Financial Models With Levy Processes and Volatility Clustering

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

• ISBN13: 9780470482353
• ISBN10: 0470482354
• Edition: 1st
• Format: Hardcover
• Copyright: 2011-02-08
• Publisher: Wiley

Summary

An in-depth guide to understanding probability distributions and financial modeling for the purposes of investment managementIn Financial Models with L_vy 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 L_vy 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 eventsFinancial Models with L_vy Processes and Volatility Clustering is a thorough guide to classical probability distribution methods and brand new methodologies for financial modeling.

Author Biography

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.

Table of Contents

Introduction

The need for better financial modeling of asset prices

The family of stable distribution and its properties

Option pricing with volatility clustering

Model dependencies

Monte Carlo

Organization of the book

Probability distributions

Basic concepts

Discrete probability distributions

Continuous probability distributions

Statistic moments and quantiles

Characteristic function

Joint probability distributions

Summary

Stable and tempered stable distributions

¿-Stable distribution

Tempered stable distributions

Infinitely divisible distributions

Exponential Moments

Summary

Appendix

Stochastic Processes in Continuous Time

Some preliminaries

Poisson Process

Pure jump process

Brownian motion

Time-Changed Brownian motion

Lévy process

Summary

Conditional Expectation and Change of Measure

Events, ¿-fields, and filtration

Conditional expectation

Change of measures

Summary

Exponential Lévy Models

Exponential Lévy Models

Fitting ¿-stable and tempered stable distributions

Illustration: Parameter estimation for tempered stable distributions

Summary

Appendix: Numerical approximation of probability density and cumulative distribution functions

Option Pricing in Exponential Lévy Models

Option contract

Boundary conditions for the price of an option

No-arbitrage pricing and equivalent martingale measure

Option pricing under the Black-Scholes model

European option pricing under exponential tempered stable

Models

The subordinated stock price model

Summary

Simulation

Random number generators

Simulation techniques for Lévy processes

Tempered stable processes

Tempered infinitely divisible processes

Time-changed Brownian motion

Monte Carlo methods

Appendix

Multi-tail t distribution

Introduction

Principal component analysis

Estimating parameters

Empirical results

Conclusion

Non-Gaussian portfolio allocation

Introduction

Multi-factor linear model

Modeling dependencies

Average value-at-risk

Optimal portfolios

The algorithm

An empirical test

Conclusions

Normal GARCH models

Introduction

GARCH dynamics with normal innovation

Market estimation

Risk-neutral estimation

Conclusions

Smoothly truncated stable GARCH models

Introduction

A Generalized NGARCH Option Pricing Model

Empirical Analysis

Conclusion

Infinitely divisible GARCH models

Stock price dynamic

Risk-neutral dynamic

Non-normal infinitely divisible GARCH

Simulate infinitely divisible GARCH

Appendix

Option Pricing with Monte Carlo

Introduction

Data set

Performance of Option Pricing Models

Conclusions

American Option Pricing with Monte Carlo Methods

American option pricing in discrete time

The Least Squares Monte Carlo method

LSM method in GARCH option pricing model

Empirical illustration

Summary

Table of Contents provided by Publisher. All Rights Reserved.