The Distribution of Stock Return Volatility

Andersen, Torben G.; Bollerslev, Tim; Diebold, Francis X.; Ebens, Heiko

doi:10.3386/w7933

Cited by 517 publications

(159 citation statements)

References 82 publications

(33 reference statements)

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“…[26][27][28][29][30]). We are concerned in this section with two types of single-stock variables defined over fixed time intervals: returns, defined here as logarithmic relative price changes, and volumes, i.e., numbers of shares traded.…”

Section: Application To Financial Returns and Volumesmentioning

confidence: 99%

Nonextensive statistical mechanics and economics

Tsallis

Anteneodo

Borland³

et al. 2003

Physica A: Statistical Mechanics and its Applications

189

145

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Ergodicity, this is to say, dynamics whose time averages coincide with ensemble averages, naturally leads to Boltzmann-Gibbs (BG) statistical mechanics, hence to standard thermodynamics. This formalism has been at the basis of an enormous success in describing, among others, the particular stationary state corresponding to thermal equilibrium. There are, however, vast classes of complex systems which accomodate quite badly, or even not at all, within the BG formalism. Such dynamical systems exhibit, in one way or another, nonergodic aspects. In order to be able to theoretically study at least some of these systems, a formalism was proposed 14 years ago, which is sometimes referred to as nonextensive statistical mechanics. We briefly introduce this formalism, its foundations and applications. Furthermore, we provide some bridging to important economical phenomena, such as option pricing, return and volume distributions observed in the financial markets, and the fascinating and ubiquitous concept of risk aversion. One may summarize the whole approach by saying that BG statistical mechanics is based on the entropy SBG = −k i pi ln pi, and typically provides exponential laws for describing stationary states and basic time-dependent phenomena, while nonextensive statistical mechanics is instead based on the entropic form Sq = k(1 − i p q i )/(q − 1) (with S1 = SBG), and typically provides, for the same type of description, (asymptotic) power laws.Connections between dynamics and thermodynamics are far from being completely elucidated. Frequently, statistical mechanics is presented as a self-contained body, which could dispense dynamics from its formulation. This is an unfounded assumption (see, for instance, [1] and references therein). Questions still remain open even for one of its most well established equilibrium concepts, namely the Boltzmann-Gibbs (BG) factor e −Ei/kT , where E i is the energy associated with the ith microscopic state of a conservative Hamiltonian system, k is Boltzmann constant, and T the absolute temperature. For example, no theorem exists stating the necessary and sufficient conditions for the use of this celebrated and ubiquitous factor to be justified. In the mathematician F. Takens' words [2]:The values of p i are determined by the following dogma: if the energy of the system in the i th state is E i and if the temperature of the system is T then: One possible reason for this essential point having been poorly emphasized is that when dealing with short-range interacting systems, BG thermodynamical equilibrium may be formulated without much referring to the underlying dynamics of its constituents. One rarely finds in textbooks much more than a quick mention to ergodicity. A full analysis of the microscopic dynamical requirements for ergodicity to be ensured is still lacking, in spite of the pioneering studies of N. Krylov [3]. In his words:In Another possibly concomitant reason no doubt is the enormous success, since more than one century, of BG statistical mechanics for very many systems....

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Section: Application To Financial Returns and Volumesmentioning

confidence: 99%

Nonextensive statistical mechanics and economics

Tsallis

Anteneodo

Borland³

et al. 2003

Physica A: Statistical Mechanics and its Applications

189

145

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“…This is true even if an MA(1) model is used to filter returns as in Andersen, Bollerslev, Diebold, and Ebens (2001).…”

Section: R 2 T[c] R 2 T[c]mentioning

confidence: 99%

The specification of GARCH models with stochastic covariates

Fleming

Kirby

Ostdiek

2008

Journal of Futures Markets

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A number of studies investigate whether various stochastic variables explain changes in return volatility by specifying the variables as covariates in a GARCH(1, 1) or EGARCH(1, 1) model. The authors show that these models impose an implicit constraint that can obscure the true role of the covariates in the analysis. They illustrate the problem by reconsidering the role of contemporaneous trading volume in explaining ARCH effects in daily stock returns. Once the constraint imposed in earlier research is relaxed, it is found that specifying volume as a covariate does little to diminish the importance of lagged squared returns in capturing the dynamics of volatility. INTRODUCTIONThe extensive use of generalized autoregressive conditional heteroscedasticity (GARCH) models in financial economics is testimony to their success in capturing volatility dynamics. As low-order GARCH and exponential GARCH (EGARCH) models typically perform well relative to more complex specifications (see, e.g., Hansen & Lunde, 2005a), researchers often use these models to investigate the relation between changes in return volatility and various stochastic variables. In particular, they assess whether the stochastic variables explain changes in volatility by including the variables as covariates in a GARCH(1, 1) or EGARCH(1, 1) model. The variables considered in the literature include interest rate levels (Engle & Patton, 2001;Glosten, Jagannathan, & Runkle, 1993), interest rate spreads (Dominguez, 1998;Hagiwara & Herce, 1999), forward-spot spreads (Hodrick, 1989), implied volatilities (Blair, Poon, & Taylor, 2001;Day & Lewis, 1992;Lamoureux & Lastrapes, 1993), futures open interest (Girma & Mougoue, 2002), a proxy for the information flow during the overnight market closure (Gallo & Pacini, 2000), and contemporaneous trading volume (Fujihara & Mougoue, 1997;Lamoureux & Lastrapes, 1990;Marsh & Wagner, 2005).Here a closer look is taken at the specification of GARCH models with stochastic covariates, highlighting a specification issue that makes it difficult to draw reliable inferences from many of the models considered in the literature. These models impose an implicit constraint that requires the coefficients on the lagged squared returns and the lagged stochastic variables to decline with the lag length at the same rate. This constraint is problematic for cases in which the stochastic variables provide little information about future return volatility beyond that contained in lagged squared returns. Obtaining precise fitted volatilities in such cases requires giving little weight to the lagged stochastic variables, but owing to the implicit constraint, this also requires giving little weight to lagged squared returns. Hence, if there is a strong contemporaneous relation between the stochastic variables and return volatility, the covariates can drive ARCH effects out of the fitted models regardless of whether they capture volatility persistence.The problem is illustrated by reconsidering the role of contemporaneous trading volume in explai...

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“…Andersen et al (2000c) find log realizations of aggregated squared returns almost gaussian. Thus, as well as the four variations of absolute and squared returns, log realizations of these measures are also included.…”

Section: Insert Table I Herementioning

confidence: 99%

“…This paper follows the analysis completed in a series of papers and chooses 5-minute intervals (for example, Andersen et al (2001) and Andersen et al (2000c)). Although the assets chosen for analysis are the most actively traded UK futures, the level of trading activity may become an issue for other thinly traded assets.…”

Section: Data Featuresmentioning

confidence: 99%

Minimum capital requirement calculations for UK futures

Cotter

2003

Journal of Futures Markets

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Key to the imposition of appropriate minimum capital requirements on a daily basis requires accurate volatility estimation. Here, measures are presented based on discrete estimation of aggregated high frequency UK futures realisations underpinned by a continuous time framework. Squared and absolute returns are incorporated into the measurement process so as to rely on the quadratic variation of a diffusion process and be robust in the presence of fat tails. The realized volatility estimates incorporate the long memory property. The dynamics of the volatility variable are adequately captured. Resulting rescaled returns are applied to minimum capital requirement calculations.

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The Distribution of Stock Return Volatility

Cited by 517 publications

References 82 publications

Nonextensive statistical mechanics and economics

Nonextensive statistical mechanics and economics

The specification of GARCH models with stochastic covariates

Minimum capital requirement calculations for UK futures

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