This volume presents current research focusing on heavy tailed distributions in finance. The contributions cover methodological issues, i.e., probabilistic, statistical and econometric modelling under non- Gaussian assumptions, as well as the applications of the stable and other non -Gaussian models in finance and risk management.
The Handbooks in Finance are intended to be a definitive source for comprehensive and accessible information in the field of finance. Each individual volume in the series should present an accurate self-contained survey of a sub-field of finance, suitable for use by finance and economics professors and lecturers, professional researchers, graduate students and as a teaching supplement. The goal is to have a broad group of outstanding volumes in various areas of finance. The Handbook of Heavy Tailed Distributions in Finance is the first handbook to be published in this series.This volume presents current research focusing on heavy tailed distributions in finance. The contributions cover methodological issues, i.e., probabilistic, statistical and econometric modelling under non- Gaussian assumptions, as well as the applications of the stable and other non -Gaussian models in finance and risk management.
Front Cover 1
Handbook of Heavy Tailed Distributions in Finance 4
Copyright Page 5
Contents 14
Introduction to the Series 6
Contents of the Handbook 8
Preface 10
Chapter 1. Heavy Tails in Finance for Independent or Multifractal Price Increments 26
Abstract 29
1. Introduction: A path that led to model price by Brownian motion (Wiener or fractional) of a multifractal trading time 30
2. Background: the Bernoulli binomial measure and two random variants: shuffled and canonical 33
3. Definition of the two-valued canonical multifractals 36
4. The limit random variable O = µ([0, 1]), its distribution and the star functional equation 38
5. The function t(q): motivation and form of the graph 40
6. When u > 1, the moment EOq diverges if q exceeds a critical exponent qcrit satisfying t(q)= 0
7. The quantity a: the original Holder exponent and beyond 46
8. The full function f (a) and the function p(ß) 48
9. The fractal dimension D = t (1) = 2[–pulog2 u– (1 – p)v log2 v] and multifractal concentration 52
10. A noteworthy and unexpected separation of roles, between the “dimension spectrum”and the total mass . the former is ruled by the accessible a for which f (a) >
11. A broad form of the multifractal formalism that allows a < 0 and f (a) <
Acknowledgments 57
References 57
Chapter 2. Financial Risk and Heavy Tails 60
Abstract 61
1. Introduction 62
2. Historical perspective 63
3. Value at risk 70
4. Risk measures 83
5. Portfolios and dependence 86
6. Univariate extreme value theory 102
7. Stable Paretian models 119
Acknowledgments 125
References 126
Chapter 3. Modeling Financial Data with Stable Distributions 130
Abstract 131
1. Basic facts about stable distributions 132
2. Appropriateness of stable models 136
3. Computation, simulation, estimation and diagnostics 138
4. Applications to financial data 139
5. Multivariate stable distributions 141
6. Multivariate computation, simulation, estimation and diagnostics 146
7. Multivariate application 149
8. Classes of multivariate stable distributions 151
9. Operator stable distributions 153
10. Discussion 153
References 154
Chapter 4. Statistical Issues in Modeling Multivariate Stable Portfolios 156
Abstract 157
1. Introduction 158
2. Multivariate stable laws 159
3. Estimation of the index of stability 166
4. Estimation of the stable spectral measure 172
5. Estimation of the scale parameter 176
6. Extensions to other stable models 177
7. Applications 182
Acknowledgment 187
References 187
Chapter 5. Jump-Diffusion Models 194
Abstract 195
Keywords 195
1. Introduction 196
2. Preliminaries 198
3. Market models with jump-diffusions 207
4. Martingale measures: Existence and uniqueness (Market price of risk and market completion) 211
5. Hedging in jump-diffusion market models 218
6. Pricing in jump-diffusion models 226
References 232
Chapter 6. Hyperbolic Processes in Finance 236
Abstract 237
1. Hyperbolic and related distributions 238
2. Lévy processes 252
3. Stochastic differential equations 255
4. Stochastic volatility models 263
Acknowledgment 267
Appendix 268
References 269
Chapter 7. Stable Modeling of Market and Credit Value at Risk 274
Abstract 275
1. Introduction 276
2. “Normal” modeling of VaR 278
3. A finance-oriented description of stable distributions 280
4. VaR estimates for stable distributed financial returns 289
5. Stable modeling and risk assessment for individual credit returns 308
6. Portfolio credit risk for independent credit returns 312
7. Stable modeling of portfolio risk for symmetric dependent credit returns 315
8. Stable modeling of portfolio risk for skewed dependent credit returns 321
9. One-factor model of portfolio credit risk 324
10. Credit risk evaluation for portfolio assets 325
11. Portfolio credit risk 330
12. Conclusions 334
Appendix A. Stable modeling of credit returns in figures 336
Appendix B. Tables 342
Appendix C. OLS credit risk evaluation for portfolio assets in figures 345
Appendix D. GARCH credit risk evaluation for portfolio assets in figures 349
Acknowledgments 351
References 351
Chapter 8. Modelling Dependence with Copulas and Applications to Risk Management 354
1. Introduction 356
2. Copulas 357
3. Dependence concepts 366
4. Marshall–Olkin copulas 376
5. Elliptical copulas 382
6. Archimedean copulas 390
7. Modelling extremal events in practice 402
References 408
Chapter 9. Prediction of Financial Downside-Risk with Heavy-Tailed Conditional Distributions 410
Abstract 411
1. Introduction 412
2. GARCH-stable processes 413
3. Modeling exchange-rate returns 414
4. Prediction of densities and downside risk 423
5. Conclusions 427
References 428
Chapter 10. Stable Non-Gaussian Models for Credit Risk Management 430
Abstract 431
1. Stable modeling in credit risk – recent advances 432
2. A one-factor model for stable credit returns 433
3. Comparison of empirical results 440
4. The detection and measurement of long-range dependence 447
5. Conclusion 464
References 465
Chapter 11. Multifactor Stochastic Variance Models in Risk Management: Maximum Entropy Approach and Lévy Processes 468
Abstract 469
1. Review of market risk models 470
2. Single-factor stochastic variance model 475
3. Multifactor stochastic variance model 488
Acknowledgment 502
References 502
Chapter 12. Modelling the Term Structure of Monetary Rates 506
Abstract 507
1. Introduction 508
2. The mathematical framework 509
3. The tree representation 519
4. The econometric analysis 525
5. Conclusions 531
References 532
Chapter 13. Asset Liability Management: A Review and Some New Results in the Presence of Heavy Tails 534
Abstract 535
1. Introduction 536
Part I: Review of the stochastic programming ALM literature 538
2. Stochastic programming ALM models 538
3. Multistage stochastic ALM programming with decision rules 548
4. Scenario generation 549
Part II: Stable asset allocation 553
5. Stable distribution 553
6. Multistage stable asset allocation model with decision rules 557
7. Conclusion 567
References 568
Chapter 14. Portfolio Choice Theory with Non-Gaussian Distributed Returns 572
Abstract 573
1. Introduction 574
2. Choices determined by a finite number of parameters 578
3. The asymptotic distributional classification of portfolio choices 586
4. A first comparison between the normal multivariate distributional assumption and the stable sub-Gaussian one 599
5. Conclusions 606
Acknowledgment 607
Appendix A: Proofs 607
Appendix B: Tables 610
References 615
Chapter 15. Portfolio Modeling with Heavy Tailed Random Vectors 620
Abstract 621
Keywords 621
1. Introduction 622
2. Heavy tails 622
3. Central limit theorems 625
4. Matrix scaling 632
5. The spectral decomposition 635
6. Sample covariance matrix 638
7. Dependent random vectors 641
8. Tail estimation 644
9. Tail estimator proof for dependent random vectors 651
10. Conclusions 661
References 662
Chapter 16. Long Range Dependence in Heavy Tailed Stochastic Processes 666
Abstract 667
Keywords 667
1. Introduction 668
2. What is long range dependence? 669
3. Tails and rare events 671
4. Some classes of heavy tailed processes 674
5. Rare events, associated functionals and long range dependence 677
References 686
Author Index 688
Subject Index 700
10 A noteworthy and unexpected separation of roles, between the “dimensionspectrum” and the total mass Ω; the former is ruled by the accessible α forwhich f(α) > 0, the latter, by the inaccessible α for which f(α) < 0
Brought together, Sections 4, 7, 8, and 9 imply, in plain words, that what you do not necessarily see may affect you significantly. This section serves to underline that the notion of canonical multifractal is very subtle and deserves to be well-understood and further discussed.
10.1 Definitions of the “accessible ranges” of the variables: qs from min* to max*and αs from max* to max*; the accessible functions τ* (q)and f* (α)
Mandelbrot (1995) worked to introduce to the function *(α)=max{0,f(α)}. That is,
• In the interval αmm*,αmax*] where (α)>0, *(α)=f(α);
• When (α)≤0, *(α)=0.
The graph of *(α) is identical to that of (α) except that the "tails" with <0 are truncated so that *>0. In terms of (q), the equality (α)=0 corresponds to lines that are tangent to the graph of (q) and also go through 0,0). In the most general case, those lines' slopes are mm* and max* and the points of contact are denoted by max* (satisfying 0) and mm* (satisfying 0). Therefore, the function *(α) corresponds to the following truncated function *(q).
• When <qmm*, *(q)=αmax*q;
• When mm*<q<qmax*, *(q)=τ(q);
• When >qmax*, *(q)=αmm*q.
In other words, the graph of * is identical to that of except that beyond max* or mm* it follows the tangents that go through the origins. Therefore it is straight.
For the TVCM, one has either max*=αmax with mm*=-∞, or mm*=αmm with max*=∞.
10.2 A confrontation
Section 4 noted that the largest values of Ω([0, 1]) are generated when a sample cascade begins with a few large values. Section 7 noted that the value of Ω([0, 1]) irrespective of size-ceases, for →∞, to have any impact on . Section 8 noted that, again for →∞, values of such that (α)<0 have a vanishing probability of being observed. Section 9. 1 followed up by defining the accessible function (α). Section 9 retumed to large values of ([0,1]) and noted their association with crit<∞. The values of they involve satisfy <0, hence a fortiori (α)<0. Those values do not occur in multifractal decomposition, yet they are extremely important.
10.3 The simplest cases where f(α) > 0 for all α, as exemplified by the canonical binomial
Here, the large values of Ω are ruled by the left-most part of the graph of f(α). That is, the same graph controls those large values and the distribution of Ω([0,1]) among the 1/dt intervals of length dt.
10.4 The extreme case where f(α) < 0 and α > 0 both occur, as exemplified by TVCM when u > 1
Due to the inequality (α)<α, the graph of (α) never intersects the quadrant where <0 and >0. The key unexpected fact is that the portions of (α) within other quad- rants play more or less separate roles. In the TVCM case, those quadrants are parts of one (analytically simple) function. But in general they are nearly independent of each other.
The function (α) was defined as having a graph that lies in the non-anomalous quadrant >0 and >0. This determines completely the multifractal decomposition of our TVCM measure, in particular, the dimension and the exponents mm*, max*, mm* and max*.
To the contrary, qcrit is entirely determined by the doubly anomalous left tail located in the quadrant characterized by f(α) < 0 and α < 0. A priori, it was quite unexpected that this quadrant should exist and play any role, least of all a central role, in the theory of multifractals. But in fact, qcrit has a major effect on the distribution, hence the value of the total measure in an interval.
10.5 The intermediate case where αmin > 0 but f(α) < 0 for some values of α
When p < 1/2, but u < 1 so that qcrit = ∞and all moments are finite, large values of ¼have a much lower probability than when u > 1. As always, however, their probability distribution continues to be determined by the left tail of the probability graph where f < 0.
11 A broad form of the multifractal formalism that allows α < 0 and f(α) < 0
The collection of rules that relate τ(q) to f(α) is called “multifractal formalism”. TVCM was specifically designed to understand multifractals directly, thus avoiding all formalism. However, general random multifractals more than TVCM demand their own broad multi-fractal formalism. Once again, the most widely known form of the multifractal formalism does not allow randomness and yields f(α) > 0, but the broad formalism first introduced in (Mandelbrot 1974a, b) concerns a generalized function for which f(α) <0 is allowed.
11.1 The broad “multifractal formalism” confirms the form of f(α) and allows f(α) < 0 for some α
Through a point on the graph of coordinates q and τ(q), draw the tangent to that graph. Under wide conditions, the tangent’s slope is α(q) and its intercept by the ordinate axis is −f(q). Thus
(q)=dτ(q)dqand−f(q)=τ(q)−qdτ(q)dq.
Through the quantities α(q) and f(q), a function α(q) is defined by using q as parameter.
The slope f ′(a) is the inverse of the function α(q). The tangent of slope f ′(a) intersects the line α = 0 at the point of ordinate −τ(a). The D(q) tangent’s equation being−τ(q) + qα, its intersection with the bisector satisfies the condition −τ + q = α, hence D = τ(q)/(q − 1). This is the critical embedding dimension discussed in Section 5.4.
11.2 The Legendre and inverse Legendre transforms and the thermodynamical analogy
The transforms that replace q and τ(q) by α and f(α), or conversely, are due...
| Erscheint lt. Verlag | 5.3.2003 |
|---|---|
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Statistik |
| Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik | |
| Recht / Steuern ► Wirtschaftsrecht | |
| Wirtschaft ► Betriebswirtschaft / Management ► Finanzierung | |
| Betriebswirtschaft / Management ► Spezielle Betriebswirtschaftslehre ► Bankbetriebslehre | |
| Wirtschaft ► Volkswirtschaftslehre ► Ökonometrie | |
| ISBN-10 | 0-08-055773-2 / 0080557732 |
| ISBN-13 | 978-0-08-055773-1 / 9780080557731 |
| Haben Sie eine Frage zum Produkt? |
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