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Multiple Time Scales and the Exponential Ornstein-Uhlenbeck Stochastic Volatility Model

Multiple Time Scales and the Exponential Ornstein-Uhlenbeck Stochastic Volatility Model
Author: Jaume Masoliver
Publisher:
Total Pages: 24
Release: 2005
Genre:
ISBN:

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We study the exponential Ornstein-Uhlenbeck stochastic volatility model and observe that the model shows a multiscale behavior in the volatility autocorrelation. It also exhibits a leverage correlation and a probability profile for the stationary volatility which are consistent with market observations. All these features make the model quite appealing since it appears to be more complete than other stochastic volatility models also based on a two-dimensional diffusion. We finally present an approximate solution for the return probability density designed to capture the kurtosis and skewness effects.


Multiple Time Scales in Volatility and Leverage Correlations

Multiple Time Scales in Volatility and Leverage Correlations
Author: Josep Perelló
Publisher:
Total Pages: 19
Release: 2013
Genre:
ISBN:

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Financial time series exhibit two different type of non linear correlations: (i) volatility autocorrelations that have a very long range memory, on the order of years, and (ii) asymmetric return-volatility (or 'leverage') correlations that are much shorter ranged. Different stochastic volatility models have been proposed in the past to account for both these correlations. However, in these models, the decay of the correlations is exponential, with a single time scale for both the volatility and the leverage correlations, at variance with observations. We extend the linear Ornstein-Uhlenbeck stochastic volatility model by assuming that the mean reverting level is itself random. We find that the resulting three-dimensional diffusion process can account for different correlation time scales. We show that the results are in good agreement with a century of the Dow Jones index daily returns (1900-2000), with the exception of crash days.


Handbook of Quantitative Finance and Risk Management

Handbook of Quantitative Finance and Risk Management
Author: Cheng-Few Lee
Publisher: Springer Science & Business Media
Total Pages: 1700
Release: 2010-06-14
Genre: Business & Economics
ISBN: 0387771174

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Quantitative finance is a combination of economics, accounting, statistics, econometrics, mathematics, stochastic process, and computer science and technology. Increasingly, the tools of financial analysis are being applied to assess, monitor, and mitigate risk, especially in the context of globalization, market volatility, and economic crisis. This two-volume handbook, comprised of over 100 chapters, is the most comprehensive resource in the field to date, integrating the most current theory, methodology, policy, and practical applications. Showcasing contributions from an international array of experts, the Handbook of Quantitative Finance and Risk Management is unparalleled in the breadth and depth of its coverage. Volume 1 presents an overview of quantitative finance and risk management research, covering the essential theories, policies, and empirical methodologies used in the field. Chapters provide in-depth discussion of portfolio theory and investment analysis. Volume 2 covers options and option pricing theory and risk management. Volume 3 presents a wide variety of models and analytical tools. Throughout, the handbook offers illustrative case examples, worked equations, and extensive references; additional features include chapter abstracts, keywords, and author and subject indices. From "arbitrage" to "yield spreads," the Handbook of Quantitative Finance and Risk Management will serve as an essential resource for academics, educators, students, policymakers, and practitioners.


Multivariate Continuous Time Stochastic Volatility Models Driven by a Lévy Process

Multivariate Continuous Time Stochastic Volatility Models Driven by a Lévy Process
Author:
Publisher:
Total Pages:
Release: 2007
Genre:
ISBN:

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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.


Essentials of Econophysics Modelling

Essentials of Econophysics Modelling
Author: Frantisek Slanina
Publisher: Oxford University Press, USA
Total Pages: 427
Release: 2014
Genre: Business & Economics
ISBN: 0199299684

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This book is a course in methods and models rooted in physics and used in modelling economic and social phenomena. It covers the discipline of econophysics, which creates an interface between physics and economics. Besides the main theme, it touches on the theory of complex networks and simulations of social phenomena in general. After a brief historical introduction, the book starts with a list of basic empirical data and proceeds to thorough investigation of mathematical and computer models. Many of the models are based on hypotheses of the behaviour of simplified agents. These comprise strategic thinking, imitation, herding, and the gem of econophysics, the so-called minority game. At the same time, many other models view the economic processes as interactions of inanimate particles. Here, the methods of physics are especially useful. Examples of systems modelled in such a way include books of stock-market orders, and redistribution of wealth among individuals. Network effects are investigated in the interaction of economic agents. The book also describes how to model phenomena like cooperation and emergence of consensus. The book will be of benefit to graduate students and researchers in both Physics and Economics.


Extreme Times for Volatility Processes

Extreme Times for Volatility Processes
Author: Josep Perelló
Publisher:
Total Pages: 29
Release: 2007
Genre:
ISBN:

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Extreme times techniques, generally applied to nonequilibrium statistical mechanical processes, are also useful for a better understanding of financial markets. We present a detailed study on the mean first-passage time for the volatility of return time series. The empirical results extracted from daily data of major indices seem to follow the same law regardless of the kind of index thus suggesting an universal pattern. The empirical mean first-passage time to a certain level L is fairly different from that of the Wiener process showing a dissimilar behavior depending on whether L is higher or lower than the average volatility. All of this indicates a more complex dynamics in which a reverting force drives volatility toward its mean value. We thus present the mean first-passage time expressions of the most common stochastic volatility models whose approach is comparable to the random diffusion description. We discuss asymptotic approximations of these models and confront them to empirical results with a good agreement with the exponential Ornstein-Uhlenbeck model.


Derivatives in Financial Markets with Stochastic Volatility

Derivatives in Financial Markets with Stochastic Volatility
Author: Jean-Pierre Fouque
Publisher: Cambridge University Press
Total Pages: 222
Release: 2000-07-03
Genre: Business & Economics
ISBN: 9780521791632

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This book, first published in 2000, addresses pricing and hedging derivative securities in uncertain and changing market volatility.


Extremal Behavior of Stochastic Volatility Models

Extremal Behavior of Stochastic Volatility Models
Author:
Publisher:
Total Pages:
Release: 2005
Genre:
ISBN:

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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


Applied Stochastic Differential Equations

Applied Stochastic Differential Equations
Author: Simo Särkkä
Publisher: Cambridge University Press
Total Pages: 327
Release: 2019-05-02
Genre: Business & Economics
ISBN: 1316510085

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With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.


Discrete Time Series, Processes, and Applications in Finance

Discrete Time Series, Processes, and Applications in Finance
Author: Gilles Zumbach
Publisher: Springer Science & Business Media
Total Pages: 326
Release: 2012-09-26
Genre: Business & Economics
ISBN: 3642317413

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This book surveys empirical properties of financial time series, discusses their mathematical basis, and describes uses in risk evaluation, option pricing or portfolio construction. The author introduces and assesses a range of processes against the benchmark.