E-book

# Financial Models with Levy Processes and Volatility Clustering

ISBN: 978-0-470-93726-6
400 pages
February 2011

## Description

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.

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Preface.

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 &alpha;-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, s-fields, and filtration.

5.2 Conditional expectation.

5.3 Change of measures.

5.4 Summary.

Chapter 6 Exponential Lévy Models.

6.1 Exponential Lévy Models.

6.2 Fitting a-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 The 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.

Chapter 9 Multi-Tail t-distribution.

9.1 Introduction.

9.2 Principal component analysis.

9.3 Estimating parameters.

9.4 Empirical results.

9.5 Conclusion.

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 Conclusion.

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.

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.

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## Author Information

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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