Financial Models with Levy Processes and Volatility Clustering

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

• Publisher: Wiley
• Year: 2011
• ISBN:
  9780470937266
• Edition: 1
• Topic: Business
• Subtopic: Finance
• Index: Business

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.

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 stable Paretian distributions or Lévy stable distributions.1 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.2 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 (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.3

The μ and σ parameters are measures of central location and scale, respectively. The parameter α determines the tail weight or the distribution's kurtosis with 0 < α ≤ 2. ...

Table of contents

• Cover
• Series
• Title Page
• Copyright
• Dedication
• Preface
• About the Authors
• Chapter 1: Introduction
• Chapter 2: Probability Distributions
• Chapter 3: Stable and Tempered Stable Distributions
• Chapter 4: Stochastic Processes in Continuous Time
• Chapter 5: Conditional Expectation and Change of Measure
• Chapter 6: Exponential Lévy model
• Chapter 7: Option Pricing in Exponential Lévy Models
• Chapter 8: Simulation
• Chapter 9: Multi-Tail t-Distribution
• Chapter 10: Non-Gaussian Portfolio Allocation
• Chapter 11: Normal GARCH Models
• Chapter 12: Smoothly Truncated Stable GARCH Models
• Chapter 13: Infinitely Divisible GARCH Models
• Chapter 14: Option Pricing with Monte Carlo Methods
• Chapter 15: American Option Pricing with Monte Carlo Methods
• Index