1 10 0 10 −1 10 −1 1
Page 487 and 488:
κ 22 20 18 16 14 12 10 8 6 4 2 Exc
Page 489 and 490:
Return μ 0.12 0.11 0.1 0.09 0.08 0
Page 491 and 492:
Conclusions et Perspectives L’obj
Page 493 and 494:
Conclusion 493 en univers gaussien
Page 495 and 496:
Annexe A Evaluation de la conduite
Page 497 and 498:
A.2. Eléments de contexte 497 bora
Page 499 and 500:
A.4. Compétences acquises et ensei
Page 501 and 502:
A.5. Conclusion 501 de la recherche
Page 503 and 504:
Bibliographie ACERBI, C. (2002) :
Page 505 and 506:
Bibliographie 505 BOUCHAUD, J. P. E
Page 507 and 508:
Bibliographie 507 DE FINETTI, B. (1
Page 509 and 510:
Bibliographie 509 GRANGER, C. W. ET
Page 511 and 512:
Bibliographie 511 LILLO, F. ET R. N
Page 513 and 514:
Bibliographie 513 PATTON, A. (2001)
Page 515:
Bibliographie 515 SUSMEL, R. (1996)
show all

statistique, théorie et gestion de portefeuille - Docs at ISFA

Create successful ePaper yourself

Delete template?

Save as template?