Über die Autorin bzw. den Autor

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.

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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.

Aus dem Klappentext

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.

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Financial Models with Levy Processes and Volatility Clustering

John Wiley & Sons

Chapter One

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...