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Author: Peipei Shi Publisher: ISBN: Category : Languages : en Pages :
Book Description
Tempered stable random variables have a LePage like series representation, which was first introduced by Rosi¶nski. In this dissertation, we study the accuracy of the Rosi¶nski representation as determined by the convergence rates of the series. We also study estimators of parameters of certain tempered stable distributions and construct their confidence intervals. Finally, we present several simulation results for the Gamma-tempered random variable.
Author: Peipei Shi Publisher: ISBN: Category : Languages : en Pages :
Book Description
Tempered stable random variables have a LePage like series representation, which was first introduced by Rosi¶nski. In this dissertation, we study the accuracy of the Rosi¶nski representation as determined by the convergence rates of the series. We also study estimators of parameters of certain tempered stable distributions and construct their confidence intervals. Finally, we present several simulation results for the Gamma-tempered random variable.
Author: Svetlozar T. Rachev Publisher: John Wiley & Sons ISBN: 0470937262 Category : Business & Economics Languages : en Pages : 316
Book 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.
Author: Michael Grabchak Publisher: ISBN: Category : Languages : en Pages : 176
Book Description
It has been observed that data often appears to be well approximated by infinite variance stable distributions in some central region, but the tails of the distribution are actually lighter. Tempered stable distributions, which were introduced in [Ros07], are a rich class of models that attempt to capture this type of behavior. We will define certain generalizations of these models, which allow for more flexible structure. We will then derive a number of important results about them. In particular, we will give necessary and sufficient conditions for when they have regularly varying tails. We will also classify the possible weak limits of sequences of tempered stable distributions, and give necessary and sufficient conditions for convergence. These two properties will help us to categorize the long and short time behavior of their corresponding Lévy processes. We also attempt to explain why such models appear in applications. The use of stable distributions is justified by the central limit theorem, which says that stable distributions are the only possible limits of scaled and shifted sums of iid random variables. While this does not apply to tempered stable distributions, we will show that they may provide a good approximation to such sums for large, but not too large, aggregation levels. We base this explanation on the prelimit theorems of [KRS99] and [KRS00]. We then generalize them to d-dimensions.
Author: V. M. Zolotarev Publisher: American Mathematical Soc. ISBN: 0821845195 Category : Mathematics Languages : en Pages : 298
Book Description
This is the first book specifically devoted to a systematic exposition of the essential facts known about the properties of stable distributions. In addition to its main focus on the analytic properties of stable laws, the book also includes examples of the occurrence of stable distributions in applied problems and a chapter on the problem of statistical estimation of the parameters determining stable laws. A valuable feature of the book is the author's use of several formally different ways of expressing characteristic functions corresponding to these laws.
Author: Siem Jan Koopman Publisher: Oxford University Press ISBN: 0199683662 Category : Business & Economics Languages : en Pages : 389
Book Description
Presents original and up-to-date studies in unobserved components (UC) time series models from both theoretical and methodological perspectives.
Author: Michele Leonardo Bianchi Publisher: World Scientific ISBN: 9813276215 Category : Business & Economics Languages : en Pages : 598
Book Description
The study of heavy-tailed distributions allows researchers to represent phenomena that occasionally exhibit very large deviations from the mean. The dynamics underlying these phenomena is an interesting theoretical subject, but the study of their statistical properties is in itself a very useful endeavor from the point of view of managing assets and controlling risk. In this book, the authors are primarily concerned with the statistical properties of heavy-tailed distributions and with the processes that exhibit jumps. A detailed overview with a Matlab implementation of heavy-tailed models applied in asset management and risk managements is presented. The book is not intended as a theoretical treatise on probability or statistics, but as a tool to understand the main concepts regarding heavy-tailed random variables and processes as applied to real-world applications in finance. Accordingly, the authors review approaches and methodologies whose realization will be useful for developing new methods for forecasting of financial variables where extreme events are not treated as anomalies, but as intrinsic parts of the economic process.
Author: Michael Grabchak Publisher: Springer ISBN: 3319249274 Category : Mathematics Languages : en Pages : 127
Book Description
This brief is concerned with tempered stable distributions and their associated Levy processes. It is a good text for researchers interested in learning about tempered stable distributions. A tempered stable distribution is one which takes a stable distribution and modifies its tails to make them lighter. The motivation for this class comes from the fact that infinite variance stable distributions appear to provide a good fit to data in a variety of situations, but the extremely heavy tails of these models are not realistic for most real world applications. The idea of using distributions that modify the tails of stable models to make them lighter seems to have originated in the influential paper of Mantegna and Stanley (1994). Since then, these distributions have been extended and generalized in a variety of ways. They have been applied to a wide variety of areas including mathematical finance, biostatistics,computer science, and physics.
Author: Gareth W. Peters Publisher: John Wiley & Sons ISBN: 1118909542 Category : Mathematics Languages : en Pages : 667
Book Description
ADVANCES IN HEAVY TAILED RISK MODELING A cutting-edge guide for the theories, applications, and statistical methodologies essential to heavy tailed risk modeling Focusing on the quantitative aspects of heavy tailed loss processes in operational risk and relevant insurance analytics, Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk presents comprehensive coverage of the latest research on the theories and applications in risk measurement and modeling techniques. Featuring a unique balance of mathematical and statistical perspectives, the handbook begins by introducing the motivation for heavy tailed risk processes. A companion with Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk, the handbook provides a complete framework for all aspects of operational risk management and includes: Clear coverage on advanced topics such as splice loss models, extreme value theory, heavy tailed closed form loss distribution approach models, flexible heavy tailed risk models, risk measures, and higher order asymptotic approximations of risk measures for capital estimation An exploration of the characterization and estimation of risk and insurance modeling, which includes sub-exponential models, alpha-stable models, and tempered alpha stable models An extended discussion of the core concepts of risk measurement and capital estimation as well as the details on numerical approaches to evaluation of heavy tailed loss process model capital estimates Numerous detailed examples of real-world methods and practices of operational risk modeling used by both financial and non-financial institutions Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk is an excellent reference for risk management practitioners, quantitative analysts, financial engineers, and risk managers. The handbook is also useful for graduate-level courses on heavy tailed processes, advanced risk management, and actuarial science.
Author: Xiaonan Zhang Publisher: ISBN: 9780438718722 Category : Mathematical statistics Languages : en Pages : 43
Book Description
Stable distributions are a class of distributions allowing heavy tail and skewness. Most of stable distributions lack closed-form expression for their densities, so that estimating parameters is a challenging problem. When distributions have no closed-form density expression, their characteristic function becomes a useful alternative to define the unique distribution. We use the empirical characteristic function, i.e. the sample analog of the characteristic function, for estimation and goodness-of-fit tests for data. The existing fixed interval empirical characteristic function method works well when the alpha parameter is large but performs poorly for small values of alpha. This study offers a modification based on an adaptive grid to improve the estimation result for small alpha parameter without having a negative effect on the estimation of the other parameters. Goodness-of-fit tests based on the empirical characteristic function are also given and compared to classical tests based on the empirical cumulative distribution function.
Author: Peipei Shi
Author: Peipei Shi
Author: Michael Grabchak
Author: Svetlozar T. Rachev
Author: Michael Grabchak
Author: V. M. Zolotarev
Author: Siem Jan Koopman
Author: Michele Leonardo Bianchi
Author: Michael Grabchak
Author: Gareth W. Peters
Author: Xiaonan Zhang