Author: Robert Josef Stelzer
Publisher:
ISBN:
Category :
Languages : en
Pages : 249

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Book Description

Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Book Description
Several multivariate stochastic models in continuous time are introduced and their probabilistic and statistical properties are studied in detail. All models are driven by Lévy processes and can generally be used to model multidimensional time series of observations. In this thesis the focus is on various stochastic volatility models for financial data. Firstly, multidimensional continuous-time autoregressive moving-average (CARMA) processes are considered and, based upon them, a multivariate continuous-time exponential GARCH model (ECOGARCH). Thereafter, positive semi-definite Ornstein-Uhlenbeck type processes are introduced and the behaviour of the square root (and similar transformations) of stochastic processes of finite variation, which take values in the positive semi-definite matrices and can be represented as the sum of an integral with respect to time and another integral with respect to an extended Poisson random measure, is analysed in general. The positive semi-definite Ornstein-Uhlenbeck type processes form the basis for the definition of a multivariate extension of the popular stochastic volatility model of Barndorff-Nielsen and Shephard. After a detailed theoretical study this model is estimated for some observed stock price series. As a further model with stochastic volatility multivariate continuous time GARCH (COGARCH) processes are introduced and their probabilistic and statistical properties are analysed.

Author: Jaya P. N. Bishwal
Publisher:
ISBN: 9783031038624
Category :
Languages : en
Pages : 0

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Book Description
This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.

Author: Peter J. Brockwell
Publisher: Walter de Gruyter GmbH & Co KG
ISBN: 3111325032
Category : Mathematics
Languages : en
Pages : 522

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Book Description
This book provides a self-contained account of continuous-parameter time series, starting with second-order models. Integration with respect to orthogonal increment processes, spectral theory and linear prediction are treated in detail. Lévy-driven models are incorporated, extending coverage to allow for infinite variance, a variety of marginal distributions and sample paths having jumps. The necessary theory of Lévy processes and integration of deterministic functions with respect to these processes is developed at length. Special emphasis is given to the analysis of continuous-time ARMA processes.

Author: Torben Gustav Andersen
Publisher: Springer Science & Business Media
ISBN: 3540712976
Category : Business & Economics
Languages : en
Pages : 1045

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Book Description
The Handbook of Financial Time Series gives an up-to-date overview of the field and covers all relevant topics both from a statistical and an econometrical point of view. There are many fine contributions, and a preamble by Nobel Prize winner Robert F. Engle.

Author: Ole E Barndorff-Nielsen
Publisher: Springer Science & Business Media
ISBN: 1461201977
Category : Mathematics
Languages : en
Pages : 414

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Book Description
A Lévy process is a continuous-time analogue of a random walk, and as such, is at the cradle of modern theories of stochastic processes. Martingales, Markov processes, and diffusions are extensions and generalizations of these processes. In the past, representatives of the Lévy class were considered most useful for applications to either Brownian motion or the Poisson process. Nowadays the need for modeling jumps, bursts, extremes and other irregular behavior of phenomena in nature and society has led to a renaissance of the theory of general Lévy processes. Researchers and practitioners in fields as diverse as physics, meteorology, statistics, insurance, and finance have rediscovered the simplicity of Lévy processes and their enormous flexibility in modeling tails, dependence and path behavior. This volume, with an excellent introductory preface, describes the state-of-the-art of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts present surveys of recent developments, or focus on some most promising applications. Despite its special character, every topic is aimed at the non- specialist, keen on learning about the new exciting face of a rather aged class of processes. An extensive bibliography at the end of each article makes this an invaluable comprehensive reference text. For the researcher and graduate student, every article contains open problems and points out directions for futurearch. The accessible nature of the work makes this an ideal introductory text for graduate seminars in applied probability, stochastic processes, physics, finance, and telecommunications, and a unique guide to the world of Lévy processes.

Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Book Description
Empirical volatility changes in time and exhibits tails, which are heavier than those of normal distributions. Moreover, empirical volatility has - sometimes quite substantial - upwards jumps and clusters on high levels. We investigate classical and non-classical stochastic volatility models with respect to their extreme behavior: subexponential Lévy driven MA processes in the maximum domain of attraction of the Gumbel distribution, regularly varying mixed MA processes, Ornstein-Uhlenbeck processes with exponentially decreasing tails and COGARCH processes. The basic volatility models of this thesis are subexponential Lévy driven MA processes $Y(t)=\int_{-\infty}^{\infty}f(t-s)\, dL(s)$ for $t\in \R$ where f is a deterministic function and L is a Lévy process. In Chapter 1 we study the extremal behavior of subexponential MA processes in the maximum domain of attraction of the Gumbel distribution and in Chapter 2 of the Fréchet distribution. The behavior is quite different in these different regimes. For both classes we give sufficient conditions for the kernel function f, such that a stationary version of the MA process Y exists, which preserves the infinitely divisibility of L. We calculate the tail behavior of the stationary distribution, which is again subexponential and in the same maximum domain of attraction as the driving Lévy process L. Hence they capture heavy tails and volatility jumps. Our investigation on the extremal behavior of Y is based on a discrete-time skeleton of Y chosen to incorporate those times, where large jumps of the Lévy process L and extremes of the kernel function f occur. Adding marks to this discrete-time skeleton, we obtain, by the weak limit of marked point processes, complete information about the extremal behavior. A complementary result guarantees the convergence of running maxima. Both models have volatility clusters. Regularly varying MA processes have long high level excursion in contrast to subexp.

Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :

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Book Description
Empirical volatility changes in time and exhibits tails, which are heavier than normal. Moreover, empirical volatility has - sometimes quite substantial - upwards jumps and clusters on high levels. We investigate classical and nonclassical stochastic volatility models with respect to their extreme behavior. We show that classical stochastic volatility models driven by Brownian motion can model heavy tails, but obviously they are not able to model volatility jumps. Such phenomena can be modelled by Lévy driven volatility processes as, for instance, by Lévy driven Ornstein-Uhlenbeck models. They can capture heavy tails and volatility jumps. Also volatility clusters can be found in such models, provided the driving Lévy process has regularly varying tails. This results then in a volatility model with similarly heavy tails. As the last class of stochastic volatility models, we investigate a continuous time GARCH(1,1) model. Driven by an arbitrary Lévy process it exhibits regularly varying tails, volatility upwards jumps and clusters on high levels. -- COGARCH ; extreme value theory ; generalized Cox-Ingersoll-Ross model ; Lévy process ; Ornstein-Uhlenbeck process ; Poisson approximation ; regular variation ;stochastic volatility model ; subexponential distribution ; tail behavior ; volatility cluster

Author: George Tauchen
Publisher:
ISBN:
Category :
Languages : en
Pages : 41

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Book Description
We develop simulation schemes for the new classes of non-Gaussian pure jump Levy processes for stochastic volatility. We write the price and volatility processes as integrals against a vector Levy process, which then makes series approximation methods directly applicable. These methods entail simulation of the Levy increments and formation of weighted sums of the increments; they do not require a closed-form expression for a tail mass function nor specification of a copula function. We also present a new, and apparently quite flexible, bivariate mixture of gammas model for the driving Levy process. Within this setup, it is quite straightforward to generate simulations from a Levy-driven CARMA stochastic volatility model augmented by a pure-jump price component. Simulations reveal the wide range of different types of financial price processes that can be generated in this manner, including processes with persistent stochastic volatility, dynamic leverage, and jumps.

Author: Jaya P. N. Bishwal
Publisher: Springer Nature
ISBN: 3031038614
Category : Mathematics
Languages : en
Pages : 634

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Book Description
This book develops alternative methods to estimate the unknown parameters in stochastic volatility models, offering a new approach to test model accuracy. While there is ample research to document stochastic differential equation models driven by Brownian motion based on discrete observations of the underlying diffusion process, these traditional methods often fail to estimate the unknown parameters in the unobserved volatility processes. This text studies the second order rate of weak convergence to normality to obtain refined inference results like confidence interval, as well as nontraditional continuous time stochastic volatility models driven by fractional Levy processes. By incorporating jumps and long memory into the volatility process, these new methods will help better predict option pricing and stock market crash risk. Some simulation algorithms for numerical experiments are provided.