Financial Models with Levy Processes and Volatility Clustering E-Book

Svetlozar T. Rachev

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Frank J. Fabozzi Series
• Sprache: Englisch

Seitenzahl: 510

Veröffentlichungsjahr: 2011

Beschreibung

An in-depth guide to understanding probability distributions and financial modeling for the purposes of investment management In Financial Models with Lévy Processes and Volatility Clustering, the expert author team provides a framework to model the behavior of stock returns in both a univariate and a multivariate setting, providing you with practical applications to option pricing and portfolio management. They also explain the reasons for working with non-normal distribution in financial modeling and the best methodologies for employing it. The book's framework includes the basics of probability distributions and explains the alpha-stable distribution and the tempered stable distribution. The authors also explore discrete time option pricing models, beginning with the classical normal model with volatility clustering to more recent models that consider both volatility clustering and heavy tails. * Reviews the basics of probability distributions * Analyzes a continuous time option pricing model (the so-called exponential Lévy model) * Defines a discrete time model with volatility clustering and how to price options using Monte Carlo methods * Studies two multivariate settings that are suitable to explain joint extreme events Financial Models with Lévy Processes and Volatility Clustering is a thorough guide to classical probability distribution methods and brand new methodologies for financial modeling.

Leseprobe

Contents

Cover

Series

Title Page

Copyright

Dedication

Preface

About the Authors

Chapter 1: Introduction

1.1 The Need for Better Financial Modeling of Asset Prices

1.2 The Family of Stable Distribution and its Properties

1.3 Option Pricing with Volatility Clustering

1.4 Model Dependencies

1.5 Monte Carlo

1.6 Organization of the Book

Chapter 2: Probability Distributions

2.1 Basic Concepts

2.2 Discrete Probability Distributions

2.3 Continuous Probability Distributions

2.4 Statistic Moments and Quantiles

2.5 Characteristic Function

2.6 Joint Probability Distributions

2.7 Summary

Chapter 3: Stable and Tempered Stable Distributions

3.1 α-Stable Distribution

3.2 Tempered Stable Distributions

3.3 Infinitely Divisible Distributions

3.4 Summary

3.5 Appendix

Chapter 4: Stochastic Processes in Continuous Time

4.1 Some Preliminaries

4.2 Poisson Process

4.3 Pure Jump Process

4.4 Brownian Motion

4.5 Time-Changed Brownian Motion

4.6 Lévy Process

4.7 Summary

Chapter 5: Conditional Expectation and Change of Measure

5.1 Events, σ-Fields, and Filtration

5.2 Conditional expectation

5.3 Change of measures

5.4 Summary

Chapter 6: Exponential Lévy model

6.1 Exponential Lévy model

6.2 Fitting α-Stable and Tempered Stable Distributions

6.3 Illustration: Parameter Estimation for Tempered Stable Distributions

6.4 Summary

6.5 Appendix: Numerical Approximation of Probability Density and Cumulative Distribution Functions

Chapter 7: Option Pricing in Exponential Lévy Models

7.1 Option contract

7.2 Boundary conditions for the price of an option

7.3 No-arbitrage pricing and equivalent martingale measure

7.4 Option pricing under the Black-Scholes model

7.5 European option pricing under exponential tempered stable models

7.6 Subordinated Stock Price Model

7.7 Summary

Chapter 8: Simulation

8.1 Random Number Generators

8.2 Simulation Techniques for Lévy Processes

8.3 Tempered Stable Processes

8.4 Tempered Infinitely Divisible Processes

8.5 Time-Changed Brownian Motion

8.6 Monte Carlo Methods

Appendix

Chapter 9: Multi-Tail t-Distribution

9.1 Introduction

9.2 Principal Component Analysis

9.3 Estimating Parameters

9.4 Empirical Results

9.5 Summary

Chapter 10: Non-Gaussian Portfolio Allocation

10.1 Introduction

10.2 Multifactor Linear Model

10.3 Modeling Dependencies

10.4 Average Value-at-Risk

10.5 Optimal portfolios

10.6 The algorithm

10.7 An empirical test

10.8 Summary

Chapter 11: Normal GARCH Models

11.1 Introduction

11.2 GARCH dynamics with normal innovation

11.3 Market Estimation

11.4 Risk-neutral estimation

11.5 Summary

Chapter 12: Smoothly Truncated Stable GARCH Models

12.1 Introduction

12.2 A Generalized NGARCH Option Pricing Model

12.3 Empirical Analysis

12.4 Summary

Chapter 13: Infinitely Divisible GARCH Models

13.1 Stock Price Dynamic

13.2 Risk-Neutral Dynamic

13.3 Non-Normal Infinitely Divisible GARCH

13.4 Simulate Infinitely Divisible GARCH

Appendix

Chapter 14: Option Pricing with Monte Carlo Methods

14.1 Introduction

14.2 Data set

14.3 Performance of Option Pricing Models

14.4 Summary

Chapter 15: American Option Pricing with Monte Carlo Methods

15.1 American option pricing in discrete time

15.2 The Least Squares Monte Carlo method

15.3 LSM method in GARCH option pricing model

15.4 Empirical illustration

15.5 Summary

Index

The Frank J. Fabozzi Series

Fixed Income Securities, Second Edition by Frank J. Fabozzi

Focus on Value: A Corporate and Investor Guide to Wealth Creation by James L. Grant and James A. Abate

Handbook of Global Fixed Income Calculations by Dragomir Krgin

Managing a Corporate Bond Portfolio by Leland E. Crabbe and Frank J. Fabozzi

Real Options and Option-Embedded Securities by William T. Moore

Capital Budgeting: Theory and Practice by Pamela P. Peterson and Frank J. Fabozzi

The Exchange-Traded Funds Manual by Gary L. Gastineau

Professional Perspectives on Fixed Income Portfolio Management, Volume 3 edited by Frank J. Fabozzi

Investing in Emerging Fixed Income Markets edited by Frank J. Fabozzi and Efstathia Pilarinu

Handbook of Alternative Assets by Mark J. P. Anson

The Global Money Markets by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

The Handbook of Financial Instruments edited by Frank J. Fabozzi

Interest Rate, Term Structure, and Valuation Modeling edited by Frank J. Fabozzi

Investment Performance Measurement by Bruce J. Feibel

The Handbook of Equity Style Management edited by T. Daniel Coggin and Frank J. Fabozzi

The Theory and Practice of Investment Management edited by Frank J. Fabozzi and Harry M. Markowitz

Foundations of Economic Value Added, Second Edition by James L. Grant

Financial Management and Analysis, Second Edition by Frank J. Fabozzi and Pamela P. Peterson

Measuring and Controlling Interest Rate and Credit Risk, Second Edition by Frank J. Fabozzi, Steven V. Mann, and Moorad Choudhry

Professional Perspectives on Fixed Income Portfolio Management, Volume 4 edited by Frank J. Fabozzi

The Handbook of European Fixed Income Securities edited by Frank J. Fabozzi and Moorad Choudhry

The Handbook of European Structured Financial Products edited by Frank J. Fabozzi and Moorad Choudhry

The Mathematics of Financial Modeling and Investment Management by Sergio M. Focardi and Frank J. Fabozzi

Short Selling: Strategies, Risks, and Rewards edited by Frank J. Fabozzi

The Real Estate Investment Handbook by G. Timothy Haight and Daniel Singer

Market Neutral Strategies edited by Bruce I. Jacobs and Kenneth N. Lévy

Securities Finance: Securities Lending and Repurchase Agreements edited by Frank J. Fabozzi and Steven V. Mann

Fat-Tailed and Skewed Asset Return Distributions by Svetlozar T. Rachev, Christian Menn, and Frank J. Fabozzi

Financial Modeling of the Equity Market: From CAPM to Cointegration by Frank J. Fabozzi, Sergio M. Focardi, and Petter N. Kolm

Advanced Bond Portfolio Management: Best Practices in Modeling and Strategies edited by Frank J. Fabozzi, Lionel Martellini, and Philippe Priaulet

Analysis of Financial Statements, Second Edition by Pamela P. Peterson and Frank J. Fabozzi

Collateralized Debt Obligations: Structures and Analysis, Second Edition by Douglas J. Lucas, Laurie S. Goodman, and Frank J. Fabozzi

Handbook of Alternative Assets, Second Edition by Mark J. P. Anson

Introduction to Structured Finance by Frank J. Fabozzi, Henry A. Davis, and Moorad Choudhry

Financial Econometrics by Svetlozar T. Rachev, Stefan Mittnik, Frank J. Fabozzi, Sergio M. Focardi, and Teo Jašić

Developments in Collateralized Debt Obligations: New Products and Insights by Douglas J. Lucas, Laurie S. Goodman, Frank J. Fabozzi, and Rebecca J. Manning

Robust Portfolio Optimization and Management by Frank J. Fabozzi, Petter N. Kolm, Dessislava A. Pachamanova, and Sergio M. Focardi

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimizations by Svetlozar T. Rachev, Stogan V. Stoyanov, and Frank J. Fabozzi

How to Select Investment Managers and Evaluate Performance by G. Timothy Haight, Stephen O. Morrell, and Glenn E. Ross

Bayesian Methods in Finance by Svetlozar T. Rachev, John S. J. Hsu, Biliana S. Bagasheva, and Frank J. Fabozzi

The Handbook of Municipal Bonds edited by Sylvan G. Feldstein and Frank J. Fabozzi

Subprime Mortgage Credit Derivatives by Laurie S. Goodman, Shumin Li, Douglas J. Lucas, Thomas A. Zimmerman, and Frank J. Fabozzi

Introduction to Securitization by Frank J. Fabozzi and Vinod Kothari

Structured Products and Related Credit Derivatives edited by Brian P. Lancaster, Glenn M. Schultz, and Frank J. Fabozzi

Handbook of Finance: Volume I: Financial Markets and Instruments edited by Frank J. Fabozzi

Handbook of Finance: Volume II: Financial Management and Asset Management edited by Frank J. Fabozzi

Handbook of Finance: Volume III: Valuation, Financial Modeling, and Quantitative Tools edited by Frank J. Fabozzi

Finance: Capital Markets, Financial Management, and Investment Management by Frank J. Fabozzi and Pamela Peterson-Drake

Active Private Equity Real Estate Strategy edited by David J. Lynn

Foundations and Applications of the Time Value of Money by Pamela Peterson-Drake and Frank J. Fabozzi

Leveraged Finance: Concepts, Methods, and Trading of High-Yield Bonds, Loans, and Derivatives by Stephen Antczak, Douglas Lucas, and Frank J. Fabozzi

Modern Financial Systems: Theory and Applications by Edwin Neave

Institutional Investment Management: Equity and Bond Portfolio Strategies and Applications by Frank J. Fabozzi

Quantitative Equity Investing: Techniques and Strategies by Frank J. Fabozzi, Sergio M. Focardi, and Petter N. Kolm

Simulation and Optimization in Finance: Modeling with MATLAB, @Risk, or VBA by Dessislava A. Pachamanova and Frank J. Fabozzi

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Library of Congress Cataloging-in-Publication Data:

Financial models with Lévy processes and volatility clustering / Svetlozar T. Rachev … [et al.]. p. cm.—(The Frank J. Fabozzi series) Includes index. ISBN 978-0-470-48235-3 (cloth); 978-0-470-93716-7 (ebk); 978-0-470-93726-6 (ebk); 978-1-118-00670-2 (ebk) 1. Capital assets pricing model. 2. Lévy processes. 3. Finance—Mathematical models. 4. Probabilities. I. Rachev, S. T. (Svetlozar Todorov) HG4637.F56 2011 332′.0415015192—dc22 2010033299

STR

To my grandchildren Iliana, Zoya, and Svetlozar

YSK

To my wife Myung-Ja and my son Minseob

MLB

To my wife Giorgia

FJF

To my wife Donna and my children Francesco, Patricia, and Karly

Preface

Carl Frederick Gauss, born in 1777, is one of the foremost mathematicians the world has known. Labeled the “prince of mathematicians” and viewed by some as on par with Sir Isaac Newton, the various works of Gauss have influenced a wide range of fields in mathematics and science. Although very few in the finance profession are familiar with his great contributions and body of work—which are published by the by the Royal Society of Göttingen in seven quatro volumes—most are familiar with his important work in probability theory that bears his name: the Gaussian distribution. The more popular name for this distribution is the normal distribution and was also referred to as the “bell curve” in 1733 by Abraham de Moivre, who first discovered this distribution based on his empirical work. Every finance professional who has taken a probability and statistics course has had a heavy dose of the Gaussian distribution and probably can still recite some properties of this distribution.

The normal distribution has found many applications in the natural sciences and social sciences. However, there are those who have long warned about the misuse of the normal distribution, particularly in the social sciences. In a 1981 article in Humanity and Society, Ted Goertzel and Joseph Fashing (“The Myth of the Normal Curve: A Theoretical Critique and Examination of its Role in Teaching and Research”) argue that

The myth of the bell curve has occupied a central place in the theory of inequality … Apologists for inequality in all spheres of social life have used the theory of the bell curve, explicitly and implicitly, in developing moral rationalizations to justify the status quo. While the misuse of the bell curve has perhaps been most frequent in the field of education, it is also common in other areas of social science and social welfare.

A good example is in the best-selling book by Richard Herrnstein and Charles Murray, The Bell Curve, published in 1994 with the subtitle Intelligence and Class Structure in American Life. The authors argue based on their empirical evidence that in trying to predict an individual's income or job performance, intelligence is a better predictor than the educational level or socioeconomic status of that individual's parents. Even the likelihood to commit a crime or to exhibit other antisocial behavior is better predicted by intelligence, as measured by IQ, than other potential explanatory factors. The policy implications drawn from the book are so profound that they set off a flood of books both attacking and supporting the findings of Herrnstein and Murray.

In finance, where the normal distribution was the underlying assumption in describing asset returns in major financial theories such as the capital asset pricing theory and option pricing theory, the attack came in the early 1960s from Benoit Mandelbrot, a mathematician at IBM's Thomas J. Watson Research Center. Although primarily known for his work in fractal geometry, the finance profession was introduced to his study of returns on commodity prices and interest rate movements that strongly rejected the assumption that asset returns are normally distributed. The mainstream financial models at the time relied on the work of Louis Bachelier, a French mathematician who at the beginning of the 20th century was the first to formulate random walk models for stock prices. Bachelier's work assumed that relative price changes followed a normal distribution. Mandelbrot, however, was not the first to attack the use of the normal distribution in finance. As he notes, Wesley Clair Mitchell, an American economist who taught at Columbia University and founded the National Bureau of Economic Research, was the first to do so in 1914. The bottom line is that the findings of Mandelbrot that empirical distributions do not follow a normal distribution led a leading financial economist, Paul Cootner of MIT, to warn the academic community that Mandelbrot's finding may mean that “past econometric work is meaningless.”

The overwhelming empirical evidence of asset returns in real-world financial markets is that they are not normally distributed. In commenting on the normal distribution in the context of its use in the social sciences, “Earnest Ernest” wrote the following in the November 10, 1974, in the Philadelphia Inquirer:

Surely the hallowed bell-shaped curve has cracked from top to bottom. Perhaps, like the Liberty Bell, it should be enshrined somewhere as a memorial to more heroic days.

Finance professionals should heed the same advice when using the normal distribution in asset pricing, portfolio management, and risk management.

In Mandelbrot's attack on the normal distribution, he suggested that asset returns are more appropriately described by a non-normal stable distribution referred to as a stable Paretian distribution or alpha-stable distribution (α-stable distribution), so-named because the tails of this distribution have Pareto power-type decay. The reason for describing this distribution as “non-normal stable” is because the normal distribution is a special case of the stable distribution. Because of the work by Paul Lévy, a French mathematician who introduced and characterized the non-normal stable distribution, this distribution is also referred to as the Lévy stable distribution and the Pareto-Lévy stable distribution. (There is another important contribution to probability theory by Lévy that we apply to financial modeling in this book. More specifically, we will apply the Lévy processes, a continuous-stochastic process.)

There are two other facts about asset return distributions that have been supported by empirical evidence. First, distributions have been observed to be skewed or nonsymmetric. That is, unlike in the case of the normal distribution where there is a mirror imaging of the two sides of the probability distribution, typically in a skewed distribution, one tail of the distribution is much longer (i.e., has greater probability of extreme values occurring) than the other tail of the probability distribution. Probability distributions with this attribute are referred to as having fat tails or heavy tails. The second finding is the tendency of large changes in asset prices (either positive or negative) to be followed by large changes, and small changes to be followed by small changes. This attribute of asset return distributions is referred to as volatility clustering.

In this book, we consider these well-established facts about asset return distributions in providing a framework for modeling the behavior of stock returns. In particular, we provide applications to the financial modeling used in asset pricing, option pricing, and portfolio/risk management. In addition to explaining how one can employ non-normal distributions, we also provide coverage of several topics that are of special interest to finance professionals.

We begin by explaining the need for better financial modeling, followed by the basics of probability distributions—the different types of probability distributions (discrete and continuous), specific types of probability distributions, parameters of a probability distribution, and joint probability distributions. The definition of the stable Pareto distribution (we adopted the term α-stable distribution in this book) that Mandelbrot suggested is described. Although this distribution has certain desirable properties and is superior to the normal distribution, it is not suitable in certain financial modeling applications such as the modeling of option prices because the mean, variance, and exponential moments of the return distribution have to exist. For this reason, we introduce distributions that we believe are better suited for financial modeling, distributions obtained by tempering the tail properties of the α-stable distribution: the smoothly truncated stable distribution and various types of tempered stable distributions. Because of their important role in the applications in this book, we review continuous-time stochastic processes with emphasis on Lévy processes.

There are chapters covering the so-called exponential Lévy model, and we study this continuous-time option pricing model and analyze the change of measure problem. Prices of plain vanilla options are calculated with both analytical and Monte Carlo methods.

After examples dealing with the simulation of non-normal random numbers, we study two multivariate settings that are suitable to explain joint extreme events. In the first approach, we describe a multivariate random variable for joint extreme events, and in the second we model the joint behavior of log-returns of stocks by considering a feasible dependence structure together with marginals able to explain volatility clutering.

Then we get into the core of the book where we deal with examples of discrete-time option pricing models. Starting from the classic normal model with volatility clustering, we progress to the more recent models that jointly consider volatility clustering and heavy tails. We conclude with a non-normal GARCH model to price American options.

We would like to thank Sebastian Kring and Markus Höchstötter for their coauthorship of Chapter 9 and Christian Menn for his coauthorship of Chapter 12. We also thank Stoyan Stoyanov for providing the MATLAB code for the skew t-copula.

The authors acknowledge that the views expressed in this book are their own and do not necessarily reflect those of their employers.

SVETLOZAR (ZARI) T. RACHEV YOUNG SHIN (AARON) KIM MICHELE LEONARDO BIANCHI FRANK J. FABOZZI July 2010

About the Authors

Svetlozar (Zari) T. Rachev completed his Ph.D. Degree in 1979 from Moscow State (Lomonosov) University, and his Doctor of Science Degree in 1986 from Steklov Mathematical Institute in Moscow. Currently, he is Chair-Professor in Statistics, Econometrics and Mathematical Finance at the Karlsruhe Institute of Technology (KIT) in the School of Economics and Business Engineering, and Professor Emeritus at the University of California, Santa Barbara in the Department of Statistics and Applied Probability. Professor Rachev has published 14 monographs, 10 handbooks and special-edited volumes, and more than 300 research articles. His recently coauthored books published by John Wiley & Sons in mathematical finance and financial econometrics include Financial Econometrics: From Basics to Advanced Modeling Techniques (2007) and Bayesian Methods in Finance (2008). He is cofounder of Bravo Risk Management Group, specializing in financial risk-management software. Bravo Group was acquired by FinAnalytica for which he currently serves as Chief-Scientist.

Young Shin (Aaron) Kim studied at the Department of Mathematics, Sogang University, in Seoul, Korea, where he received his doctorate degree in 2005. Currently, he is a scientific assistant in the Department of Statistics, Econometrics and Mathematical Finance at Karlsruhe Institute of Technology (KIT). His current professional and research interests are in the area of Lévy processes, including tempered stable processes, time-varying volatility models, and their applications to finance.

Michele Leonardo Bianchi is an analyst in the Division of Risk and Financial Innovation Analysis at the Specialized Intermediaries Supervision Department of the Bank of Italy. Dr. Bianchi has authored articles on quantitative finance, probability theory, and nonlinear optimization. He earned an Italian “Laurea” in Mathematics in 2005 from the University of Pisa and completed his Ph.D. in Computational Methods for Economic and Financial Decisions and Forecasting in 2009 from the University of Bergamo.

Frank J. Fabozzi is Professor in the Practice of Finance in the School of Management and Becton Fellow at Yale University. He is an Affiliated Professor at the University of Karlsruhe's Institute of Statistics, Econometrics and Mathematical Finance. Prior to joining the Yale faculty, he was a Visiting Professor of Finance in the Sloan School at MIT. Professor Fabozzi is a Fellow of the International Center for Finance at Yale University and on the Advisory Council for the Department of Operations Research and Financial Engineering at Princeton University. He is the editor of the Journal of Portfolio Management and an associate editor of Quantitative Finance. He is a trustee for the BlackRock family of closed-end funds. In 2002, he was inducted into the Fixed Income Analysts Society's Hall of Fame and is the 2007 recipient of the C. Stewart Sheppard Award given by the CFA Institute. His recently coauthored books published by Wiley include Institutional Investment Management (2009), Quantitative Equity Investing (2010), Bayesian Methods in Finance (2008), Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures (2008), Financial Modeling of the Equity Market: From CAPM to Cointegration (2008), Robust Portfolio Optimization and Management (2007), and Financial Econometrics: From Basics to Advanced Modeling Techniques (2007). Professor Fabozzi earned a doctorate in economics from the City University of New York in 1972. He earned the designations of Chartered Financial Analyst and Certified Public Accountant.

Chapter 1

Introduction

1.1 The Need for Better Financial Modeling of Asset Prices

Major debacles in financial markets since the mid-1990s such as the Asian financial crisis in 1997, the bursting of the dot-com bubble in 2000, the subprime mortgage crisis that began in the summer of 2007, and the days surrounding the bankruptcy of Lehman Brothers in September 2008 are constant reminders to risk managers, portfolio managers, and regulators of how often extreme events occur. These major disruptions in the financial markets have led researchers to increase their efforts to improve the flexibility and statistical reliability of existing models that seek to capture the dynamics of economic and financial variables. Even if a catastrophe cannot be predicted, the objective of risk managers, portfolio managers, and regulators is to limit the potential damages.

The failure of financial models has been identified by some market observers as a major contributor—indeed some have argued that it is the single most important contributor—for the latest global financial crisis. The allegation is that financial models used by risk managers, portfolio managers, and even regulators simply did not reflect the realities of real-world financial markets. More specifically, the underlying assumption regarding asset returns and prices failed to reflect real-world movements of these quantities. Pinpointing the criticism more precisely, it is argued that the underlying assumption made in most financial models is that distributions of prices and returns are normally distributed, popularly referred to as the “normal model.” This probability distribution—also referred to as the Gaussian distribution and in lay terms the “bell curve”—is the one that dominates the teaching curriculum in probability and statistics courses in all business schools. Despite its popularity, the normal model flies in the face of what has been well documented regarding asset prices and returns. The preponderance of the empirical evidence has led to the following three stylized facts regarding financial time series for asset returns: (1) they have fat tails (heavy tails), (2) they may be skewed, and (3) they exhibit volatility clustering.

The “tails” of the distribution are where the extreme values occur. Empirical distributions for stock prices and returns have found that the extreme values are more likely than would be predicted by the normal distribution. This means that between periods where the market exhibits relatively modest changes in prices and returns, there will be periods where there are changes that are much higher (i.e., crashes and booms) than predicted by the normal distribution. This is not only of concern to financial theorists, but also to practitioners who are, in view of the frequency of sharp market down turns in the equity markets noted earlier, troubled by, in the words of hoppe (1999), the “… compelling evidence that something is rotten in the foundation of the statistical edifice … used, for example, to produce probability estimates for financial risk assessment.” Fat tails can help explain larger price fluctuations for stocks over short time periods than can be explained by changes in fundamental economic variables as observed by shiller (1981).

The normal distribution is a symmetric distribution. That is, it is a distribution where the shape of the left side of the probability distribution is the mirror image of the right side of the probability distribution. For a skewed distribution, also referred to as a nonsymmetric distribution, there is no such mirror imaging of the two sides of the probability distribution. Instead, typically in a skewed distribution one tail of the distribution is much longer (i.e., has greater probability of extreme values occurring) than the other tail of the probability distribution, which, of course, is what we referred to as fat tails. Volatility clustering behavior refers to the tendency of large changes in asset prices (either positive or negative) to be followed by large changes, and small changes to be followed by small changes.

The attack on the normal model is by no means recent. The first fundamental attack on the assumption that price or return distribution are not normally distributed was in the 1960s by mandelbrot (1963). He strongly rejected normality as a distributional model for asset returns based on his study of commodity returns and interest rates. Mandlebrot conjectured that financial returns are more appropriately described by a non-normal stable distribution. Since a normal distribution is a special case of the stable distribution, to distinguish between Gaussian and non-Gaussian stable distributions, the latter are often referred to as distributions or distributions. We will describe these distributions later in this book.