Stable Paretian Models in Finance

The authors reconsider the problem of parametrically specifying distribution suitable for asset-return models. They describe alternative distributions, showing how they can be estimated and applied to stock-index and exchange-rate data. The implications for options pricing are also investigated.

Svetlozar Rachev is Chair-Professor in the School of Economics and Business Engineering at the University of Karlsruhe, and Professor Statistics and Economics at the University of California, Santa Barbara. He has published five monographs and more than 200 research articles. His research areas include mathematical and empirical finance, econometrics, probability, and statistics. He is a Fellow of the Institute of Mathematical Statistics, Elected Member of the International Statistical Institute, Foreign Member of the Russian Academy of Natural Sciences, and holds an honorary doctorate degree from ST. Petersburg Technical University. <BR> Stefan Mittnik is Professor of Statistics and Empirical Economics at the University of Kiel and Director of the Institute of Statistics Econometrics. His academic and consulting work covers the areas of empirical finance, forecasting financial risk, portfolio management, computational finance, econometrics, and time series analysis. <P>

Foreword

xiii

Preface

xv

Introduction

1

(24)

Stable Models in Finance

1

(3)

An Empirical Application

4

(13)

Overview

17

(8)

Univariate Stable Distributions

25

(58)

Definitions and Main Properties of Univariate Stable Distributions

26

(8)

Univariate Geometric Stable Distributions

34

(16)

A Contaminated Geometric Stable Law

50

(1)

Distributions Arising in the General Random Summation Scheme

51

(26)

Stable and Geo-stable Central Pre-limit Theorems and Their Applications

77

(6)

Identification, Estimation and Goodness of Fit

83

(66)

Regression-type Estimators

83

(5)

Visual Identification

88

(2)

Maximum Likelihood Estimation

90

(1)

Efficient Estimators for the Parameters of Paretianstable and Geometric Laws

91

(8)

Statistical Inference for Laplace--Weibull Mixture Models

99

(7)

Tail Estimation of the Stable Index α

106

(13)

FFT-approximation and ML-Estimation of stable Paretian laws

119

(30)

Empirical Comparison

149

(32)

Modeling the unconditional Distribution of Highly Volatile Exchange-rate Time Series

149

(7)

Unconditional Distributional Models for the Nikkei Index

156

(12)

Conditional Distributional Models for Nikkei Index

168

(13)

Subordinated, Fractional Stable and Stable ARIMA Processes

181

(88)

Subordinated Processes

182

(12)

Fractional Stable Processes

194

(3)

ARMA and ARIMA Models with Infinite--Variance Innovations

197

(19)

Subordinated Models: Evidence for Heavy Tailed Distributions and Long--Range Dependence

216

(10)

The Heavy--Tailedness and Long--Range Dependence in the USD--CHF Exchange Rate Time Series

226

(26)

The Heavy-Tailedness and Long--Range Dependence in the High--Frequency Deutsche Bank Price Record

252

(17)

ARCH--type and Shot Noise Processes

269

(58)

Relationship Between Unconditional Stable and ARCH-type Models

269

(6)

The Stable Paretian GARCH Model

275

(19)

Numerical Solution of Stochastic Differential Equations with Applications to ARCH/GARCH Modeling

294

(10)

Shot Noise Processes for Modeling Asset Returns

304

(5)

Conditionally Exponential Dependence Model for Financial Returns

309

(4)

Proofs of the Results on Numerical Solution of Stochastic Differential Equations

313

(14)

Multivariate Stable Models

327

(30)

Multivariate (α, +)-stable and Operator Stable Distributions

327

(15)

Multivariate Max-stable and Min-stable Models. The Weibull-Marshall-Olkin distributions

342

(5)

Weibull and Weibull-Marshall-Olkin Distributions: An Application to Stock Returns Distributions

347

(2)

Multivariate (α, M)-stable Distributions

349

(2)

Multivariate Geo-stable Distributions

351

(6)

Estimation, Association, Risk, and Symmetry of Stable Portfolios

357

(42)

Overview

357

(1)

Estimation of the Index of Stability and the Spectral Measure

358

(8)

A Test for Association

366

(2)

The Risk and Covariation Matrix of Stable Paretian Portfolios

368

(6)

Testing Multivariate Symmetry

374

(13)

Test of dependence between two asset-return series

387

(12)

Asset--Pricing and Portfolio Theory Under Stable Paretian Laws

399

(66)

Preliminaries to Stable CAPM and APT: Covariation, Lp Spaces and Risk

399

(10)

Stable Paretian Asset Pricing

409

(7)

Testing Stable Paretian Asset--Pricing Models

416

(3)

Stable Paretian Portfolio Theory

419

(3)

Lemmas on James--Orthogonality

422

(2)

The Stable Paretian Approach to Safety-first Analysis and Portfolio Choice Theory

424

(6)

The Portfolio Choice Problem

430

(10)

Stochastic Bounds and Two Parameter Safety-first Analysis

440

(17)

Safety-first Analysis with More Parameters

457

(8)

Risk Management: Value at Risk for Heavy-Tailed Distributed Rating

465

(44)

Introduction: Value at Risk (VAR) and the New Bank Capital Requirements for Market Risk

465

(3)

Computation of VAR

468

(7)

Components of VAR Methodologies

475

(5)

Evaluation of VAR Methods: Strength and Weaknesses

480

(3)

Testing VAR Measures

483

(9)

Stable Modeling of VAR

492

(17)

Option Pricing Under Alternative Stable Models

509

(40)

The Option-Pricing Problem

509

(9)

Option Pricing for Generalized Binomial Model

518

(4)

Option Pricing for the Generalized Mandelbrot-Taylor Model

522

(4)

Option valuation for subordinated asset pricing model

526

(10)

Appendix: Option Pricing with Heavy-Tailed Distributed Returns

536

(6)

Appendix: Option Pricing with Returns in the Domain of Attraction of the Normal Law

542

(4)

Appendix: Term Structure of Interest Rates Driven by Stable Motion

546

(3)

Option Pricing for Infinitely Divisible Return Models

549

(54)

The Problem

549

(1)

Limits of the Binomial Option Pricing Model

550

(9)

A Random Number of Price Changes

559

(6)

Convergence of the Binomial Pricing Formula

565

(3)

Black-Scholes Formulas when the Number of Price Movements is Random

568

(9)

Some Examples

577

(2)

An Alternative Randomization and Continuous Trading

579

(3)

An Example of the difference Between Continuous and Discrete Option Valuation; the Hyperbolic Model

582

(5)

The Rachev-Ruschendorf approach

587

(2)

Generalized Hyperbolic Distributions

589

(2)

Option Pricing in Discrete Models

591

(3)

Option Pricing in Continuous Time Models

594

(3)

Empirical Analysis

597

(6)

Numerical Results on Option Pricing: Modeling and Forecasting

603

(60)

Overview

603

(2)

Description of the Models

605

(9)

The Binomial Model with Non-identically Distributed Jumps

614

(3)

Empirical Analysis: the DAX--Options Market

617

(46)

Stable Models in Econometrics

663

(60)

Empirical Evidence for the Stable Paretian Against the Gaussian Assumption

665

(6)

Stable Paretian Econometrics: Modifications of Test Statistics

671

(31)

Testing for Structural Breaks

702

(15)

Test for Outlier in Heavy-tailed Samples; Proof of the Main Theorem

717

(6)

Stable Paretian Econometrics: Unit-Root Theory and Cointegrated Models

723

(22)

Statistical Inference in Time Series with Unit-Root:

723

(4)

Statistical Inference in Regression with Integrated Variables under Stable Assumption:

727

(12)

Appendix. Some Facts on Levy Processes

739

(6)

References

745

(84)

Indexes

829

(1)

Author-Index

829

(9)

Subject-Index

838

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Stable Paretian Models in Finance

Summary

Author Biography

Table of Contents

Supplemental Materials