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: 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: 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: 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: John P. Nolan Publisher: Springer Nature ISBN: 3030529150 Category : Mathematics Languages : en Pages : 342
Book Description
This textbook highlights the many practical uses of stable distributions, exploring the theory, numerical algorithms, and statistical methods used to work with stable laws. Because of the author’s accessible and comprehensive approach, readers will be able to understand and use these methods. Both mathematicians and non-mathematicians will find this a valuable resource for more accurately modelling and predicting large values in a number of real-world scenarios. Beginning with an introductory chapter that explains key ideas about stable laws, readers will be prepared for the more advanced topics that appear later. The following chapters present the theory of stable distributions, a wide range of applications, and statistical methods, with the final chapters focusing on regression, signal processing, and related distributions. Each chapter ends with a number of carefully chosen exercises. Links to free software are included as well, where readers can put these methods into practice. Univariate Stable Distributions is ideal for advanced undergraduate or graduate students in mathematics, as well as many other fields, such as statistics, economics, engineering, physics, and more. It will also appeal to researchers in probability theory who seek an authoritative reference on stable distributions.
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: Robert Adler Publisher: Springer Science & Business Media ISBN: 9780817639518 Category : Mathematics Languages : en Pages : 560
Book Description
Twenty-four contributions, intended for a wide audience from various disciplines, cover a variety of applications of heavy-tailed modeling involving telecommunications, the Web, insurance, and finance. Along with discussion of specific applications are several papers devoted to time series analysis, regression, classical signal/noise detection problems, and the general structure of stable processes, viewed from a modeling standpoint. Emphasis is placed on developments in handling the numerical problems associated with stable distribution (a main technical difficulty until recently). No index. Annotation copyrighted by Book News, Inc., Portland, OR
Author: Michael Grabchak
Author: Michele Leonardo Bianchi
Author: Michael Grabchak
Author: V. M. Zolotarev
Author: Peipei Shi
Author: John P. Nolan
Author: Maria Coca
Author: Michael Grabchak
Author: Svetlozar T. Rachev
Author: Robert Adler