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Handbook of Heavy Tailed Distributions in Finance

Book Description

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.

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright page
  5. Introduction to the Series
  6. Contents of the Handbook
  7. Preface
  8. Chapter 1: Heavy Tails in Finance for Independent or Multifractal Price Increments
    1. Abstract
    2. 1 Introduction: A path that led to model price by Brownian motion (Wiener or fractional) of a multifractal trading time
    3. 2 Background: the Bernoulli binomial measure and two random variants: shuffled and canonical
    4. 3 Definition of the two-valued canonical multifractals
    5. 4 The limit random variable <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#937;</span> = = <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#181;</span> ([0,1]), its distribution and the star functionalequation ([0,1]), its distribution and the star functionalequation
    6. 5 The function <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#964;</span>((<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">q</span>): motivation and form of the graph): motivation and form of the graph
    7. 6 When <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">u</span> &gt; 1, the moment > 1, the moment <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">E&#937;<sup>q</sup></span> diverges if diverges if <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">q</span> exceeds a critical exponent q exceeds a critical exponent q<sub xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML">crit</sub>satisfying satisfying <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#964;</span>((<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">q</span>) = 0; ) = 0; <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#937;</span> follows a power-law distribution of exponent q follows a power-law distribution of exponent q<sub xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML">crit</sub>
    8. 7 The quantity α: the original Hölder exponent and beyond
    9. 8 The full function <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">f</span>((<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#945;</span>) and the function ) and the function <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#961;</span>((<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#945;</span>))
    10. 9 The fractal dimension <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">D</span> = &#964;&#8242;(1) = 2[&#8211; = τ′(1) = 2[– <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">pu</span> log log<sub xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML">2</sub> <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">u</span> &#8211; (1 &#8211; – (1 – <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">p</span>))<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">v</span> log log<sub xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML">2</sub> <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">v</span>] and multifractal concentration] and multifractal concentration
    11. 10 A noteworthy and unexpected separation of roles, between the “dimensionspectrum” and the total mass <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#937;</span>; the former is ruled by the accessible ; the former is ruled by the accessible <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#945;</span> forwhich forwhich <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">f</span>((<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#945;</span>) &gt; 0, the latter, by the inaccessible ) > 0, the latter, by the inaccessible <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#945;</span> for which for which <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">f</span>((<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#945;</span>) &lt; 0) < 0
    12. 11 A broad form of the multifractal formalism that allows <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#945;</span> &lt; 0 and < 0 and <span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">f</span>((<span xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML" class="italic">&#945;</span>) &lt; 0) < 0
    13. Acknowledgments
  9. Chapter 2: Financial Risk and Heavy Tails
    1. Abstract
    2. Contents
    3. 1 Introduction
    4. 2 Historical perspective
    5. 3 Value at risk
    6. 4 Risk measures
    7. 5 Portfolios and dependence
    8. 6 Univariate extreme value theory
    9. 7 Stable Paretian models
    10. Acknowledgments
  10. Chapter 3: Modeling Financial Data with Stable Distributions
    1. Abstract
    2. 1 Basic facts about stable distribution
    3. 2 Appropriateness of stable models
    4. 3 Computation, simulation, estimation and diagnostics
    5. 4 Applications to financial data
    6. 5 Multivariate stable distributions
    7. 6 Multivariate computation, simulation, estimation and diagnostics
    8. 7 Multivariate application
    9. 8 Classes of multivariate stable distributions
    10. 9 Operator stable distributions
    11. 10 Discussion
  11. Chapter 4: Statistical issues in modeling multivariate stable portfolios
    1. Abstract
    2. 1 Introduction
    3. 2 Multivariate stable laws
    4. 3 Estimation of the index of stability
    5. 4 Estimation of the stable spectral measure
    6. 5 Estimation of the scale parameter
    7. 6 Extensions to other stable models
    8. 7 Applications
    9. Acknowledgment
  12. Chapter 5: Jump-diffusion models
    1. Abstract
    2. 1 Introduction
    3. 2 Preliminaries
    4. 3 Market models with jump-diffusions
    5. 4 Martingale measures: Existence and uniqueness (Market price of risk and market completion)
    6. 5 Hedging jump-diffusion market models
    7. 6 Pricing in jump-diffusion models
  13. Chapter 6: Hyperbolic Processes in Finance
    1. Abstract
    2. 1 Hyperbolic and related distributions
    3. 2 Lévy processes
    4. 3 Stochastic differential equations
    5. 4 Stochastic volatility models
    6. Acknowledgment
    7. Appendix
  14. Chapter 7: Stable Modeling of Market and Credit Value at Risk
    1. Abstract
    2. 1 Introduction
    3. 2 “Normal” modeling of VaR
    4. 3 A finance-oriented description of stable distributions
    5. 4 VaR estimates for stable distributed financial returns
    6. 5 Stable modeling and risk assessment for individual credit returns
    7. 6 Portfolio credit risk for independent credit returns
    8. 7 Stable modeling of portfolio risk for symmetric dependent credit returns
    9. 8 Stable modeling of portfolio risk for skewed dependent credit returns
    10. 9 One-factor model of portfolio credit risk
    11. 10 Credit risk evaluation for portfolio assets
    12. 11 Portfolio credit risk
    13. 12 Conclusions
    14. Appendix A Stable modeling of credit returns in figures
    15. Appendix B Tables
    16. Appendix C OLS credit risk evaluation for portfolio assets in figures
    17. Appendix D GARCH credit risk evaluation for portfolio assets in figures
    18. Acknowledgments
  15. Chapter 8: Modelling dependence with copulas and applications to risk management
    1. 1 Introduction
    2. 2 Copulas
    3. 3 Dependence concepts
    4. 4 Marshall-Olkin copulas
    5. 5 Elliptical copulas
    6. 6 Archimedean copulas
    7. 7 Modelling extremal events in practice
  16. Chapter 9: Prediction of Financial Downside-Risk with Heavy-Tailed Conditional Distributions
    1. Abstract
    2. 1 Introduction
    3. 2 GARCH-stable processes
    4. 3 Modeling exchange-rate returns
    5. 4 Prediction of densities and downside risk
    6. 5 Conclusions
  17. Chapter 10: Stable Non-Gaussian Models for Credit Risk Management
    1. Abstract
    2. 1 Stable modeling in credit risk – recent advances
    3. 2 A one-factor model for stable credit returns
    4. 3 Comparison of empirical results
    5. 4 The detection and measurement of long-range dependence
    6. 5 Conclusion
  18. Chapter 11: Multifactor stochastic variance models in risk management Maximum entropy approach and Lévy processes<sup xmlns="http://www.w3.org/1999/xhtml" xmlns:epub="http://www.idpf.org/2007/ops" xmlns:mml="http://www.w3.org/1998/Math/MathML">*</sup>
    1. Abstract
    2. 1 Review of market risk models
    3. 2 Single-factor stochastic variance model
    4. 3 Multifactor stochastic variance model
    5. Acknowledgment
  19. Chapter 12: Modelling the Term Structure of Monetary Rates*
    1. Abstract
    2. 1 Introduction
    3. 2 The mathematical framework
    4. 3 The tree representation
    5. 4 The econometric analysis
    6. 5 Conclusions
  20. Chapter 13: Asset Liability Management: A Review and Some New Results in the Presence of Heavy Tails
    1. Abstract
    2. 1 Introduction
    3. Part I: Review of the stochastic programming ALM literature
    4. Part II: Stable asset allocation
  21. Chapter 14: Portfolio Choice Theory With Non-Gaussian Distributed Returns
    1. Abstract
    2. 1 Introduction
    3. 2 Choices determined by a finite number of parameters
    4. 3 The asymptotic distributional classification of portfolio choices
    5. 4 A first comparison between the normal multivariate distributional assumption and the stable sub-Gaussian one
    6. 5 Conclusions
    7. Acknowledgment
    8. Appendix A: Proofs
    9. Appendix B: Tables
  22. Chapter 15: Portfolio Modeling With Heavy Tailed Random Vectors
    1. Abstract
    2. 1 Introduction
    3. 2 Heavy tails
    4. 3 Central limit theorems
    5. 4 Matrix scaling
    6. 5 The spectral decomposition
    7. 6 Sample covariance matrix
    8. 7 Dependent random vectors
    9. 8 Tail estimation
    10. 9 Tail estimator proof for dependent random vectors
    11. 10 Conclusions
  23. Chapter 16: Long Range Dependence in Heavy Tailed Stochastic Processes
    1. Abstract
    2. 1 Introduction
    3. 2 What is long range dependence?
    4. 3 Tails and rare events
    5. 4 Some classes of heavy tailed processes
    6. 5 Rare events, associated functionals and long range dependence
  24. Author Index
  25. Subject Index