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ITISE 2018

International Conference on Time Series and Forecasting

Proceedings of Papers

19-21 September 2018

Granada (Spain)

Editors and Chairs

Olga Valenzuela

Fernando Rojas

Héctor Pomares

Ignacio Rojas

I.S.B.N: 978-84-17293-57-4

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sanctions established in the laws.

Power laws in stock market and fractal

complexity of S&P500 and DAX

Anna Krakovsk´a

Institute of Measurement Science, Slovak Academy of Sciences,

D´ubravsk´a Cesta 9, 842 19 Bratislava, Slovak Republic

krakovska@savba.sk

Abstract. In this study the current research evidence about the role of

power laws and fractals in stock market data is being reviewed. Then

the fractal complexity of stock indexes in US (S&P 500) and Germany

(DAX) is estimated. Daily closing prices from 1950 to 2017 are used for

calculations.

The results indicate a slightly downward trend in Hurst exponent during

the investigated decades, meaning modest decrease in the spectral power-

law exponent and increase in the fractal complexity of the indices.

Possible links between the ﬁnancial crises and changes in complexity are

also discussed and questioned.

Keywords: power law, fractal dimension, Hurst exponent, S&P 500,

DAX

1 Introduction

Complexity can be generated in a variety of ways. In chaos theory, irregular,

hardly predictable behaviour can arise from a few nonlinear diﬀerential equa-

tions. In agent based models, large number of individual objects governed by a

few simple rules are capable of generating remarkably complicated formations.

Moreover, complexity may be spatial or temporal - it may involve complicated

patterns that do not change over time, or may appear as a surprising time vary-

ing behaviour.

In terms of complexity, the stock market is a great place to look at. It is

composed of many decision-making individuals with similar motivations. The

market is open to the environment, and its dynamics has interesting features.

Moreover, the development of this remarkable system is well documented. In the

form of stock indexes and price series, we have decades of data to study.

However, if we see the stock market as a system of complex behaviour, then

we must call into question the standard random walk model with step sizes that

vary according to a normal distribution. However, questioning the normal or

log-normal distributions in economics, or in any real-world data, is not easy.

Gaussians seem to be all around us. Frequently mentioned examples are distri-

butions of height, weight, measurement errors. We even have the central limit

theorem that explains why normal distributions are so common. The theorem

Proceedings ITISE 2018. Granada, 19-21 September, 2018.

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2 Power laws in stock market and fractal complexity of S&P500 and DAX

states that the normal distribution arises whenever a large number of indepen-

dent and identically distributed random variables with ﬁnite variances are added

together.

In economics, Gaussian distribution of stock returns and the correspond-

ing random walk model are the essence of the eﬃcient market hypothesis [1].

According to this idea, the investors have all information about the expected

earnings of the stocks. They should always agree about the right price. The

prices can change a little bit after the companies release new estimates of the

future earnings. But no such things as bubbles and crises should happen.

Nevertheless, normal distribution may not be as typical as it has long been

thought of. Already in 1960 Benoit Mandelbrot pointed out that large move-

ments in prices are much more common than would be predicted from a normal

distribution [2]. As an alternative, he suggested the heavy tailed L´evy distribu-

tions. The importance of this result was under-valued by the economists until

the appearance of modern risk management methods around the 1990s. At that

time, the quantitative estimation of the heavy tailed distributions started to be

taken much more seriously. Mantegna and Stanley have shown that for the S&P

500 the central part of the distribution corresponds to the L´evy stable process

[3]. Scaling behaviour has been observed for time intervals spanning from min-

utes to weeks. The tails of the distribution deviated from that for a L´evy process

and were approximately exponential, ensuring - as one would expect for the price

returns - the ﬁnite variance of the distribution.

Although heavy-tailed distributions are already accepted in ﬁnance, discus-

sions about their origins are still actively going on. For proponents of complex

systems, the heavy tails, implying occasional market crashes, and also the for-

mation of bubbles appear as a natural and inherent feature of the system. The

ﬁrst artiﬁcial models that conﬁrm this, were designed in the nineties [4]. In the

models - computer simulated markets - adaptive learning agents (traders) are

trying to predict price movements and buy or sell based on their forecasts. The

subsequent change in the price then gets fed back to the traders and is used

for further decision about buying or selling. The feedback makes the market a

speciﬁc dynamic system. Every emergence of obvious predictability and thus the

vision of proﬁt opportunity is immediately revealed by a huge number of traders.

The massive attempt to use the new discovery immediately leads to canceling

or even reversing the pattern. But it is remarkable that in models (and also in

reality) sometimes the exact opposite occurs - market participants keep a certain

pattern alive. Usually it is a sustained rise in stock prices, known as a bubble.

Bubbles are accompanied by an unreasonable increase in the price-earning (P/E)

ratio of the shares concerned. The bubbles inevitably crash after some time, but

the height of peaks are hard to predict.

Excessive increase in P/E ratio can be caused by unrealistic expectations of

future earnings. However, it can also be triggered by herding behaviour. Nate

Silver emphasizes in his book [5] that, in the 1960s, only about 15 percent of

stocks were held by institutions rather than individuals. By 2007, the percentage

had risen to 68 percent. When a trader does not risk his own money, then it may

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Power laws in stock market and fractal complexity of S&P500 and DAX 3

be quite rational for him to stay with the herd when the market goes up. When

the trouble comes, the investor can blame the unpredictability of the market or,

at best, admit collective guilt.

Herding behaviour, along with fat tailed distributions, volatility persistence,

uncorrelated returns and other stylized facts were replicated by a plethora of

agent-based models to date (see e.g. [6] and references therein). What is inter-

esting is that the systems often converge to reality-like complex behavior, even

though they have started with very simple initial condition of huge number of

agents and a couple of trivial rules. Bak et al. have argued that under rather gen-

eral conditions the preferred state is the one for which there is no characteristic

spatial or temporal scale size [7]. Objects with no spatial scale are fractals (term

coined by Mandelbrot in 1970s), and ﬂuctuations with no temporal scale are gov-

erned by power laws. For the above phenomenon, in 1980s Bak et al. introduced

the term self-organized criticality. In ﬁnance it immediately attracted increased

attention to the power-law process as to the next candidate for modeling the

distribution of returns.

Our variable of interest xhas a so called power-law distribution if

p(x)∝1/xα.

Power-law distributions are long-tailed, giving a relatively large probability

p(x) to extreme events. They diverge at zero, so there must be a minimum value

xmin >0 for which the power-law behaviour holds. If α > 1, then the median

can be computed as 21/(α−1)xmin [8]. A power-law process has a well-deﬁned

mean only if α > 2, and it has a ﬁnite variance only if α > 3.

Power laws are very common in nature. For example both the gravitational

and electrical force decrease inversely with the square of the distance from the

mass or charge and thus have no characteristic length scale (until the size of the

molecules). Let us also mention the Zipf’s law from 1929, valid for the frequency

of words in diﬀerent languages, and for many other phenomena [9]. Another

similar law emerged in the 1930s from seismological research of Gutenberg and

Richter [10]. Power laws are also ubiquitous in economics. Just remember the

Pareto principle from 1896, concerning the income distribution [11].

It is unclear why power laws should be so common. They do not emerge

from the generalized central limit theorem as easily as the normal distribution.

A generalization of the central limit theorem due to Kolmogorov and Gnedenko

states that the sum of a number of random variables with power-law tails tends

to a stable distribution only for 1 < α ≤3 [12]. If α > 3 then the sum converges

to the Gaussian distribution. However, most identiﬁed power laws in nature are

below 3 and they are so common that, sometimes, it would seem that power

laws are ”the new normal”.

Self-organized criticality has been one of the ﬁrst major attempt to explain

the abundance of power laws. Bak et al. declared discovery of a robust mech-

anism of spontaneous emergence of complexity from simple local interactions

- a promising source of ubiquitous complexity. However, the idea has ﬁnally

shown serious weaknesses and considerable eﬀorts have been made to ﬁnd new

candidates for universal mechanisms for generating power laws [13], [14].

Proceedings ITISE 2018. Granada, 19-21 September, 2018.

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4 Power laws in stock market and fractal complexity of S&P500 and DAX

The attraction of power laws probably lies in the fact that processes with

power-law distribution are fractals. It means that they are self-similar or self

aﬃne and hence scale-free. For example, graphs of share prices over minutes,

days, or weeks have a very similar overall form, displaying statistical self-aﬃnity,

which means that a rescaled version of a small part of the graph has the same

statistical distribution as the larger part. Power-law distributions are accompa-

nying signs of fractals and vice versa. Evidence of inevitable ubiquity of fractals

would be impressive.

However, not just the search for a universal mechanism is unsuccessful. We

often fail even with a seemingly less ambitious goal of demonstrating that some

data follows a power law. The natural procedure is to take the histogram and plot

it on a log-log scale. If it looks linear then the estimate of the slope is considered

the exponent α. Unfortunately, the data may be well-ﬁtted by a power law, and

another function might be an even better ﬁt.

One of the latest studies on this topic comes from Broido and Clauset [15].

The authors compare the ﬁtted power-law distributions to alternative heavy-

tailed distributions, like the log-normal or the stretched-exponential. The ap-

proximations are compared using a likelihood ratio test. The ﬁndings of the

authors are clear right from the title of the article - Scale-free networks are rare.

The provocative title has prompted an immediate response from Albert-L´aszl´o

Barab´asi, one of the advocates of ubiquitous scale-free networks [16]. He argues

that the ﬁtting diﬃculties are not a reason to dismiss the idea of the scale-free

networks. Pure power law only emerges in idealized models. In real networks, we

have to admit additional eﬀects. Then, the power law tends to coexist with some

corrective function, leading to power laws with exponential cutoﬀs, stretched ex-

ponentials, and so on.

The idea behind the Barab´asi–Albert networks driven by only growth and the

so called preferential attachment is that nodes with higher degree receive more

new links than nodes with lower degree [17]. Intuitively this makes sense. Web

pages with many incoming links are easier to ﬁnd, so even more new Web pages

will link to them. In networks, this ”the rich get richer” principle of preferential

attachment leads to scale-free degree distributions. From this point of view the

stock market can be seen as a network of traded companies (nodes, vertices).

In the network, pairs of vertices are linked by edges, if the level of correlation

between the corresponding two price series is above a speciﬁed threshold.

Is the preferential attachment, resulting in power-law distributions, the right

model for the price returns? Or should we prefer one of many other proposed

models and mechanisms? It is not obvious how to decide.

Also, it is still not even clear whether it is appropriate to see the price series

and indexes as power-law processes and fractals. On the one hand, price returns

certainly have some kind of fat-tailed distributions, but verifying the speciﬁc

power-law distribution could prove to be be infeasible. On the other hand, we

have the remarkable indicators given by Mandelbrot: the apparent statistical

self-aﬃnity in data and also the volatility persistence that is strongly supportive

Proceedings ITISE 2018. Granada, 19-21 September, 2018.

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Power laws in stock market and fractal complexity of S&P500 and DAX 5

of a long memory property. These are typical features of fractal (power-law)

processes.

After this relatively extensive introductory review, the rest of the article is

organized as follows. In the next two sections we introduce the data sets and

the methods used for estimation of the fractal complexity. The fourth section

presents and discusses the results. Finally, the ﬁndings are summarized in Con-

clusion.

2 Methods

Although the idea of a power-law distribution in stock prices is still not justiﬁed,

use of fractal characteristics such as the Hurst exponent [18], has been quite

common for some time.

The Hurst exponent His usually referred to as a measure of persistence

but in fact it is also one of the fractal complexity measures. Hcan be derived

directly from the best-known fractal characteristic, which is the fractal dimension

D. Fractal dimension determines the irregularity of the signal. It tells us how

smooth or rough the trace of the graph is. Dcan take values between 1 and 2.

The more the graph ﬁlls the plane the closer it approaches the value of 2.

Let us also mention the possibility of spectral representation of the investi-

gated scale-free process:

S(f)∝1/fβ,

where S(f) is the power spectral density for the frequency components of the

signal. Processes with this power-law relationship are called 1/f noises.

When we have self-aﬃne time series, some of the properties as D, decay of au-

tocorrelation or power spectrum, persistence, etc. are elegantly connected. Above

all, the next relation between the fractal dimension D, decay of autocorrelation

γ, Hurst exponent, and spectral decay βholds:

D=2 + γ

2= 2 −H=5−β

2(1)

for 1 ≤β≤3 [19], [20].

The consequence of (1) is that if you estimate one of the characteristics, you

also get the others. For example, in this study, we estimated Dand used it to

get the estimate of the Hurst exponent and spectral decay.

For computation of D, we chose the Higuchi method, introduced in [21].

The method stood out from our earlier comparison of four diﬀerent estimation

techniques [22].

The Higuchi’s method scans ﬂuctuations of the signal by investigating the

deﬁned length of the curve for diﬀerent magniﬁcations of the time axis of the

signal. We take time-series X(1), X(2), ..., X (N),and make klagged time series

that start from m-th place (m= 1,2, ..., k) with gap of the size k:

Xm

k=X(m), X(m+k), ..., X m+N−m

kk

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6 Power laws in stock market and fractal complexity of S&P500 and DAX

Then Higuchi deﬁnes a length of the curve of Xm

kas follows:

Lm

k=

[N−m)

k]

X

i=1

|X(m+ik)−X(m+ (i−1)k|N−1

N−m

kk2

When the length of the curve is calculated (and normalized) for every mand k,

we get an L(k) as the mean of all lengths Lm

k. If L(k)∼k−D, then the curve is

a fractal with dimension D.

To estimate the Higuchi dimension, we used a Matlab code shared by J.

Monge- ´

Alvarez [23].

Real data, however, can only be investigated over a restricted range of scales.

We have to determine how high should the value kgo. The recommendation is

to compute the estimates for increasing kand use the value where the estimates

reach a plateau. For the indexes analyzed in this study, the optimal choice was

found to be k= 4.

After estimating the Higuchi fractal dimension D, we use Eq. (1) to get the

corresponding Hurst exponent: H= 2 −D.

When we estimate Hor other fractal characteristics from the ﬁnancial series,

we actually work with a hypothesis that the given series is a power-law process

and thus a fractal.

Finding H > 0.5 indicates that the increments of the process are positively

correlated, meaning that a high value in the series will probably be followed by

another high value (persistence). Variation increases at a faster rate than what

is expected for the Gaussian case, so that large jumps are possible, and the series

may travel a larger distance than a random walk would imply.

On the other hand, H < 0.5 means negative correlation and long-term ten-

dency to switch between high and low values (anti-persistence).

A value of H= 0.5 means that the investigated process has independent

increments (e.g. ordinary Brownian motion), or the absolute values of the auto-

correlations decay exponentially quickly to zero, unlike the above-mentioned case

of typically power-law decay. In stock returns data, H= 0.5 can be considered

to support the eﬃcient market hypothesis.

3 Data

In this study, the daily closing values of S&P 500 and DAX indexes are examined.

The S&P 500 is an American stock market index based on 500 large compa-

nies listed on the New York Stock Exchange or the Nasdaq Stock Market. Daily

closing values (17110) of S&P 500 from January 3, 1950 to December 29, 2017

were downloaded from https://ﬁnance.yahoo.com.

The DAX is a German stock market index consisting of the 30 major com-

panies trading on the Frankfurt Stock Exchange. Daily closing values (14578)

of DAX from January 4, 1960 to December 29, 2017 were downloaded from

https://stooq.com.

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Power laws in stock market and fractal complexity of S&P500 and DAX 7

For the computation of the fractal characteristics, the daily closing values of

the indexes were used. For the plot of a histogram and the corresponding ﬁt of a

normal density function, ordinary returns expressed as a percentage were used.

4 Results and Discussion

Fig. 1 shows the empirical distribution of the returns of S&P 500 for years 1950–

2017. The estimated σis 0.9608. We can see, that the distribution has fatter tails

than predicted by the Gaussian. Events that should theoretically be extremely

rare, such as price changes by more than 3 percent (above 3σ), are actually quite

common. The result suggests that the market does not follow a random walk.

-10 -5 0 5 10

100

101

102

103

104

histogram and a normal distribution fit

histogram

normal fit

Fig. 1. Histogram for the daily returns of the S&P 500 during the period 1950–2017

compared to the Gaussian normal distribution. In order to amplify the heavy tails of

the distributions, semilogarithmic plot is used.

Fig. 2 shows estimates of Hurst exponent for the S&P 500 during the period

1950–2017. Each value of His estimated from 250 previous daily closing values

of the index. Most of the time, the H > 0.5 indicates signiﬁcantly persistent

behaviour. However, Hshows a slightly downward trend over the years and

during the last two decades it oscillates quite close to the value of 0.5 typical for

the ordinary random walk. Similar decreasing pattern has been observed when

high frequency (1-minute) data were used for S&P 500 index for the period

1983–2009 [24]. The similarity of the results obtained from daily and minute

data supports the possible presence of scale-free processes.

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8 Power laws in stock market and fractal complexity of S&P500 and DAX

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Hurst exponent

0

500

1000

1500

2000

2500

S&P 500

Hurst exponent

S&P 500

Fig. 2. Hurst exponent for the S&P 500 during the period 1950–2017. Each value of

His estimated from the previous year of daily closing values.

Fig. 3 shows estimates of Hurst exponent for the German DAX during the

period 1960–2017. Again, each value of His estimated from 250 previous daily

closing values of the index. Also in this case the Hurst exponent shows a moderate

downward trend over the investigated years.

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 2020

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Hurst exponent

0

2000

4000

6000

8000

10000

12000

14000

DAX

Hurst exponent

DAX

Fig. 3. Hurst exponent for the DAX during the period 1960–2017. Each value of His

estimated from the previous year of daily closing values.

In case of S&P 500, the highest values of Hwere found for the years 1971–

1972. Left part of the Fig. 5 shows a closer look at these years. The index is

relatively smooth and persistent. The estimation of the Higuchi dimension is

D≈1.3, so based on Eq. (1), H≈0.7 and the signal resembles 1/f noise with

an exponent β≈2.4.

The lowest values of Hwere found for the turbulent years of 2008–2009.

Right part of the Fig. 6 shows S&P 500 and Hfor these two years. The index

here is mostly anti-persistent and it looks more rough and space-ﬁlling than

in the previous case of 1971–1972. The corresponding estimates of the fractal

complexity measures are D≈1.6, H≈0.4, β≈1.8.

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Power laws in stock market and fractal complexity of S&P500 and DAX 9

Can we learn more from the course of the Hurst exponent?

The most important question is, of course, whether Hcan help to predict the

coming crash. By stock market crash we understand panic selling and double-

digit percentage losses in a stock market index over a period of few days. Al-

though long-term market predictions are undoubtedly illusory, some authors

believe that limited short-term trend predictions are possible [25], [26].

To discuss this topic, let us look closer at the Hurst exponent’s course over

some speciﬁc 2-year periods.

1960-01-01 1961-01-01 1962-01-01

0

0.2

0.4

0.6

0.8

1

Hurst exponent

50

55

60

65

70

75

S&P 500

Hurst exponent

S&P 500

1962-01-01 1963-01-01 1964-01-01

0

0.2

0.4

0.6

0.8

1

Hurst exponent

50

55

60

65

70

75

80

S&P 500

Hurst exponent

S&P 500

Fig. 4. Hurst exponent (blue) and the S&P 500 (red) during the period 1960–1961 (on

the left) and during 1962–1963 (on the right). Each value of His estimated from 50

previous days.

1971-01-01 1972-01-01 1973-01-01

0

0.2

0.4

0.6

0.8

1

Hurst exponent

90

95

100

105

110

115

120

S&P 500

Hurst exponent

S&P 500

1987-01-01 1988-01-01 1989-01-01

0

0.2

0.4

0.6

0.8

1

Hurst exponent

220

240

260

280

300

320

340

S&P 500

Hurst exponent

S&P 500

Fig. 5. Hurst exponent (blue) and the S&P 500 (red) during the period 1971–1972 (on

the left) and during 1987–1988 (on the right). Each value of His estimated from 50

previous days.

The right part of Fig. 6 shows estimates of Hurst exponent for the S&P

500 around the ﬁnancial crisis in September 2008. In the ﬁrst half of 2008, the

stock market was at its peak with H > 0.5 suggesting a slight persistence in

the data. In such situation, the short-term predictability of the market can be

increased, traders may become conﬁdent about their strategies and the liquidity,

but also the vulnerability of the market grows. What seems to be interesting is

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10 Power laws in stock market and fractal complexity of S&P500 and DAX

1989-01-01 1990-01-01 1991-01-01

0

0.2

0.4

0.6

0.8

1

Hurst exponent

260

280

300

320

340

360

380

S&P 500

Hurst exponent

S&P 500

2008-01-01 2009-01-01 2010-01-01

0

0.2

0.4

0.6

0.8

1

Hurst exponent

700

800

900

1000

1100

1200

1300

1400

S&P 500

Hurst exponent

S&P 500

Fig. 6. Hurst exponent (blue) and the S&P 500 (red) during the period 1989–1990 (on

the left) and during 2008–2009 (on the right). Each value of His estimated from 50

previous days.

that well before the crash Hbegan to fall sharply. Unusually low values of the

exponent (H < 0.3), shortly before the crash, could be interpreted as the onset

of strong anti-persistence and therefore extreme nervousness in the market. Do

we see a useful prognostic pattern here? Unfortunately, analysis of other similar

situations does not conﬁrm this.

Let us see, for example, the strong market decline in 1987. On Black Monday

(October 19, 1987), the S&P 500 dropped more than 20%. The crash came

after years of strong economic optimism and unprecedented market increase.

Estimates of the Hurst exponent for period 1987–1988 can be seen on the right

graph of Fig. 5. Similarly, as in 2008, we can see the stock market at its peak

with H > 0.5 suggesting persistence in data. However, this time, no warning

sign of nervousness in the form of a falling Hurst exponent is present.

In other situations, on the contrary, the market looks nervous in the sense

that it is anti-persistent, but it still rises. See for example Fig. 4, with periods

of rising market, accompanied by Hwell below 0.5. Another signiﬁcant example

is the rapid market growth in 2017 with S&P 500 booming and anti-persistent

at the same time.

5 Conclusion

In this study, we estimated fractal complexity characteristics from daily closing

values of US and German stock market indexes. Estimates were mostly found in

the following ranges:

1.3−1.6 for the fractal dimension D,

0.4−0.7 for the Hurst exponent H,

1.8−2.4 for the spectral decay β.

In case of S&P 500, the results showed change from the strongly persistent

behaviour until 1970s to mostly anti-persistent behaviour of the last two decades.

Then we looked for patterns that are potentially useful for predicting forth-

coming danger of stock collapses. However, the prognostic value of the measures

of fractal complexity was not conﬁrmed.

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Power laws in stock market and fractal complexity of S&P500 and DAX 11

We also discussed the role of the power-law processes in stock market mod-

eling. If we accepted power-law processes as inevitable consequences of the evo-

lution of the market as a large complex system, then we would have to accept

that:

•The market is very susceptible to any disturbance. Minor shocks can lead to

ﬂuctuations of all large sizes, just as it is with earthquakes and avalanches.

Catastrophes can occur for no obvious reason, either due to the internal

dynamics of the system or triggered by sudden external events like terrorist

attacks. But collapses are fundamentally due to the unstable position and

they are unavoidable.

•It is not possible to predict the long-term course of the market.

•It is not possible to get rid of the ﬂuctuations through some sort of economic

regulations.

Nevertheless, the most robust and eﬃcient state of the economy could be the

one with occasional ﬂuctuations of extreme sizes and durations. It is probably

not the best imaginable state, but it might be the best achievable state.

The questions that we continue to ask are: Are power laws and fractals typ-

ical for the stock market dynamics? If so, is there a universal mechanism that

inevitably leads to these power laws?

An overview of current literature shows that we do not have deﬁnitive an-

swers. It seems that there are many diﬀerent ways that fractals and power laws

can arise, and that the long sought universal mechanism may not exist. And we

still can not rule out the possibility that ubiquitous power laws (fractals) are

just mere idealization. As we mentioned above, many published claims for the

existence of power laws are almost surely wrong. Maybe what we see are mostly

just heavy-tailed distributions. They still indicate a very diﬀerent phenomenon

from Gaussian or exponential, and force us to look at the rare events in a new

way. However, there is no doubt that they are not nearly as exciting as power

laws and fractals.

Acknowledgments. This work was supported by the Slovak Grant Agency for

Science (grants no. 2/0011/16) and by the Slovak Research and Development

Agency (grant no. APVV-15-0295). I would also like to thank Hana Krakovsk´a

for her help with the computational part of the paper.

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journal of Finance. 25, 2, 383–417 (1970)

2. Mandelbrot, B.: The Pareto-L´evy law and the distribution of income. International

Economic Review. 1, 2, 79–106 (1960)

3. Mantegna, R.N., Stanley, H.E.: Scaling behaviour in the dynamics of an economic

index. Nature. 376, 6535, 46 (1995)

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