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Power Laws and Financial Markets. Sergio Da Silva Department of Economics, Federal University of Rio Grande Do Sul Raul Matsushita Department of Statistics, University of Brasilia Iram Gleria Department of Physics, Federal University of Alagoas Annibal Figueiredo
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Power LawsandFinancial Markets Sergio Da Silva Department of Economics, Federal University of Rio Grande Do Sul Raul Matsushita Department of Statistics, University of Brasilia Iram Gleria Department of Physics, Federal University of Alagoas Annibal Figueiredo Department of Physics, University of Brasilia Pushpa Rathie Department of Statistics, University of Brasilia
Power LawsIntuition A non-normal scale-free power law means that there is no such thing as a typical event, and that there is no qualitative difference between the larger and smaller fluctuations Upheavals are not unusual A big event need not have a cause The causes that trigger a small change on one occasion may initiate a devastating change on another, and no analysis of the conditions at the initial point will suffice to predict the event
Power LawsCritical Numbers and Critical Exponents Gutenberg-Richter power law: the number of earthquakes realeasing energy E is inversely proportional to E2 So double the energy of the earthquake and it becomes four times as rare critical number = 2critical exponent There is nothing sacred about which number is used to specify a power law What really matters is that there are different power laws, and yet all of them share the same special, self-similar character
Power LawsEarthquakes If the size of an earthquake is doubled, these quakes become four times less frequent The bigger the quake, the rarer it is The distribution is scale invariant, that is, what triggers small and large quakes is precisely the same
Power LawsExtinctions Same as for earthquakes: every time the size of an extinction is doubled, it becomes four times as rare
Power LawsAvalanches One can predict the likely frequency of avalanches, but not when they will happen or what size each will be It may come as no surprise that big avalanches occur less frequently than small ones What is surprising is that there is a power law: each time the size of an avalanche of rice grains is doubled, it becomes twice as rare
Power LawsFracture If one throws frozen potatoes at a wall, they will break into fragments of varying size If one collects all the pieces up, from the microscopic ones to the large, and puts them into different piles according to weight, a power law for fracture emerges: each time the weight of the fragments is reduced by two, there will be six times as many
Power LawsForest Fires When the area covered by a fire is doubled, it becomes about 2.48 times as rare
Power LawsSpreading of Diseases Same as for forest fires: when the area covered by a disease is doubled, it becomes about 2.48 times as rare
Power LawsWars Every time the number of deaths is doubled, wars of that size become 2.62 times less common Such a power law means that when a war starts out no one knows how big it will become There seem to be no special conditions to trigger a great conflict Likewise revolutions are moments that got lucky
Power LawsDistribution of Paper Citations If the number of citations is doubled, the number of papers receiving that many falls off by about eight So there is no typical number of citations for a paper
Power LawsStock Markets Index α of the Lévy is the negative inverse of the power law slope of the probability of return to the origin This shows how to reveal self-similarity in a non-Gaussian scaling α = 2: Gaussian scaling α < 2: non-Gaussian scaling For the S&P 500 stock index α = 1.4 For the Bovespa index α = 1.6
Power LawsDistribution of Wealth Pareto law: if one counts how many people in America have a net worth of a billion dollars, one will find that about four times as many have a net worth of about half a billion Four times as many again are worth a quarter of a billion, and so on
Power LawsLog-Log Plots Newtonian law of motion governing free fall can be thought of as a power law Dropping an object from a tower
Power LawsDrop Time versus Height of Free Fall The relation between height and drop time is not linear
Pareto-Lévy Distributions I Since returns of financial series are usually larger than those implied by a Gaussian distribution, research interest has revisited the hypothesis of a stable Pareto-Lévy distribution Ordinary stable Lévy distributions have fat power-law tails that decay more slowly than an exponential decay Such a property can capture extreme events, and that is plausible for financial data But it also generates an infinite variance, which is implausible
Pareto-Lévy Distributions II Truncated Lévy flights are an attempt to overturn such a drawback The standard Lévy distribution is thus abruptly cut to zero at a cutoff point The TLF is not stable though, but has finite variance and slowly converges to a Gaussian process as implied by the central limit theorem A canonical example of the use of the truncated Lévy flight for real-world financial data is that of Mantegna and Stanley for the S&P 500
S&P 500Probability Density Functions Collapsed onto the ∆t = 1 Distribution
S&P 500Comparison of the ∆t = 1 Distribution with a Theoretical Lévy and a Gaussian
BovespaPower Law for the Means of Increasing Return Time Lags
BovespaProbability Density Functions Collapsed onto the ∆t = 1 Distribution
BovespaComparison of the ∆t = 1 Distribution with a Theoretical Lévy
Daily Real-Dollar RatePower Law for the Means of Increasing Return Time Lags
Daily Real-Dollar RatePower Law in the Standard Deviation II
Daily Real-Dollar RatePower Law in the Relative LZ Complexity
15-Minute Real-Dollar RatePower Law in the Standard Deviation
15-Minute Real-Dollar RatePower Law in the Hurst Exponent II
15-Minute Real-Dollar RatePower Law in the Hurst Exponent III
15-Minute Real-Dollar RatePower Law in the Autocorrelation Time
15-Minute Real-Dollar RatePower Law in the Relative LZ Complexity
S&P 500Monthly, Jan 1871-Jan 2003Power Law in the Standard Deviation
