Scaling behaviour in the dynamics of an economic index

Mantegna, Rosario N.; Stanley, H. Eugene

doi:10.1038/376046a0

Cited by 1,653 publications

(1,219 citation statements)

References 22 publications

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“…The decay time in this economic example is short (4 min), so one cannot easily make money on these correlations (14,15). A little less well known is the measure of the volatility (15,16).…”

Section: Statistical Features Of Price Fluctuationsmentioning

confidence: 98%

Self-organized complexity in economics and finance

Stanley

Amaral

Buldyrev

et al. 2002

Proc. Natl. Acad. Sci. U.S.A.

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This article discusses some of the similarities between work being done by economists and by physicists seeking to contribute to economics. We also mention some of the differences in the approaches taken and seek to justify these different approaches by developing the argument that by approaching the same problem from different points of view, new results might emerge. In particular, we review two newly discovered scaling results that appear to be universal, in the sense that they hold for widely different economies as well as for different time periods: (i) the fluctuation of price changes of any stock market is characterized by a probability density function, which is a simple power law with exponent ؊4 extending over 10 2 SDs (a factor of 10 8 on the y axis); this result is analogous to the Gutenberg-Richter power law describing the histogram of earthquakes of a given strength; and (ii) for a wide range of economic organizations, the histogram shows how size of organization is inversely correlated to fluctuations in size with an exponent Ϸ0.2. Neither of these two new empirical laws has a firm theoretical foundation. We also discuss results that are reminiscent of phase transitions in spin systems, where the divergent behavior of the response function at the critical point (zero magnetic field) leads to large fluctuations.

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“…The decay time in this economic example is short (4 min), so one cannot easily make money on these correlations (14,15). A little less well known is the measure of the volatility (15,16).…”

Section: Statistical Features Of Price Fluctuationsmentioning

confidence: 98%

Self-organized complexity in economics and finance

Stanley

Amaral

Buldyrev

et al. 2002

Proc. Natl. Acad. Sci. U.S.A.

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“…Historically, the central limit theorem led to the first paradigm in terms of Gaussian pdf's that was first put in doubt by Mandelbrot [41] when he proposed to use Lévy distributions, that are characterised by a fat tail decaying as a power law with index µ between 0 and 2. Recently, physicists have characterised more precisely the distribution of market price variations [42,43] and found that a power law truncated by an exponential provides a reasonable fit at short time scales (much less than one day), while at larger time scales the distributions cross over progressively to the Gaussian distribution which becomes approximately correct for monthly and larger scale price variations. Alternative representations exist in terms of a superposition of Gaussian pdf's corresponding to cascade models inspired from an analogy with turbulence [44].…”

Section: -4 Daily Forex Us-mark and Us-franc Price Variationsmentioning

confidence: 99%

“…The second model is the curved shifted linear fractal log S n = log S 0 -a log (n+A) , (14) and the last model we consider is the lognormal distribution (with standard deviation s and most probable value m). We have chosen the parameters of these four models in such a way that they approach each other the most closely in the interval of ranks 5-500, as shown in Figure 1.…”

Section: -Evidence Of Stretched Exponentials and Comparison With Othmentioning

confidence: 99%

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Stretched exponential distributions in nature and economy: “fat tails” with characteristic scales

Laherrere¹,

1998

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To account quantitatively for many reported « natural » fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok temperature variations, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also briefly discuss its potential for the distribution of earthquake sizes and fault displacements and earth temperature variations over the last 400 000 years. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions. 1-IntroductionFrequency or probability distribution functions (pdf) that decay as a power law of their argument P(x) dx = P 0 x -(1+µ) dx (1) have acquired a special status in the last decade. They are sometimes called ``fractal'' (even if this term is more appropriate for the description of self-similar geometrical objects rather than statistical distributions). A power law distribution characterizes the absence of a characteristic size : independently of the value of x, the number of realizations larger dans λx is λ -µ times the number of realizations larger than x. In contrast, an exponential for instance or any other functional dependence does not enjoy this self-similarity, as the existence of a characteristic scale destroys this continuous scale invariance property [1]. In words, a power law pdf is such that there is the same proportion of smaller and larger events, whatever the size one is looking at within the power law range.The asymptotic existence of power laws is a well-established fact in statistical physics and critical phenomena with exact solutions available for the 2D Ising model, for selfavoiding walks, for lattice animals, etc.[2], with an abondance of numerical evidence for instance for the distribution of percolation clusters at criticality [3] and for many other models in statistical physics. There is in addition the observation from numerical simulations that simple ``sandpile'' models of spatio-temporal dynamics with strong non-linear behavior [4] give power law distributions of avalanche sizes. Furthermore, precise experiments on critical phenomena confirm the asymptotic existence of power laws, for instance on superfluid helium at the lambda point and on binary mixtures [5]. These are th...

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“…West et al [1997], Barabási and Albert [1999], Newman [2005]). In finance, there is one scaling law that has been widely reported (Müller et al [1990], Mantegna and Stanley [1995], Galluccio et al [1997], Guillaume et al [1997], Ballocchi et al [1999], , Corsi et al [2001], Di Matteo et al [2005]): the size of the average absolute price change (return) is scale-invariant to the time interval of its occurrence. This scaling law has been applied to risk management and volatility modelling (see Ghashghaie et al [1996], Gabaix et al [2003], Sornette [2000], Di Matteo [2007]) even though there has been no consensus amongst researchers for why the scaling law exists (e.g., Bouchaud [2001], Barndorff-Nielsen and Prause [2001], Farmer and Lillo [2004], Lux [2006], Joulin et al [2008]).…”

Section: Introductionmentioning

confidence: 99%

Patterns in high-frequency FX data: discovery of 12 empirical scaling laws

Glattfelder¹,

Dupuis

Olsen

2011

Quantitative Finance

144

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We have discovered 12 independent new empirical scaling laws in foreign exchange data-series that hold for close to three orders of magnitude and across 13 currency exchange rates. Our statistical analysis crucially depends on an event-based approach that measures the relationship between different types of events. The scaling laws give an accurate estimation of the length of the price-curve coastline, which turns out to be surprisingly long. The new laws substantially extend the catalogue of stylised facts and sharply constrain the space of possible theoretical explanations of the market mechanisms.

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Scaling behaviour in the dynamics of an economic index

Cited by 1,653 publications

References 22 publications

Self-organized complexity in economics and finance

Self-organized complexity in economics and finance

Stretched exponential distributions in nature and economy: “fat tails” with characteristic scales

Patterns in high-frequency FX data: discovery of 12 empirical scaling laws

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