University of Ljubljana

Faculty of Mathematics and Physics

Jadranska 19, Ljubljana

Seminar

SIMILARITIES A�D DIFFERE�CES

BETWEE� TURBULE�CE A�D FI�A�CIAL

MARKETS

April 2009

Author: Matjaž Ivančič

Mentor: doc. dr. Primož Ziherl

Abstract: The comparative analysis of the statistical properties on financial markets and the

velocity of the air in a fully turbulent state shows that an interaction between economics and

statistical physics may be useful - i.e., it may be fruitful to pursue analogies and differences

between two various stochastic models developed in economics and the approaches used in

turbulence theory. The exchange of concepts, models and techniques of data analysis offers an

opportunity to characterize qualitatively and quantitatively analogies and differences between

these two stochastic processes.

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

2

Introduction

One of the criticisms of physicists working with economic systems is that this kind of activity

cannot be a branch of physics because the equation of motion of the process is unknown. But

if this criterion were so important, several current research fields in physics would be

disqualified, such as the modelling of friction and many studies in the area of granular matter.

Moreover, a number of problems in physics that are described by a well-defined equation,

such as turbulence, are not analytically solvable even with sophisticated mathematical and

physical tools [1].

On a qualitative level, turbulence and financial markets are attractively similar. The complex

statistical behaviour of velocity increments on a certain length scale in turbulent flows is

assumed to be due to a cascading process. The energy, which is fed into the system on large

scales, is continuously transported towards smaller scales due to the inherent instability of

vortices of a given scale towards perturbations on smaller scales. Finally, the energy is

dissipated at the smallest scale. A similar mechanism has been proposed for financial markets,

where the energy cascade was replaced by a flow of information. Initially, the assumption of a

cascading process in financial markets was based on similarities in the empirical description

of the probability density functions (PDFs) of price and velocity increments. The analogy

between turbulence and finance (Table I) has inspired many further studies, but has also been

criticized for being too superficial [1, 2].

Hydrodynamic Turbulence Financial Markets

Energy Information

Spatial distance Time delay

Intermittency (laminar periods interrupted by

turbulent burst)

Volatility clustering

Energy cascade in space hierarchy Information cascade in time hierarchy

Table I) Postulated correspondence between fully developed three-dimensional turbulence and financial markets [2]

In this seminar, we discuss the fully developed turbulence hydrodynamics in fluid flow in

comparison with stochastic modelling of stock prices. Our aim is to show that cross-

fertilization between the turbulence and the economic index might be useful, but we shall find

that the formal correspondence between turbulence and financial systems is not supported by

qualitative calculations.

Financial markets

A financial market is a mechanism that allows people to easily buy and sell financial

securities, commodities and other replaceable items of value at low transaction costs and at

prices that reflect the efficient-market hypothesis. Financial markets have evolved

significantly over several hundred years and are undergoing constant innovation to improve

liquidity.

University of Ljubljana

Faculty of Mathematics and Physics

Both general markets and specialized markets

buyers and sellers in one place, thus making it easier for them to find each other. An economy

which relies primarily on interactions between buyers and sellers to allocate resources is

known as a market economy in contrast either to a command economy or to a non

economy such as a gift economy. The financial markets can be divided into different subtypes

[3]:

• Foreign exchange markets

• Commodity markets, w

• Capital markets, which consist of:

o Stock markets,

common stock, and enable the subsequent trading thereof.

o Bond markets, which provide financing through t

enable the subsequent trading thereof.

• Money markets, which provide short term debt financing and investment.

• Derivatives markets, which provide instruments for the management of financial risk.

o Futures markets

products at some future date; see also forward market.

• Insurance markets, which facilitate the redistribution of various risks.

If we look at different economic indexes [Fig. 1] we can find that different types of mar

have a similar complex price dynamic. Although, we can presume that a study of one index

will give us a general view of all financial markets.

The availability of high-frequency data for financial markets has made it possible to study

market dynamics on timescales of less than a day. For stock market and foreign exchange

rates have shown that there is a net flow of information

Fig. 1/a) Charts derived from Ref. [4] for a 2 year period sh

exchange rate (foreign exchange market).

Turbulence in Financial M

April 2008

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Both general markets and specialized markets exist. Markets work by placing many interested

buyers and sellers in one place, thus making it easier for them to find each other. An economy

which relies primarily on interactions between buyers and sellers to allocate resources is

my in contrast either to a command economy or to a non

economy such as a gift economy. The financial markets can be divided into different subtypes

Foreign exchange markets, which facilitate the trading of foreign exchange

, which facilitate the trading of commodities.

which consist of:

which provide financing through the issuance of shares or

common stock, and enable the subsequent trading thereof.

, which provide financing through the issuance of bonds, and

enable the subsequent trading thereof.

, which provide short term debt financing and investment.

, which provide instruments for the management of financial risk.

Futures markets, which provide standardised forward contracts for trading

products at some future date; see also forward market.

, which facilitate the redistribution of various risks.

If we look at different economic indexes [Fig. 1] we can find that different types of mar

have a similar complex price dynamic. Although, we can presume that a study of one index

will give us a general view of all financial markets.

frequency data for financial markets has made it possible to study

on timescales of less than a day. For stock market and foreign exchange

rates have shown that there is a net flow of information from long to short timescales [3

] for a 2 year period showing a moderate complexity of the chart

).

Turbulence in Financial Markets

Matjaž Ivančič

exist. Markets work by placing many interested

buyers and sellers in one place, thus making it easier for them to find each other. An economy

which relies primarily on interactions between buyers and sellers to allocate resources is

my in contrast either to a command economy or to a non-market

economy such as a gift economy. The financial markets can be divided into different subtypes

he trading of foreign exchange.

which provide financing through the issuance of shares or

he issuance of bonds, and

, which provide short term debt financing and investment.

, which provide instruments for the management of financial risk.

dardised forward contracts for trading

, which facilitate the redistribution of various risks.

If we look at different economic indexes [Fig. 1] we can find that different types of markets

have a similar complex price dynamic. Although, we can presume that a study of one index

frequency data for financial markets has made it possible to study

on timescales of less than a day. For stock market and foreign exchange

from long to short timescales [3, 4, 5].

chart dynamics of USD-CHF

University of Ljubljana

Faculty of Mathematics and Physics

Fig. 2/b) Charts derived from Ref. [4] for a 2 year period showing a moderate complexity of the price dynamics

(commodity market).

Fig. 3/c) Charts derived from Ref. [4] for a 2 year period showing a moderate complexity of the

Apple Inc. (capital market). Due to similar influences and volatility we can presume that the whole

described by a single equation of motion.

Turbulence

Turbulence is a well-defined physical problem which

physics. Among the approaches that have been

arguments based on dimensional analysis, statistical modelling

Consider a simple system that exhibits turbulence, a fluid of kinematic viscosity

with velocity � in a pipe of diameter

complexity of this flowing fluid is the Reynolds number,

When �� reaches a particular threshold value, the complexities of the fluid ex

suddenly becomes turbulent (Fig. 2

Turbulence in Financial M

April 2008

4

] for a 2 year period showing a moderate complexity of the price dynamics

] for a 2 year period showing a moderate complexity of the

Apple Inc. (capital market). Due to similar influences and volatility we can presume that the whole

equation of motion.

defined physical problem which remains one of the great challenges in

physics. Among the approaches that have been proposed are analytical approaches, scaling

uments based on dimensional analysis, statistical modelling and numerical simulations [

Consider a simple system that exhibits turbulence, a fluid of kinematic viscosity

in a pipe of diameter �. The control parameter whose value determines the

complexity of this flowing fluid is the Reynolds number,

�� � ��� �

reaches a particular threshold value, the complexities of the fluid ex

(Fig. 2) [7].

Turbulence in Financial Markets

Matjaž Ivančič

] for a 2 year period showing a moderate complexity of the price dynamics of gold

] for a 2 year period showing a moderate complexity of the shares price dynamics of

Apple Inc. (capital market). Due to similar influences and volatility we can presume that the whole financial market is

one of the great challenges in

are analytical approaches, scaling

and numerical simulations [5].

Consider a simple system that exhibits turbulence, a fluid of kinematic viscosity � flowing

. The control parameter whose value determines the

(1)

reaches a particular threshold value, the complexities of the fluid explode as it

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

5

Fig. 4) Solution of the 2D vorticity equation proposed by P. Bartosz from McMaster University, Hamilton, Canada. As we

can see above, increasing the fluid velocity causes a major complexity in the motion of the fluid past the obstacle. In a) where

the Re = 10 the flow is laminar. As the Re increases turbulence occurs. From b) (Re = 75) through c) (Re = 100) to d) (Re =

150) the flow becomes turbulent and the von Karman vortex street appears. For Re = 250 (e) and 500 (f) where the turbulent

cascade is clearly seen [8].

The equations describing the time evolution of an incompressible fluid have been known

since Navier’s work published in 1823 [4], which led to the Navier-Stokes equation,

�� ��, � + ���, � ∙ ∇���, � � −∇� + �∇���, �, (2)

Usually one assumes that the fluid is incompressible:

∇ ∙ ��, � � 0� (3)

Here ��, � is the velocity at position � and time and � is the pressure. The Navier-Stokes

equation completely characterizes the fully developed turbulence, a technical term indicating

turbulence at high Reynolds number. For very large values of Re no analytical or even

numerical solutions of Eqs. (1) and (2) are known.

a)

b)

c)

d)

e)

f)

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

6

In 1942, a breakthrough in the description of fully developed turbulence was achieved by

Kolmogorov [1]. He showed that in the limit of infinite Reynolds numbers, in the inertial

range (the range of length scales over which energy is transferred and dissipation due to

molecular viscosity is negligible), the mean square velocity increment

��Δ������� � ����� + �� − ������� (4)

behaves approximately as

���������~�� �� (5)

where � is the distance of two points where the velocity is measured. The distances are smaller

than the overall dimension where the fluid turbulent behaviour occurs and larger than the

typical length below the kinetic energy is dissipated into heat.

Although the Kolmogorov theory describes well the second-order of velocity increment

�������2� and provides the exact relation for the third-order �������3� moments observed in

experiments, its fails to describe the higher moments and the intermittent behaviour of

velocity increments. In the experimental studies of fully developed turbulence,

experimentalists usually measure the velocity as a function of time [1].

Parallel analysis of price dynamics and fluid velocity

Thanks to the amounts of data from stock markets it is possible to make a good evaluation of

the price dynamics and compare it to measurements of a turbulent fluid. Those data can

provide us with a better comprehension of the speculative price dynamics similar to the

energy cascade of a hydrodynamic turbulence.

Several statistical techniques (e.g. the measure of the probability density function, the

measure of the spectral density, etc.) commonly used in the study of the stochastic processes

have been used for a long time in turbulence [6]. Moreover, recently there have been many

attempts to identify stochastic processes whose statistical properties are close to those

observed in turbulence. In their paper, Mantegna and Stanley [6] report analogies and

differences between the quantitative measures of fluctuations in an economic index and the

fluctuations in velocity of a fluid in a fully turbulent state. They observe non-Gaussian

statistics and intermittency for both processes but the time evolution of the second moment

and the shape of the probability density functions stock market dynamics and fluid turbulence

are not the same.

The economic data set studied consists of all 1447514 records of the S&P 500 cash index

recorded during the period January 1984 December 1989 [7]. In this analysis, the "trading

time" extends from the opening until the closing of the day, and then continues with the

opening of the next trading day. The time intervals between successive records are not fixed:

the average value between successive records is close to 1 min during 1984 and 1985 and

close to 15 s during the period 1986-1989. From this database is selected the complete set of

non-overlapping records separated by a time interval ∆ ± $Δ (where $ is the tolerance and

always less than 0.035 s). We denote the value of the S&P 500 as %�� [Fig. 3(a)], and the

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

7

successive variations of the S&P index is &'(�� � %�� − %� − Δ� [Fig. 3(b)] [6]. The

complexity of the price motion clearly shows a strange behaviour with smaller and bigger

variations around the uptrend.

Fig. 5) a) Time evolution of the S&P 500, sampled with a time resolution ∆) � *+, over the period January 1984 –

December 1989, where we can notice the big leap slightly above the 6000 trading time. b) Hourly variations of the S&P 500

index in the 6-year period January 1984 – December 1989, which emphasise the big leap at t = 6000 h [6].

To make an evaluated comparison between the financial markets and the fluid turbulence

accurate data of fully developed turbulent flow is needed.

The turbulence data were provided by P. Kaylashnat and coworkers [9]. Measurements were

made in the atmospheric surface layer about 6m above a wheat canopy in the Connecticut

Agricultural Research Station. The Reynolds number ,- expressed in the Taylor microscale,

where the dissipation begins to affect the eddies, was around 1500. Velocity fluctuations were

measured using the standard hot-wire velocimeter operated in the constant temperature mode

on a DISA 55M01 anemometer. The file consists of 130000 velocity records ��� digitized

and linearized before processing [Fig. 4(a)]. The associated velocity differences .'(�� ���� − �� − Δ� are shown in Fig. 4(b).

Fig. 6) a) Time evolution of the wind velocity recorded in the atmosphere at very high Reynolds number; the Taylor

microscale Reynolds number around 1500. The time units are given in arbitrary units. (b) Velocity differences of the time

series given in (a) where we can notice the same kind of leaps that in Fig. 3 (b) [9].

%��

&��

trading time (hours) trading time (hours)

���

.��

time (hours) time (hours)

400

300

200

100

0 2000 4000 6000 8000 10000

20

10

0

-10

-20

0 2000 4000 6000 8000 10000

20

10

0

-10

-20

6

3

0

-3

-6

0 5000 100000 0 5000 100000

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

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We focus attention on the dynamics of the index variation &'(�� and on the dynamics of the

velocity difference .'(��, and denote by ��&� and ��.� the associated probability density

functions (PDFs).

By measuring the time dependence of the standard deviations 01�� and 02�� of ��&� and

��.�, we find that:

• In the case of the S&P 500 index variations [Fig. 5(a)] the time dependence of the

standard deviation, when Δ > ~ 15 min fits well the behaviour [5]

0&�Δ� ∝ Δ0�53. (6)

The exponent is close to the typical value of 0.5 observed in random processes with

independent increments.

• The velocity difference of the fully turbulent fluid shows a time dependence of the

standard deviation, fitting the behaviour [Fig. 5(b)] [6]

0.�Δ� ∝ Δ0�33 (7)

which is observed in short-time anti-correlated random processes, meaning the more

positive differences occur, the more probable is a negative difference in direction.

Fig. 7) a) Log-log plot of the standard deviation :;�<)� of the probability distribution =�;� characterizing the increments ;<)�)� as a function of <) for the S&P 500 time series. After a time intervaI of superdiffusive behaviour, when the standard

deviation is not yet linear �> < <) ≤ *A BCD� a diffusive behavior close to the one expected for a random process with

independent identically-distributed increments is observed; the measured diffusion exponent >� AE is close to the theoretical

value * F� characteristic for normal diffusion. (b) Standard deviation :G�<)� of the probability distribution =�G�

characterizing the velocity increments G<)�)� plotted double logarithmically as a function of <) for the velocity difference

time series in turbulence. After a time interval of superdiffusive behaviour �> < <) ≤ *>�, a diffusive behaviour close to the

one expected for a fluid in the inertial range is observed (the measured diffusion exponent 0.33 is close to the theoretical

value * E� predicted by Kolmogorov) [6].

Similar conclusions are reached if we measure the spectral density of the time series %�� and

���. Economic data [Fig. 6(a)] have the spectral density typical of a Brownian motion,

H�I� ∝ IJ�. For turbulence data [Fig. 6(b)] the spectral density shows a wide inertial range

(the range of length scales over which energy is transferred and dissipation due to molecular

viscosity is negligible) as H�I� ∝ IJK �� [6].

0 1�∆�

0 2�∆�

trading time (minutes) time (a.u.)

slope = 0.33 slope = 0.53

10�

10L

10M

10JL

10J�

10M 10L 10� 10� 10N

10�

10L

10M

10JL

10J�

10M 10L 10� 10� 10N

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

9

Fig. 8) a) Spectral density of the S&P 500 time series. The O−F power-law behaviour expected for a random process with

increments independent and identically distributed is observed over a frequency interval of more than 4 orders of magnitude.

b) Spectral density of the velocity time series. The O−A E� inertial range (low frequency) and the dissipative range (high

frequency) are clearly observed [6].

A different kind of analysis of the PDFs ��&� and ��.� turns out to be quite powerful for the

description of experimental results. We analyze the point of each PDF that is least affected by

the noise introduced by the finiteness of the data set – ��0� varies with Δ [6].

Figs. 7(a) and (b) show log-log plots of ��& � 0� and ��. � 0� as functions of the time

interval Δ between successive observations. The deviation from a Gaussian process is shown

by plotting on the same figure the value of �P�0� determined from the measured values of

0�Δ� by assuming that the process is Gaussian – using the equation

�Q�0� � 1R2S0�Δ� � (8)

The clear difference between ��0� and �P�0� seen in Fig. 7 shows that both PDFs have a non-

Gaussian distribution and that the detailed shapes and the scaling properties of the two PDFs

are different. Recent empirical studies of stock market indices show that financial markets can

be described by a Lévy-stable distribution commonly used in stochastic processes�11, 6�.

Sp�ctra

l D�nsit

y�dB�

Sp�ctra

l D�nsit

y�dB�

10 logLM �I� 10 logLM �I�

experimental data

slope = - 1.62

experimental data

slope = - 1.98

40

20

0

-20

-40

-60

-80

-100 -60 -50 -40 -30 -20 -10 0

40

20

0

-20

-40

-60

-80

-100 -50 -40 -30 -20 -10 0

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

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Fig. 9) a) S&P 500 index. Probability of return to the origin: =�; � >� (circles) and =c�; � >� (filled squares) [see Eq. (8)]

as functions of the time sampling interval <). The two measured quantities differ in the full interval implying that the profile

of the PDF must be non-Gaussian. A power-law behaviour is observed for the entire time interval spanning three orders of

magnitude. The slope of the best linear fit is −>� d*e ± >� >FA [6]. b) Velocity of the fully turbulent fluid. Probability of

return to the origin: =�>� (circles) and =c�>� (filled squares) (see Eq. (8)) as functions of the time sampling interval <).

Again, the two measured quantities differ across the full interval, implying that the profile of the PDF must be non-Gaussian.

However in this case, a single sealing power-law behaviour does not exist for the entire time interval spanning three orders of

magnitude. The slope of the best linear fit (which is of quite poor quality) is −>� Afe ± >� ** [6].

The PDF of any stochastic process can be defined by the Fourier transform of its

characteristic function:

��g� � 12S h I��i−jgk+∞

−∞ � (9)

The characteristic function for the Lévy-stable distribution is

lnI�� � jm − noop q1 + jr s oot tg upS

2 vw � (10)

The characteristic function tells us that Lévy-stable distributions have four parameters: p, r, n

and m. The location parameter is m, and if p is greater than 1, m is equal to the expectation of

mean of the distribution. The scale parameter is n, while the parameter r is an index of

skewness, which can take any value in the interval −1 ≤ r ≤ 1. When r � 0 the distribution

is symmetric. When r > 0 (and 1 < p < 2), the distribution is skewed right (i.e., has a long

tail to the right) and the degree of skewness increases in the interval 0 < r ≤ 1 as r

approaches 1. Similarly, when r < 0 (and 1 < p < 2), the distribution is skewed left, with

the degree of skewness increasing in the interval −1 ≤ r < 0 as r approaches −1 [11].

Of the four parameters of a Lévy-stable distribution, the characteristic exponent p is the most

important for the purpose of comparing the fit. The character exponent p determines the

height of the extreme tails of the distribution, and can take any value in the interval 0 < p ≤2. When p � 2, the relevant Lévy-stable distribution is the normal (Gaussian) distribution.

The total probability in the extreme tails increases as p moves away from 2 and toward 0.

A scaling compatible with a Lévy-stable process is observed for economic data [Fig. 7(a)] and

indeed a Lévy distribution reproduces quite well the central part of the distribution of the S&P

10�

10L

10M

10JL

∆�a� u� � ∆�minut�s�

��0�

��0�

P(0)

Pg(0)

slope = - 0.59

P(0)

Pg(0)

slope = - 0.71

10�

10L

10M

10JL

10M 10L 10� 10� 10M 10L 10� 10�

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

11

500 index variations [Fig. 8(a)] [4]. The Lévy-stable modelling with p � 1�40, r � 0,

n � 0�00375 and m � 0, obtained from the best fit of the probability of return to the origin

data, describes the data well over a three order of magnitude time interval (ranging from 1 to

1000 minutes). The tails deviate from the Lévy profile when & ≥ 0�3, ensuring a finite

variance to the stochastic process. The deviantion show us have a more complex distribution

broken at & � 0�3 [10].

No similar scaling exists for turbulence data over a wide time interval [Fig. 7(b)] [5]. By

using the measured values of ��0� and 02 and hypothesizing a stretched exponential PDF, it

is possible to describe quite well the experimental PDF of the velocity difference with a

stretched exponential distribution

��.� � |2�1 |� Γ�1 |� � �xp �− o.o|

� �� (11)

characterized by a (time-dependent) stretching exponent | and a scale factor �. Fig. 8(b)

shows the experimental probability density function measured for Δ � 1, together with a

stretched exponential distribution characterized by the parameter | � 0�61 [6].

Fig. 10) a) Experimental PDF =�;� of the S&P 500 index variations =∆)�;� observed at time intervals ∆) � * BCD (circles).

The symmetrical Lévy stable distribution of index � � *� e> and scale factor � � >� >>EdA is plotted as a solid line. The

parameters characterizing the stable distribution are obtained from the analysis of the scaling properties of the experimental

data on the probability of return to the origin =�; � >� [5]. b) Experimental PDF =�G� of the velocity difference G∆)�)� of a

fluid in fully developed turbulence observed at the highest temporal resolution available ∆) � * BCD (circles). In the figure is

also plotted as a solid line the symmetrical stretched exponential distribution of index � � >� �* and scale factor � � >� >�Ae (solid line). The characterizing parameters of the stretched exponential distribution are obtained starting from the

experimental value of the probability of return to the origin =�G � >� �6�.

Conclusion

The parallel analysis of velocity fluctuations in turbulence and index changes in financial

markets shows that the same statistical methods can be used to investigate systems with

known, but unsolvable equations of motion, and systems for which a basic mathematical

description of the process is still unknown. Despite that, in the two phenomena we find both

10 log

LM ��.�

10 log

LM ��&�

& .

experimental data

Lévy stable experimental data

streched exp.

2

0

-2

-4

-1.0 -0.5 0.0 0.5 1.0

2

1

0

-1

-2

-3

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

University of Ljubljana Turbulence in Financial Markets

Faculty of Mathematics and Physics Matjaž Ivančič

April 2008

12

• similarities: intermittency, non-Gaussian PDFs, the phenomena of information or

energy cascade, and

• differences: the PDFs have different shapes in the two systems and the probability of

return to the origin shows different behaviour. Moreover, velocity fluctuations are

anti-correlated whereas index fluctuations are essentially uncorrelated.

References

[1] R. N. Mantegna and H. E. Stanley, An introduction to econophysics: correlations and

complexity in finance (Cambridge University Press, Cambridge, 2000).

[2] J. Voit, The Statistical Mechanics of Financial Markets (Springer, Berlin, 2005).

[3] M. Levinson, Guide to Financial Markets (Bloomberg Press, New York, 2003).

[4] http://finance.yahoo.com/ (31. 3. 2009).

[5] U. Frish, Turbulence: The Legacy of A. ". Kolmogorov (Cambrige University Press,

Cambrige, 1995).

[6] T. Mizuno, S. Kurihara, M. Takayasu and H. Takayasu, Physica A 234, 296 (2003).

[7] R. N. Mantegna in H. E. Stanley, Physica A 239, 255 (1997).

[8] http://www.math.mcmaster.ca/ (31. 3. 2009).

[9] P. Kailasnath, K.R. Sreenivasan and G. Stolovitzky, Phys. Rev. Lett. 68, 2766 (1992).

[10] M. F. Shlesinger, G. M. Zaslavskya and U. Frisch, Lévy Flights and Related Topics in

Physics (Springer, Berlin, 1995).

[11] N. Oman, Study and Modeling of Price Variations in Financial Markets (Seminarji na

FMF, Ljubljana, 2008/2009).