An in-depth guide to understanding probability distributions and financial modeling for the purposes of investment management In Financial Models with Levy 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 Levy 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 Levy Processes and Volatility Clustering is a thorough guide to classical probability distribution methods and brand new methodologies for financial modeling.
Preface. About the Authors. 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. 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. 3 Stable and tempered stable distributions. 3.1 a-Stable distribution. 3.2 Tempered stable distributions. 3.3 Infinitely divisible distributions. 3.3.1 Exponential Moments. 3.4 Summary. 3.5 Appendix. 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 L'evy process. 4.7 Summary. 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. 6 Exponential L'evy Models. 6.1 Exponential L'evy 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. 7 Option Pricing in Exponential L'evy 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. 8 Simulation. 8.1 Random number generators. 8.2 Simulation techniques for L'evy processes. 8.3 Tempered stable processes. 8.4 Tempered infinitely divisible processes. 8.5 Time-changed Brownian motion. 8.6 Monte Carlo methods. Appendix. 9 Multi-tail t distribution. 9.1 Introduction. 9.2 Principal component analysis. 9.3 Estimating parameters. 9.4 Empirical results. 9.5 Conclusion. 10 Non-Gaussian portfolio allocation. 10.1 Introduction. 10.2 Multi-factor 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 Conclusions. 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 Conclusions. 12 Smoothly truncated stable GARCH models. 12.1 Introduction. 12.2 A Generalized NGARCH Option Pricing Model. 12.3 Empirical Analysis. 12.4 Conclusion. 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. Appendix. 14 Option Pricing with Monte Carlo. 14.1 Introduction. 14.2 Data set. 14.3 Performance of Option Pricing Models. 14.4 Conclusions. 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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