Stable Paretian Models in Finance

by Rachev, Svetlozar T.; Mittnik, Stefan

• ISBN13: 9780471953142
• ISBN10: 0471953148
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-06-15
• Publisher: WILEY

Summary

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.

Author Biography

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>

Table of Contents

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