Handbook of heavy tailed distributions in finance
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| Format: | Buch |
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Elsevier
2003
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| Ausgabe: | 1. ed. |
| Schriftenreihe: | Handbooks in finance
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| 245 | 1 | 0 | |a Handbook of heavy tailed distributions in finance |c ed. by Svetlozar T. Rachev |
| 250 | |a 1. ed. | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2003 | |
| 300 | |a XXIV, 680 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Handbooks in finance |v 1 | |
| 650 | 4 | |a Statistische Methodenlehre / Finanzierungstheorie / Theorie | |
| 650 | 4 | |a Finance -- Statistical methods | |
| 650 | 0 | 7 | |a Finanzmathematik |0 (DE-588)4017195-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Heavy-tailed Verteilung |0 (DE-588)4747317-4 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Heavy-tailed Verteilung |0 (DE-588)4747317-4 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Račev, Svetlozar T. |d 1951- |e Sonstige |0 (DE-588)12022979X |4 oth | |
| 830 | 0 | |a Handbooks in finance |v 1 |w (DE-604)BV017164584 |9 1 | |
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Datensatz im Suchindex
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| adam_text | CONTENTS
Introduction to the Series v
Contents of the Handbook vii
Preface by SVETLOZAR T. RACHEV ix
Chapter 1
Heavy Tails in Finance for Independent or Multifractal Price Increments
BENOIT B. MANDELBROT 1
Abstract 4
1. Introduction: A path that led to model price by Brownian motion (Wiener or
fractional) of a multifractal trading time 5
1.1. From the law of Pareto to infinite moment anomalies that contradict the Gaussian norm 5
1.2. A scientific principle: scaling invariance in finance 6
1.3. Analysis alone versus statistical analysis followed by synthesis and graphic output 7
1.4. Actual implementation of scaling invariance by multifractal functions: it requires additional
assumptions that are convenient but not a matter of principle, for example, separability and
compounding 7
2. Background: the Bernoulli binomial measure and two random variants: shuffled
and canonical 8
2.1. Definition and construction of the Bernoulli binomial measure 8
2.2. The concept of canonical random cascade and the definition of the canonical binomial measure 9
2.3. Two forms of conservation: strict and on the average 9
2.4. The term canonical is motivated by statistical thermodynamics 10
2.5. In every variant of the binomial measure one can view all finite (positive or negative) powers
together, as forming a single class of equivalence 10
2.6. The full and folded forms of the address plane 1 ]
2.7. Alternative parameters 1 1
3. Definition of the two valued canonical multifractals 11
3.1. Construction of the two valued canonical multifractal in the interval [0. 1] 11
3.2. A second special two valued canonical multifractal: the unifractal measure on the canonical
Cantor dust 1 2
3.3. Generalization of a useful new viewpoint: when considered together with their powers from
—oc to oo, all the TVCM parametrized by either p or 1 — p form a single class of equivalence 1 2
3.4. The full and folded address planes | 2
3.5. Background of the two valued canonical measures in the historical development of multi
fractals 13
xiii
xiv Contents
4. The limit random variable Q = /x([0, 1J), its distribution and the star functional
equation 13
4.1. The identity EM = 1 implies that the limit measure has the martingale property, hence the
cascade defines a limit random variable Q = ii([0, 1]) 13
4.2. Questions 14
4.3. Exact stochastic renormalizability and the star functional equation for Q 14
4.4. Metaphor for the probability of large values of Q, arising in the theory of discrete time
branching processes 14
4.5. To a large extent, the asymptotic measure Q of a TVCM is large if, and only if, the pre fractal
measure m UO. 1]) has become large during the very first few stages of the generating cascade 1 5
5. The function x{q): motivation and form of the graph 15
5.1. Motivation of r(q) 15
5.2. A generalization of the role of Q: middle and high frequency contributions to microran
domness 15
5.3. The expected partition function J] £/i (d,0 16
5.4. Form of the r(q) graph 17
5.5. Reducible and irreducible canonical multifractals 18
6. When u 1, the moment EQq diverges if q exceeds a critical exponent gcrjt
satisfying r{q) =0; Q follows a power law distribution of exponent qQX{t 18
6.1. Divergent moments, power law distributions and limits to the ability of moments to deter¬
mine a distribution 1 8
6.2. Discussion 19
6.3. An important apparent anomaly : in a TVCM, the ^ th moment of n may diverge 19
6.4. An important role of x(q): if q 1 the q lh moment of Q is finite if, and only if, x(q) 0;
the same holds for ; (d/) whenever it is a dyadic interval 19
6.5. Definition of qcrlt; proof that in the case of TVCM qcrlt is finite if, and only if. u 1 20
6.6. The exponent qcr t can be considered as a macroscopic variable of the generating process 20
7. The quantity a: the original Holder exponent and beyond 21
7.1. The Bernoulli binomial case and two forms of the Holder exponent: coarse grained (or
coarse) and fine grained • 21
7.2. In the general TVCM measure, a ^ a. and the link between a and the Holder exponent
breaks down; one consequence is that the doubly anomalous inequalities amjn 0, hence
a 0, are not excluded 22
8. The full function f(a) and the function p(a) 23
X.I. The Bernoulli binomial measure: definition and derivation of the box dimension function f(a) 23
8.2. The entropy ogive function f(a); the role of statistical thermodynamics in multifractals
and the contrast between equipartition and concentration 23
X.3. The Bernoulli binomial measure, continued: definition and derivation of a function p(a) =
f(a) — 1 that originates as a rescaled logarithm of a probability 24
X.4. Cienerali/ation of p(a) to the case of TVCM; the definition of f(a) as p{a) + I is indirect
but significant because it allows the generalized / to be negative 24
8.5. Comments in terms of probability theory 25
8.6. Distinction between center and tail theorems in probability 26
Contents xv
8.7. The reason for the anomalous inequalities /(or) 0 and a 0 is that, by the definition of a
random variable n(dt). the sample size is bounded and is prescribed intrinsically: the notion
of supersampling 26
8.8. Excluding the Bernoulli case p = 1/2. TVCM faces either one of two major anomalies :
for p —1/2, one has /(ormin) = + °g2 / 0 iln(J /( max) = 1 + logil 1 — / ) 0: for
p 1/2, the opposite signs hold 27
8.9. The minor anomalies /(ormax) 0 or /(amjn) 0 lead to sample function with a clear
ceiling or floor 27
9. The fractal dimension D = r (l) —2[—pitlog it — (I — p)v og~, v] and multi
fractal concentration 27
9.1. In the Bernoulli binomial measures weak asymptotic negligibility holds but strong asymp¬
totic negligibility fails 28
9.2. For the Bernoulli or canonical binomials, the equation /(or) = a has one and only one solu¬
tion: that solution satisfies D 0 and is the fractal dimension of the carrier of the measure 28
9.3. The notion of multifractal concentration 29
9.4. The case of TVCM with p 1 /2, allows D to be positive, negative, or zero 29
10. A noteworthy and unexpected separation of roles, between the dimension spec¬
trum and the total mass Q; the former is ruled by the accessible a for which
f(a) 0, the latter, by the inaccessible a for which /(a) 0 30
10.1. Definitions of the accessible ranges of the variables: /s from y*j to /,*,ax and as from
min to max1 tlle accessible functions r*( /) and /*(«) 30
10.2. A confrontation 30
10.3. The simplest cases where f(a) 0 for all a, as exemplified by the canonical binomial 3 1
10.4. The extreme case where /(or) 0 and a 0 both occur, as exemplified by TVCM when it 1 31
10.5. The intermediate case where am n 0 but /(or) 0 for some values of a 31
11. A broad form of the multifractal formalism that allows a 0 and f(a) 0 31
11.1. The broad multifractal formalism confirms the form of/(or) and allows /(or) 0 for some a 32
11.2. The Legendre and inverse Legendre transforms and the thermodynamical analogy 32
Acknowledgments 32
References 32
Chapter 2
Financial Risk and Heavy Tails
BRENDAN O. BRADLEY and MURAD S. TAQQU 35
Abstract 36
1. Introduction 37
2. Historical perspective 38
2.1. Risk and utility 38
2.2. Markowitz mean variance portfolio theory 39
2.3. CAPMandAPT 40
2.4. Empirical evidence 43
3. Value at risk 45
3.1. Computation of VaR 47
xvi Contents
3.2. Parameter estimation 5 1
4. Risk measures 58
4.1. Coherent risk measures 58
4.2. Expected shortfall 59
5. Portfolios and dependence 61
5.1. Copulas 61
5.2. Measures of dependence 66
5.3. Elliptical distributions 72
6. Univariate extreme value theory 77
6.1. Limit law for maxima 78
6.2. Block maxima method 80
6.3. Using the block maxima method for stress testing 82
6.4. Peaks over threshold method 83
7. Stable Paretian models 94
7.1. Stable portfolio theory 95
7.2. Stable asset pricing 98
Acknowledgments 100
References 101
Chapter 3
Modeling Financial Data with Stable Distributions
JOHN P. NOLAN 105
Abstract 106
1. Basic facts about stable distributions 107
2. Appropriateness of stable models 111
3. Computation, simulation, estimation and diagnostics 113
4. Applications to financial data 114
5. Multivariate stable distributions 116
6. Multivariate computation, simulation, estimation and diagnostics 121
7. Multivariate application 124
8. Classes of multivariate stable distributions 126
9. Operator stable distributions 128
10. Discussion 128
References 129
Chapter 4
Statistical Issues in Modeling Multivariate Stable Portfolios
TOMASZ J. KOZUBOWSKI, ANNA K. PANORSKA and SVETLOZAR T.
RACHEV 131
Abstract 132
1. Introduction 133
2. Multivariate stable laws 134
2.1. Domains of attraction 135
Contents xvii
2.2. Strictly stable and symmetric stable vectors 1 36
2.3. One dimensional case 1 36
2.4. Discrete spectral measure 137
2.5. Linear combinations and risk of a financial portfolio 138
2.6. Densities 139
2.7. An alternative parameterization 139
2.8. Association 140
3. Estimation of the index of stability 141
3.1. Estimation of univariate stable parameters 142
3.2. Estimation of the tail index a 143
4. Estimation of the stable spectral measure 147
4.1. Tail estimators 147
4.2. The empirical characteristic function method 148
4.3. The projection method 150
5. Estimation of the scale parameter 151
6. Extensions to other stable models 152
6.1. y stable laws 152
6.2. Operator stable laws 155
7. Applications 157
Acknowledgment 162
References 162
Chapter 5
Jump Diffusion Models
WOLFGANG J. RUNGGALDIER 169
Abstract 170
Keywords 170
1. Introduction 171
2. Preliminaries 173
2.1. Univariate point processes (Poisson jump processes) 173
2.2. Multivariate and marked point processes 175
2.3. Martingale representation 177
2.4. Exponential formula: generalized Ito formula 178
2.5. Absolutely continuous transformation of measures 1 80
3. Market models with jump diffusions 182
3.1. Asset price and term structure models with additive jumps 182
3.2. Jump diffusion models driven by hidden jump processes 1 84
3.3. Asset prices as diffusions sampled at the jump times of a jump process 185
4. Martingale measures: Existence and uniqueness (Market price of risk and mar¬
ket completion) 186
4.1. The case of jump diffusion asset price models 187
4.2. The case of jump diffusion term structure models 190
5. Hedging in jump diffusion market models 193
xviii Contents
5.1. Hedging when the market is completed 194
5.2. Hedging when the market is not complete 198
6. Pricing in jump diffusion models 201
6.1. General aspects 201
6.2. Computational aspects 203
References 207
Chapter 6
Hyperbolic Processes in Finance
BO MARTIN BIBBY and MICHAEL S0RENSEN 211
Abstract 212
1. Hyperbolic and related distributions 213
1.1. The generalized hyperbolic distribution 213
1.2. The generalized inverse Gaussian distribution 222
1.3. Statistical inference 226
2. Levy processes 227
3. Stochastic differential equations 230
3.1. Diffusion models 231
3.2. Statistical inference for diffusion processes 233
3.3. Ornstein Uhlenbeck processes 235
3.4. Compound processes 236
4. Stochastic volatility models 238
Acknowledgment 242
Appendix 243
References 244
Chapter 7
Stable Modeling of Market and Credit Value at Risk
SVETLOZAR T. RACHEV, EDUARDO S. SCHWARTZ and IRINA KHINDA
NOVA 249
Abstract 250
1. Introduction 251
2. Normal modeling of VaR 253
2.1. VaR for a single asset 253
2.2. Portfolio VaR 254
3. A finance oriented description of stable distributions 255
3.1. Parameters and properties of stable distributions 255
3.2. Estimation of parameters of stable distributions 259
4. VaR estimates for stable distributed financial returns 264
4.1. In sample evaluation of VaR estimates 264
4.2. Forecast evaluation of VaR estimates 279
5. Stable modeling and risk assessment for individual credit returns 283
6. Portfolio credit risk for independent credit returns 287
Contents xix
7. Stable modeling of portfolio risk for symmetric dependent credit returns 290
8. Stable modeling of portfolio risk for skewed dependent credit returns 296
9. One factor model of portfolio credit risk 299
10. Credit risk evaluation for portfolio assets 300
11. Portfolio credit risk 305
11.1. Independent credit risks 305
11.2. Symmetric dependent credit risks 305
11.3. Skewed dependent credit risks 307
12. Conclusions 309
Appendix A. Stable modeling of credit returns in figures 311
Appendix B. Tables 317
Appendix C. OLS credit risk evaluation for portfolio assets in figures 320
Appendix D. GARCH credit risk evaluation for portfolio assets in figures 324
Acknowledgments 326
References 326
Chapter 8
Modelling Dependence with Copulas and Applications to Risk Management
PAUL EMBRECHTS, FILIP L1NDSKOG and ALEXANDER MCNEIL 329
1. Introduction 331
2. Copulas 332
2.1. Mathematical introduction 332
2.2. Sklar s Theorem 334
2.3. The Frechet Hoeffding bounds for joint distribution functions 335
2.4. Copulas and random variables 336
3. Dependence concepts 341
3.1. Linear correlation 341
3.2. Perfect dependence 342
3.3. Concordance 343
3.4. Kendall s tau and Spearman s rho 345
3.5. Tail dependence 348
4. Marshall Olkin copulas 351
4.1. Bivariate Marshall Olkin copulas 352
4.2. A multivariate extension 354
4.3. A useful modelling framework 355
5. Elliptical copulas 357
5.1. Elliptical distributions 357
5.2. Gaussian copulas 360
5.3. r copulas 361
6. Archimedean copulas 365
6.1. Definitions 365
6.2. Properties 367
6.3. Kendall s tau revisited 370
xx Contents
6.4. Tail dependence revisited 371
6.5. Multivariate Archimedean copulas 373
7. Modelling extremal events in practice 377
7.1. Insurance risk 377
7.2. Market risk 380
References 383
Chapter 9
Prediction of Financial Downside Risk with Heavy Tailed Conditional Distributions
STEFAN MITTNIK and MARC S. PAOLELLA 385
Abstract 386
1. Introduction 387
2. GARCH stable processes 388
3. Modeling exchange rate returns 389
3.1. Approximate maximum likelihood estimation 390
3.2. Estimation results and volatility persistence 392
3.3. Goodness of fit 395
4. Prediction of densities and downside risk 398
5. Conclusions 402
References 403
Chapter 10
Stable Non Gaussian Models for Credit Risk Management
BERNHARD MARTIN, SVETLOZAR T. RACHEV and EDUARDO S.
SCHWARTZ 405
Abstract 406
1. Stable modeling in credit risk recent advances 407
2. A one factor model for stable credit returns 408
2.1. Credit risk evaluation for single assets 410
2.2. A stable portfolio model with independent credit returns 411
2.3. A stable portfolio model with dependent credit returns 413
3. Comparison of empirical results 415
3.1. The observed portfolio data 415
3.2. Generating comparable risk free bonds from the yield curve 415
3.3. Fitting the empirical time series for R,, K,. and (7, 416
3.4. CVaR rcsults for the independence assumption 417
3.5. CVaR results for the dependence assumption 41 8
4. The detection and measurement of long range dependence 422
4.1. fractal processes and the Hurst Exponent 423
4.2. The Aggregated Variance Method 425
4.3. Absolute Values of the Aggregated Series 426
4.4. Classical R/S analysis 426
4.5. The modified approach by Lo 427
Contents xxj
4.6. The statistic of Mansfield, Rachev and Samorodnitsky (MRS) 430
4.7. Empirical results for long range dependence in credit data 43 1
5. Conclusion 439
References 440
Chapter 11
Multifactor Stochastic Variance Models in Risk Management: Maximum Entropy
Approach and Levy Processes
ALEXANDER LEVIN and ALEXANDER TCHERNITSER 443
Abstract 444
1. Review of market risk models 445
1.1. Market risk management and Value at Risk 445
1.2. Statistical properties of the market risk factors 447
1.3. A short review of stochastic volatility models 448
2. Single factor stochastic variance model 450
2.1. Maximum entropy approach and Levy processes 450
2.2. Generalized Gamma Variance model 456
2.3. Mean reverting stochastic variance model 460
3. Multifactor stochastic variance model 463
3.1. Requirements for multifactor VaR models 463
3.2. Naive multifactor model 464
3.3. Elliptical stochastic variance model 465
3.4. Independent stochastic variances for the principal components 467
3.5. A model with correlated stochastic variances 468
3.6. Calibration for the GSV model 472
Acknowledgment 477
References 477
Chapter 12
Modelling the Term Structure of Monetary Rates
LUISAIZZI 481
Abstract 482
1. Introduction 483
2. The mathematical framework 484
2.1. Model setup and notation 484
2.2. Regularity conditions on the jump diffusion process 486
2.3. Interest rates with non identically distributed jumps 487
2.4. The smile effect and infinitely divisible distributions 490
2.5. The discrete time process 492
3. The tree representation 494
4. The econometric analysis 500
4.1. Data 500
4.2. Estimation results 501
xxii Contents
5. Conclusions 506
References 507
Chapter 13
Asset Liability Management: A Review and Some New Results in the Presence of
Heavy Tails
YESIM TOKAT, SVETLOZAR T. RACHEV and EDUARDO S. SCHWARTZ 509
Abstract 510
1. Introduction 511
Part I: Review of the stochastic programming ALM literature 513
2. Stochastic programming ALM models 513
2.1. Chance constrained model 513
2.2. Dynamic programming 515
2.3. Sequential decision analysis 516
2.4. Stochastic Linear Programming with Recourse (SLPR) 518
2.5. Dynamic generalized networks 520
2.6. Scenario optimization 521
2.7. Robust optimization 522
3. Multistage stochastic ALM programming with decision rules 523
4. Scenario generation 524
4.1. Discrete time series model 524
4.2. Continuous time model 526
Part II: Stable asset allocation 528
5. Stable distribution 528
5.1. Description of stable distributions 529
5.2. Financial modeling and estimation 530
6. Multistage stable asset allocation model with decision rules 532
6.1. Scenario generation 534
6.2. Valuation of assets 536
6.3. Computational results 537
7. Conclusion 542
References 543
Chapter 14
Portfolio Choice Theory with Non Gaussian Distributed Returns
SERGIO ORTOBELLI. ISABELLA HUBER, SVETLOZAR T. RACHEV and
EDUARDO S. SCHWARTZ 547
Abstract 548
1. Introduction 549
2. Choices determined by a finite number of parameters 553
2.1. Portfolio choice with institutional restrictions 553
2.2. Portfolio choice when unlimited short sales are allowed 558
2.3. Relations with Ross multi parameter models 560
Contents xxiii
3. The asymptotic distributional classification of portfolio choices 561
3.1. The sub Gaussian stable model 567
3.2. A three fund separation model in the domain of attraction of a stable law 570
3.3. A k + I fund separation model in the domain of attraction of a stable law 573
4. A first comparison between the normal multivariate distributional assumption
and the stable sub Gaussian one 574
4.1. An optimal allocation problem 575
4.2. Stable versus normal optimal allocation: a first comparison 578
5. Conclusions 581
Acknowledgment 582
Appendix A: Proofs 582
Appendix B: Tables 585
References 590
Chapter 15
Portfolio Modeling with Heavy Tailed Random Vectors
MARK M. MEERSCHAERT and HANS PETER SCHEFFLER 595
Abstract 596
Keywords 596
1. Introduction 597
2. Heavy tails 597
3. Central limit theorems 600
4. Matrix scaling 607
5. The spectral decomposition 610
6. Sample covariance matrix 613
7. Dependent random vectors 616
8. Tail estimation 619
9. Tail estimator proof for dependent random vectors 626
10. Conclusions 636
References 637
Chapter 16
Long Range Dependence in Heavy Tailed Stochastic Processes
BORJANA RACHEVA IOTOVA and GENNADY SAMORODNITSKY 641
Abstract 642
Keywords 642
1. Introduction 643
2. What is long range dependence? 644
3. Tails and rare events 646
4. Some classes of heavy tailed processes 649
4.1. Linear processes 649
4.2. Infinitely divisible processes 650
5. Rare events, associated functional and long range dependence 652
xxiv Contents
5.1. Unusual sample mean and long strange segments for heavy tailed linear processes 653
5.2. Ruin probability for heavy tailed linear processes 656
5.3. Rare events for stationary stable processes 657
5.4. High dimensional joint tails for a linear process with stable innovations 659
References 661
Author Index 663
Subject Index 675
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| format | Book |
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| id | DE-604.BV014746739 |
| illustrated | Illustrated |
| indexdate | 2024-12-23T15:59:59Z |
| institution | BVB |
| isbn | 0444508961 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009988759 |
| oclc_num | 249009529 |
| open_access_boolean | |
| owner | DE-N2 DE-91G DE-BY-TUM DE-521 DE-19 DE-BY-UBM DE-91 DE-BY-TUM |
| owner_facet | DE-N2 DE-91G DE-BY-TUM DE-521 DE-19 DE-BY-UBM DE-91 DE-BY-TUM |
| physical | XXIV, 680 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Elsevier |
| record_format | marc |
| series | Handbooks in finance |
| series2 | Handbooks in finance |
| spellingShingle | Handbook of heavy tailed distributions in finance Handbooks in finance Statistische Methodenlehre / Finanzierungstheorie / Theorie Finance -- Statistical methods Finanzmathematik (DE-588)4017195-4 gnd Heavy-tailed Verteilung (DE-588)4747317-4 gnd |
| subject_GND | (DE-588)4017195-4 (DE-588)4747317-4 |
| title | Handbook of heavy tailed distributions in finance |
| title_auth | Handbook of heavy tailed distributions in finance |
| title_exact_search | Handbook of heavy tailed distributions in finance |
| title_full | Handbook of heavy tailed distributions in finance ed. by Svetlozar T. Rachev |
| title_fullStr | Handbook of heavy tailed distributions in finance ed. by Svetlozar T. Rachev |
| title_full_unstemmed | Handbook of heavy tailed distributions in finance ed. by Svetlozar T. Rachev |
| title_short | Handbook of heavy tailed distributions in finance |
| title_sort | handbook of heavy tailed distributions in finance |
| topic | Statistische Methodenlehre / Finanzierungstheorie / Theorie Finance -- Statistical methods Finanzmathematik (DE-588)4017195-4 gnd Heavy-tailed Verteilung (DE-588)4747317-4 gnd |
| topic_facet | Statistische Methodenlehre / Finanzierungstheorie / Theorie Finance -- Statistical methods Finanzmathematik Heavy-tailed Verteilung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009988759&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV017164584 |
| work_keys_str_mv | AT racevsvetlozart handbookofheavytaileddistributionsinfinance |