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 Department of Physics, University of Brasilia Pushpa Rathie Department of Statistics, University of Brasilia Power Laws Intuition 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 Laws Critical 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 Laws Earthquakes 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 Laws Extinctions Same as for earthquakes:

every time the size of an extinction is doubled, it becomes four times as rare Power Laws Avalanches 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 Laws Fracture 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 Laws

Forest Fires When the area covered by a fire is doubled, it becomes about 2.48 times as rare Power Laws Spreading 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 Laws Wars 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 Laws Distribution 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 Laws Stock Markets Index of the Lvy 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

= 2: Gaussian scaling < 2: non-Gaussian scaling For the S&P 500 stock index = 1.4 For the Bovespa index = 1.6 Power Laws Distribution 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 Laws Log-Log Plots Newtonian law of motion governing free fall can be thought of as a power law

Dropping an object from a tower Height 4 8 12 16 20 24 28 32 Time 0.89 1.26 1.55 1.79 2 2.19 2.37 2.53

log(Height) 0.602059991 0.903089987 1.079181246 1.204119983 1.301029996 1.380211242 1.447158031 1.505149978 log(Time) -0.050609993 0.100370545 0.190331698 0.252853031 0.301029996 0.340444115 0.374748346 0.403120521

Power Laws Drop Time versus Height of Free Fall The relation between height and drop time is not linear 3 2.5 2 1.5 1 0.5 0 0 10 20 30

40 Power Laws Log of Drop Time versus Log of Height of Fall 0.5 y = 0.5028x - 0.3532 R2 = 1 0.4 0.3 0.2 0.1 0 -0.1 0 0.5 1

1.5 2 Power Laws Log-Log Plots The power law takes the form t c h d log t d log h log c d slope log c b is the y-intercept ( c 10 b ) From Figure 2 we get d 0.5 and log c 0.35 c 10 0.35 0.45 t 0.45 h 0.5 This is in line with Newtons law g h t2 2

2h t 9.81 t 0.45 h 0.5 Pareto-Lvy 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-Lvy distribution Ordinary stable Lvy 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-Lvy Distributions II Truncated Lvy flights are an attempt to overturn such a drawback

The standard Lvy 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 Lvy flight for real-world financial data is that of Mantegna and Stanley for the S&P 500 S&P 500 Probability Density Functions S&P 500 Power Law in the Probability of Return to the Origin S&P 500 Probability Density Functions Collapsed onto the t = 1 Distribution S&P 500 Comparison of the t = 1 Distribution with a Theoretical Lvy and a Gaussian

Bovespa Power Laws Log returns Z(t) lnY(t + t) - lnY(t) (1) where Y is a Bovespa index closing day value in dollar terms, t is time (i.e. a trading day), and t is initially one trading day. Probability of return to the origin P(Z = 0) (2) Since the peak of a distribution is not exactly located at Z = 0 to all t , we take P(Z = (t ) ) to represent the probability of return to the origin instead. If the central region of the distributions is well described by a Lvy PDF of the form 1 (3) P(W) L (W, t) exp( tq )cos(qW)dq 0

of scaling index and scale factor at t = 1 , then the probability of return to the origin is given by (1 / ) , (4) P(0) = (t)1/ where stands for the Gamma function. Bovespa Power Laws By plugging the slope value of -0.601207, the scaling index =1.66332 obtains. Lvy PDFs rescale under the transformations W Ws , (5) (t) 1/ and

L (W, t) L (Ws ,1) -1 . (6) (t) Existence of a scaling law in the probability of return to the origin justifies a scaled plot of the PDFs. By using equations (5) and (6) with the scaling index =1.66332, all the data collapse onto the t = 1 distribution. Bovespa Power Law for the Means of Increasing Return Time Lags Bovespa Probability Density Functions Bovespa Power Law in the Probability of Return to the Origin Bovespa Probability Density Functions Collapsed onto the t = 1 Distribution

Bovespa Comparison of the t = 1 Distribution with a Theoretical Lvy Daily Real-Dollar Rate Power Law in the Mean 0.5 0 -0.5 -1 -1.5 -2 -2.5 -3 -3.5 y = 1.0343x - 3.015 R2 = 0.9996

0 1 2 3 Daily Real-Dollar Rate Power Law for the Means of Increasing Return Time Lags 1 2 3 4 5 6 7 8 9 10

20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 0.00101137

0.00200142 0.003033665 0.004074478 0.005066924 0.006088324 0.007115207 0.008144736 0.009164996 0.01017602 0.02014626 0.03055909 0.04188924 0.05416851 0.06693177 0.08036166 0.09405512 0.1077238 0.1204376 0.2621603 0.3683358

0.4735743 0.5972076 0.7071869 0.8148769 0.9289032 1.061132 1.217908 0 0.301029996 0.477121255 0.602059991 0.698970004 0.77815125 0.84509804 0.903089987 0.954242509 1 1.301029996 1.477121255

1.602059991 1.698970004 1.77815125 1.84509804 1.903089987 1.954242509 2 2.301029996 2.477121255 2.602059991 2.698970004 2.77815125 2.84509804 2.903089987 2.954242509 3 -2.995089933 -2.698661114 -2.518032379

-2.389928023 -2.29525561 -2.215502244 -2.147812461 -2.089122988 -2.03786772 -1.992422048 -1.695805566 -1.514859582 -1.377897519 -1.26625311 -1.17436769 -1.094951101 -1.026617558 -0.967688335 -0.919237907 -0.581433075 -0.433756068 -0.324611874 -0.223874674

-0.150465793 -0.088907993 -0.032029541 0.025769412 0.085614483 Daily Real-Dollar Rate Power Law in the Standard Deviation I 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8

y = 0.5331x - 1.5889 R2 = 0.9993 0 0.5 1 1.5 2 Daily Real-Dollar Rate Power Law in the Standard Deviation II 0 y = 0.1924x - 0.9079 R2 = 0.876

-0.1 -0.2 -0.3 -0.4 -0.5 -0.6 2 2.2 2.4 2.6 2.8 3 Hurst Exponent

The Hurst exponent can distinguish a random series from a nonrandom series, even if the random series in non-Gaussian H = 0.5: data are uncorrelated H > 0.5: persistence, i.e. past trends persist into the future H < 0.5: antipersistence, i.e. past trends tend to reverse in the future The Hurst exponent is the inverse of the scaling exponent So data generated by a Lvy shows long memory = 1.6 H = 0.625 = 2 H = 0.5 Daily Real-Dollar Rate Power Law in the Hurst Exponent I 0 -0.05 -0.1

-0.15 -0.2 -0.25 y = 0.1692x - 0.2519 R2 = 0.9983 -0.3 0 0.2 0.4 0.6 0.8 1

Daily Real-Dollar Rate Power Law in the Hurst Exponent II 0 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 y = 0.0753x - 0.1528 R2 = 0.9687 1 1.2

1.4 1.6 1.8 2 Daily Real-Dollar Rate Power Law in the Hurst Exponent III 0.002 0 -0.002 -0.004 -0.006 -0.008 y = 0.0076x - 0.0216

R2 = 0.8159 -0.01 2 2.2 2.4 2.6 2.8 3 Daily Real-Dollar Rate Power Law in the Autocorrelation Time 3 2.5

2 1.5 1 0.5 0 -0.5 y = 0.7892x + 0.0212 R2 = 0.9677 0 1 2 3 LZ Complexity The Lempel-Ziv (LZ) index measures complexity

relative to Gaussian white noise LZ = 0: perfect predictability LZ = 1: maximal complexity (randomness) Daily Real-Dollar Rate Power Law in the Relative LZ Complexity 0.5 y = -0.3822x + 0.1525 R2 = 0.8849 0 -0.5 -1 -1.5 0 1

2 3 15-Minute Real-Dollar Rate Power Law in the Mean -0.5 -1 -1.5 -2 -2.5 -3 -3.5 -4 -4.5 y = 1.0763x - 4.1595 R2 = 0.9994

0 0.5 1 1.5 2 2.5 3 15-Minute Real-Dollar Rate Power Law in the Standard Deviation 0 -0.5 -1

-1.5 -2 y = 0.4903x - 1.9931 R2 = 0.9966 -2.5 0 1 2 3 15-Minute Real-Dollar Rate Power Law in the Hurst Exponent I -0.1 -0.15

-0.2 y = 0.1658x - 0.2807 R2 = 0.9966 -0.25 -0.3 0 0.2 0.4 0.6 0.8 1 15-Minute Real-Dollar Rate

Power Law in the Hurst Exponent II 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 y = 0.1015x - 0.2177 R2 = 0.9971 1 1.2 1.4

1.6 1.8 2 15-Minute Real-Dollar Rate Power Law in the Hurst Exponent III 0 -0.005 -0.01 -0.015 y = 0.015x - 0.0454 R2 = 0.9256 -0.02 2

2.2 2.4 2.6 2.8 3 15-Minute Real-Dollar Rate Power Law in the Autocorrelation Time 3 2.5 2 1.5 1 0.5 0

-0.5 y = 1.0111x - 0.2045 R2 = 0.9949 0 1 2 3 15-Minute Real-Dollar Rate Power Law in the Relative LZ Complexity 0.5 y = -0.48x + 0.2013 R2 = 0.9464

0 -0.5 -1 -1.5 -2 0 1 2 3 S&P 500 Monthly, Jan 1871-Jan 2003 Power Law in the Mean 3 2.5

2 1.5 1 0.5 y = 0.8854x - 0.1016 R2 = 0.9761 0 -0.5 0 1 2 3 S&P 500 Monthly, Jan 1871-Jan 2003 Power Law in the Standard Deviation

2.5 2 1.5 y = 0.5029x + 0.8702 R2 = 0.9941 1 0.5 0 0.5 1 1.5 2 2.5

3 S&P 500 Monthly, Jan 1871-Jan 2003 Power Law in the Autocorrelation Time 1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 y = 1.4945x - 0.2764 R2 = 0.912 0

0.2 0.4 0.6 0.8 S&P 500 Daily, 2 Jan 1980-31 Dec 2001 Power Law in the Mean 2.5 2 1.5 1 0.5 0 -0.5

-1 y = 1.0735x - 0.9427 R2 = 0.9977 0 0.5 1 1.5 2 2.5 3 S&P 500

Daily, 2 Jan 1980-31 Dec 2001 Power Law in the Standard Deviation 2.5 2 1.5 1 y = 0.5029x + 0.8702 R2 = 0.9941 0.5 0 0.5 1 1.5

2 2.5 3 S&P 500 Daily, 2 Jan 1980-31 Dec 2001 Power Law in the Autocorrelation Time 2 1.5 1 0.5 y = 1.0178x - 0.2331 R2 = 0.9884 0 -0.5 0

0.5 1 1.5 2 Daily Exchange Rates Data Sets Country Australia Austria Belgium Brazil Britain Canada China Denmark

Euro Area Finland France Germany Ireland India Italy Japan Malaysia Mexico Netherlands New Zealand Portugal Singapore South Africa South Korea Spain Sri Lanka Sweden Switzerland

Taiwan Thailand Venezuela Currency Australian Dollar Shilling Belgian Franc Real Pound Canadian Dollar Yuan Krone False Euro Markka Franc Deutsche Mark Pound Rupee Lira

Yen Ringgit Peso Guilder New Zealand Dollar Escudo Singapore Dollar Rand Won Peseta Rupee Krona Swiss Franc Taiwan Dollar Baht Bolivar Time Period 4Jan71 10Jan03 4Jan71 31Dec98

4Jan71 31Dec98 2Jan95 10Jan03 4Jan71 10Jan03 4Jan71 10Jan03 2Jan81 10Jan03 4Jan71 10Jan03 4Jan93 10Jan03 4Jan71 31Dec98 4Jan71 31Dec98 4Jan71 31Dec98 4Jan71 31Dec98 2Jan73 10Jan03 4Jan71 31Dec98 4Jan71 10Jan03 4Jan71 10Jan03 8Nov93 10Jan03 4Jan71 31Dec98 4Jan71 10Jan03 2Jan73 31Dec98 2Jan81 10Jan03

4Jan71 10Jan03 13Apr81 10Jan03 2Jan73 31Dec98 2Jan73 10Jan03 4Jan71 10Jan03 4Jan71 10Jan03 30Oct83 10Jan03 2Jan81 10Jan03 2Jan95 10Jan03 Data Points 8025 6999 7013 2014 8032 8038 5471 8031 2521

6976 7021 7021 7021 7525 7020 8026 8010 2300 7021 8016 6518 5531 8005 5416 6521 7172 8031 8032 4548

5428 2013 Daily Exchange Rates Parameters of the Lvy Country Australia Austria Belgium Brazil Britain Canada China Denmark Finland France Germany Ireland India Italy

Japan Malaysia Mexico Netherlands New Zealand Portugal Singapore South Africa South Korea Spain Sri Lanka Sweden Switzerland Taiwan Thailand Venezuela

1.41487 1.90185 1.56042 .89059 1.76454 2.04822 4.19286 1.39021 1.75114 1.48668 1.54737 1.61516 1.87979 1.27801 1.43542 2.78363 1.60305 1.55999 1.87623

1.33192 1.81272 3.46313 .93298 1.28282 1.22370 1.53611 1.68564 1.19228 2.03006 4.13507 .004656830 .000010368 .009849513 .003707604 .000078676 .000005979 2.5306E-11 .002288440

.00040350 .001021003 .000330146 .000150068 .000018721 .014513000 .011937000 .000000169 .000923330 .00034082 .000011118 .00514115 .000027149 1.7417E-8 .015343000 .01962400 .000898138 .000880928 .000209469 .003356487

.000033537 6.6404E-9 Daily Exchange Rates Probability Density Functions Daily Exchange Rates Power Laws in the Probability of Return to the Origin Daily Exchange Rates Probability Density Functions Collapsed onto the t = 1 Distribution Daily Exchange Rates Yuan: IFS Clumpiness Test Lvy Flights I Owing to the sharp truncation, the characteristic function of the TLF is no longer infinitely divisible as well However, it is still possible to define a TLF with a smooth cutoff that yields an infinitely divisible characteristic

function: smoothly truncated Lvy flight In such a case, the cutoff is carried out by asymptotic approximation of a stable distribution valid for large values Yet the STLF breaks down in the presence of positive feedbacks Lvy Flights II But the cutoff can still be alternatively combined with a statistical distribution factor to generate a gradually truncated Lvy flight Nevertheless that procedure also brings fatter tails The GTLF itself also breaks down if the positive feedbacks are strong enough This apparently happens because the truncation function decreases exponentially Lvy Flights III Generally the sharp cutoff of the TLF makes moment scaling approximate and valid for a finite time interval

only; for longer time horizons, scaling must break down And the breakdown depends not only on time but also on moment order Exponentially damped Lvy flight: a distribution might be assumed to deviate from the Lvy in both a smooth and gradual fashion in the presence of positive feedbacks that may increase Daily Exchange Rates Exponentially Damped Lvy Flights I Daily Exchange Rates Exponentially Damped Lvy Flights II Multiscaling Whether scaling is single or multiple depends on how a Lvy flight is broken While the abruptly truncated Lvy flight (the TLF itself) exhibits mere single scaling,

the STLF shows multiscaling An abruptly TLF fits data for daily exchange rates against the US The same data set might be well fitted by an EDLF We can then focus on the multiscaling properties stemming from the EDLF Daily Exchange Rates Multiscaling I Country Australia Austria Belgium Brazil Britain Canada China Denmark Euro Area Finland France

Germany Ireland India Italy Japan Malaysia Mexico Netherlands New Zealand Portugal Singapore South Africa South Korea Spain Sri Lanka Sweden Switzerland Taiwan Thailand Venezuela

Currency Australian Dollar Shilling Belgian Franc Real Pound Canadian Dollar Yuan Krone False Euro Markka Franc Deutsche Mark Pound Rupee Lira Yen Ringgit Peso

Guilder New Zealand Dollar Escudo Singapore Dollar Rand Won Peseta Rupee Krona Swiss Franc Taiwan Dollar Baht Bolivar Exponent Multiscaling (2) Multiscaling (1.8) Single Scaling Single Scaling Single Scaling

Single Scaling Multiscaling (1) Multiscaling (2) Multiscaling (2.2) Multiscaling (2.1) Multiscaling (2) Multiscaling (2) Multiscaling (2) Multiscaling (2) Multiscaling (2.5) Multiscaling (2) Multiscaling (1.8) Multiscaling (2) Multiscaling (2.5) Multiscaling (1.5) Multiscaling (2.5) Multiscaling (2) Multiscaling (1) Multiscaling (1.5) Multiscaling (2)

Multiscaling (~0) Multiscaling (2) Multiscaling (2.5) Multiscaling (1.5) Multiscaling (~0) Multiscaling (1) Exponent Multiscaling (2) Single Scaling Multiscaling (0.3) Single Scaling Single Scaling Single Scaling Multiscaling (1) Single Scaling Single Scaling Single Scaling Single Scaling Single Scaling

Single Scaling Multiscaling (1.4) Multiscaling (0.1) Multiscaling (0.25) Multiscaling (2) Multiscaling (1.5) Single Scaling Single Scaling Multiscaling (0.2) Single Scaling Multiscaling (2) Multiscaling (2) Multiscaling (~0) Multiscaling (~0) Multiscaling (3) Single Scaling Multiscaling (0.5) Multiscaling (0.5) Multiscaling (~0)

Daily Exchange Rates Multiscaling II Daily Exchange Rates Multiscaling III Daily Exchange Rates Multiscaling IIII Daily Exchange Rates Multiscaling V References Power Laws in General I Bak, Per (2000) 'The end of history', New Scientist, 11 November, 56-7. Bak, P., Chen, K., Scheinkman, J. and Woodford, M. (1993) 'Aggregate fluctuations from independent sectoral shocks: self-organized criticality in a model of production and invertory dynamics', Ricerche Economiche 47, 3-30. Bak, Per, Tang, Chao and Wiesenfeld, Kurt (1987) 'Self-organized criticality: an explanation of 1/f noise', Physics Review Letters 59: 381-4. Bouchaud, Jean-Philippe and Mezard, Marc (2000) 'Wealth condensation in a simple model of economy', Physica A 282, 536-45.

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[3] A. Figueiredo, I. Gleria, R. Matsushita, S. Da Silva, Autocorrelation as a source of truncated Lvy flights in foreign exchange rates, Physica A 323 (2003) 601-625. [4] R. Matsushita, P. Rathie, S. Da Silva, Exponentially damped Lvy flights, Physica A 326 (2003) 544-555. [5] R. Matsushita, I. Gleria, A. Figueiredo, S. Da Silva, Fractal structure in the Chinese yuan/US dollar rate, Econom. Bull. 7 (2003) 1-13. [6] A. Figueiredo, I. Gleria, R. Matsushita, S. Da Silva, On the origins of truncated Lvy flights, Phys. Lett. A 315 (2003) 51-60. [7] A. Figueiredo, I. Gleria, R. Matsushita, P. Rathie, S. Da Silva, Exponentially damped Lvy flights, multiscaling, and exchange rates, Physica A, forthcoming. References Our Own Work References Our Own Work References Our Own Work

References Our Own Work