Financial Models with Levy Processes and Volatility Clustering: 187 (Frank J. Fabozzi Series) - Hardcover

Rachev, Svetlozar T.; Kim, Young Shin; Bianchi, Michele L.; Fabozzi, Frank J.

Hardcover

ISBN 10: 0470482354 ISBN 13: 9780470482353

Publisher: Wiley, 2011

Synopsis

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.

About this title

About the Author

SVETLOZAR T. RACHEV is Chair-Professor in Statistics, Econometrics, and Mathematical Finance at the Karlsruhe Institute of Technology (KIT) in the School of Economics and Business Engineering; Professor Emeritus at the University of California, Santa Barbara; and Chief Scientist at FinAnalytica Inc.

YOUNG SHIN KIM is a scientific assistant in the Department of Statistics, Econometrics, and Mathematical Finance at the Karlsruhe Institute of Technology (KIT).

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.

FRANK J. FABOZZI is Professor in the Practice of Finance and Becton Fellow at the Yale School of Management and Editor of the Journal of PortfolioManagement. He is an Affiliated Professor at the University of Karlsruhe's Institute of Statistics, Econometrics, and Mathematical Finance and serves on the Advisory Council for the Department of Operations Research and Financial Engineering at Princeton University.

From the Back Cover

FINANCIAL MODELS WITH LéVY PROCESSES AND VOLATILITY CLUSTERING

The failure of financial models has been identified by some market observers as a major contributor to the global financial crisis. More specifically, it's been argued that the underlying assumption made in most of these models―that distributions of prices and returns are normally distributed―have been responsible for their undoing.

Financial crises and black swan events may not be precisely predictable by models, but improving the reliability and flexibility of those models is essential for both financial practitioners and academics intent on limiting the impact of major market crashes.

In Financial Models with Lévy Processes and Volatility Clustering, authors Svetlozar Rachev, Young Shin Kim, Michele Leonardo Bianchi, and Frank Fabozzi focus on the application of non-normal distributions for modeling the behavior of stock price returns. Opening with a brief introduction to the basics of probability distributions, this practical resource quickly moves on to:

• Address a wide array of methods for the simulation of infinitely divisible distributions and Lévy processes with a view toward option pricing.

• Discuss two approaches to deal with non-normal multivariate distributions, providing insight into portfolio allocation assuming a multi-tail t distribution and a non-Gaussian multivariate model.

• Examine discrete time option pricing models with volatility clustering―namely non-Gaussian GARCH models.

• Provide guidance on pricing American options with Monte Carlo methods.

If you want to gain a better understanding of how financial models can be used to capture the dynamics of economic and financial variables, Financial Models with Lévy Processes and Volatility Clustering is the best place to start.

From the Inside Flap

The financial crisis that began in the summer of 2007 has led to criticisms that the financial models used by risk managers, portfolio managers, and even regulators simply do not reflect the realities of today's markets. While one tool cannot be blamed for the entire global financial crisis, improving the flexibility and statistical reliability of existing models, in addition to developing better models, is essential for both financial practitioners and academics seeking to explain and prevent extreme events.

Nobody understands this better than the expert author team of Svetlozar Rachev, Young Shin Kim, Michele Leonardo Bianchi, and Frank Fabozzi, and in Financial Models with Lévy Processes and Volatility Clustering, they present a framework for modeling the behavior of stock returns in a univariate and multivariate setting providing you with practical applications to option pricing and portfolio management. They also explain the reasons for working with non-normal distributions in financial modeling and the best methodologies for employing them.

This reliable resource includes detailed discussions of 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. This practical guide:

• 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 for explaining joint extreme events

• And much more

Filled with in-depth insights and expert advice, Financial Models with Lévy Processes and Volatility Clustering is a thorough guide to both current probability distribution methods and brand new methodologies for financial modeling.

Excerpt. © Reprinted by permission. All rights reserved.

Financial Models with Levy Processes and Volatility Clustering

By Svetlozar T. Rachev Young Shim Kim Michele L. Bianchi Frank J. Fabozzi

John Wiley & Sons

Copyright © 2011 John Wiley & Sons, Ltd
All right reserved.
ISBN: 978-0-470-48235-3

Chapter One

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 stable Paretian distributions or Lévy stable distributions. We will describe these distributions later in this book.

Mandelbrot's early investigations on returns were carried further by Fama (1963a, 1963b), among others, and led to a consolidation of the hypothesis that asset returns can be better described as a stable Paretian distribution. However, there was obviously considerable concern in the finance profession by the findings of Mandelbrot and Fama. In fact, shortly after the publication of the Mandelbrot paper, Cootner (1964) expressed his concern regarding the implications of those findings for the statistical tests that had been published in prominent scholarly journals in economics and finance. He warned that (Cootner, 1964, p. 337):

Almost without exception, past econometric work is meaningless. Surely, before consigning centuries of work to the ash pile, we should like to have some assurance that all our work is truly useless. If we have permitted ourselves to be fooled for as long as this into believing that the Gaussian assumption is a workable one, is it not possible that the Paretian revolution is similarly illusory?

Although further evidence supporting Mandelbrot's empirical work was published, the "normality" assumption remains the cornerstone of many central theories in finance. The most relevant example for this book is the pricing of options or, more generally, the pricing of contingent claims. In 1900, the father of modern option pricing theory, Louis Bachelier, proposed using Brownian motion for modeling stock market prices. Inspired by his work, Samuelson (1965) formulated the log-normal model for stock prices that formed the basis for the well-known Black-Scholes option pricing model. Black and Scholes (1973) and Merton (1974) introduced pricing and hedging theory for the options market employing a stock price model based on the exponential Brownian motion. The model greatly influences the way market participants price and hedge options; in 1997, Merton and Scholes were awarded the Nobel Prize in Economic Science.

Despite the importance of option theory as formulated by Black, Scholes, and Merton, it is widely recognized that on Black Monday, October 19, 1987, the Black-Scholes formula failed. The reason for the failure of the model particularly during volatile periods is its underlying assumptions necessary to generate a closed-form solution to price options. More specifically, it is assumed that returns are normally distributed and that return volatility is constant over the option's life. The latter assumption means that regardless of an option's strike price, the implied volatility (i.e., the volatility implied by the Black-Scholes model based on observed prices in the options market) should be the same. Yet, it is now an accepted fact that in the options market, implied volatility varies depending on the strike price. In some options markets, for example, the market for individual equities, it is observed that, for options, implied volatility decreases with an option's strike price. This relationship is referred to as volatility skew. In other markets, such as index options and currency options, it is observed that at-the-money options tend to have an implied volatility that is lower than for both out-of-the-money and in-the-money options. Since graphically this relationship would show that implied volatility decreases as options move from out-of-the-money options to at-the-money options and then increase from at-the-money options to in-the-money options, this relationship between strike price and implied volatility is called volatility smile. Obviously, both volatility skew and volatility smile are inconsistent with the assumption of a constant volatility.

Consequently, since the mid-1990s there has been growing interest in non-normal models not only in academia but also among financial practitioners seeking to try to explain extreme events that occur in financial markets. Furthermore, the search for proper models to price complex financial instruments and to calibrate the observed prices of those instruments quoted in the market has motivated studies of more complex models. There is still a good deal of work to be done on financial modeling using alternative non-normal distributions that have recently been proposed in the finance literature. In this book, we explain these univariate and multivariate models (both discrete and continuous) and then show their applications to explaining stock price behavior and pricing options.

In the balance of this chapter we describe some background information that is used in the chapters ahead. At the end of the chapter we provide an overview of the book.

1.2 THE FAMILY OF STABLE DISTRIBUTION AND ITS PROPERTIES

As noted earlier, Mandelbrot and Fama observed fat tails for many asset price and return data. For assets whose returns or prices exhibit fat-tail attributes, non-normal distribution models are required to accurately model the tail behavior and compute probabilities of extreme returns. The candidates for non-normal distributions that have been proposed for modeling extreme events in addition to the α-stable Paretian distribution include mixtures of two or more normal distributions, Student t-distributions, hyperbolic distributions, and other scale mixtures of normal distributions, gamma distributions, extreme value distributions. The class of stable Paretian distributions (which includes α-stable Paretian distribution as a special case) are simply referred to as stable distributions.

Although we cover the stable distribution in considerable detail in Chapter 3, here we only briefly highlight the key features of this distribution.

1.2.1 Parameterization of the Stable Distribution

In only three cases does the density function of a stable distribution have a closed-form expression. In the general case, stable distributions are described by their characteristic function that we describe in Chapter 3. A characteristic function provides a third possibility (besides the cumulative distribution function and the probability density function) to uniquely define a probability distribution. At this point, we just state the fact that knowing the characteristic function is mathematically equivalent to knowing the probability density function or the cumulative distribution function. What is important to understand is that the characteristic function (and thus the density function) of a stable distribution is described by four parameters: μ, σ, α, and β.

The μ and σ parameters are measures of central location and scale, respectively. The parameter determines the tail weight or the distribution's kurtosis with 0 < α ≤ 2. The β determines the distribution's skewness. When the β of a stable distribution is zero, the distribution is symmetric around μ. Stable distributions allow for skewed distributions when β ≠ = 0 and fat tails; this means a high probability for extreme events relative to the normal distribution when α < 0. The value of β can range from -1 to +1. When β is positive, a stable distribution is skewed to the right; when β is negative, a stable distribution is skewed to the left. Figure 1.1 shows the effect on tail thickness of the density as well as peakedness at the origin relative to the normal distribution (collectively the "kurtosis" of the density) for the case of where μ = 0, σ = 1, and β = 0. As the values of decrease, the distribution exhibits fatter tails and more peakedness at the origin. Figure 1.1 illustrates the influence of β on the skewness of the density function for the case where α = 1.5, μ = 0, and σ = 1. Increasing (decreasing) values of β result in skewness to the right (left).

There are only four stable distributions that possess a closed-form expression for their density function. The case where α = 2 (and β = 0, which plays no role in this case) and with the re-parameterization in the scale parameter σ, yields the normal distribution. Thus, the normal distribution is one of the four special cases of the stable distribution, one that possesses a closed-form expression. The second occurs when α = 1 and β = 0. In this case we have the Cauchy distribution, which, although symmetric, is characterized by much fatter tails than the normal distribution. When we have α = 0.5 and β = 1, the resulting density function is the Levy distribution.

And finally, the fourth case with a closed-form density is the reflected Lévy distribution with parameters α = 0.5 and β = -1, so-called because this distribution can be obtained from the Lévy distribution by reflecting the graph of the density at the vertical axis.

1.2.2 Desirable Properties of the Stable Distributions

An attractive feature of stable distributions, not shared by other probability distribution models, is that they allow generalization of financial theories based on normal distributions and, thus, allow construction of a coherent and general framework for financial modeling. These generalizations are possible only because of two specific probabilistic properties that are unique to stable distributions (both normal and non-normal): (1) the stability property and (2) the Central Limit Theorem.

The stability property was briefly mentioned before and denotes the fact that the sum of two independent α-stable random variables follows—up to some correction of scale and location—again the same stable distribution. This property, which is well known for the special case of the normal distribution, becomes important in financial applications such as portfolio choice theory or when measuring returns on different time-scales. The second property, also well known for the normal distribution, generalizes to the stable case. Specifically, by the Central Limit Theorem, appropriately normalized sums of independent and identically distributed (i.i.d.) random variables with finite variance converge weakly to a normal random variable, and with infinite variance, the sums converge weakly to a stable random variable. This gives a theoretical basis for the use of stable distributions when heavy tails are present and stable distributions are the only distributional family that has its own domain of attraction—that is, a large sum of appropriately standardized i.i.d. random variables will have a distribution that converges to a stable one. This is a unique feature and its fundamental implications for financial modeling are the following: If changes in a stock price, interest rate, or any other financial variable are driven by many independently occurring small shocks, then the only appropriate distributional model for these changes is a stable model (normal or non-normal stable).

1.2.3 Considerations in the Use of the Stable Distribution

Despite the empirical evidence rejecting the normal distribution and in support of the stable distribution, there have been several barriers to the application of stable distribution models, both conceptual and technical. The major problem is that the variance of the stable non-normal distributions equals infinity. This fact can be explained by the tail behavior of stable distributions. One can show that the density function of a stable distribution with index of stability α "behaves like" |x|-α-1 and consequently all moments E|X|^p with p ≥ α do not exist. In particular, the mean only exists for α > 1.

(Continues...)

Excerpted from Financial Models with Levy Processes and Volatility Clusteringby Svetlozar T. Rachev Young Shim Kim Michele L. Bianchi Frank J. Fabozzi Copyright © 2011 by John Wiley & Sons, Ltd. Excerpted by permission of John Wiley & Sons. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Edition details

• Publisher: Wiley
• Publication date: 2011
• Language: English
• ISBN 10: 0470482354
• ISBN 13: 9780470482353
• Binding: Hardcover
• Edition number: 1
• Number of pages: 416