Bio-box on Carl Freidrich Gauss and L Bachelier · 2 1 THE RANDOM WALK 1.1 What is a Random Walk?...

Econophysics. Sinha, Chatterjee, Chakraborti and ChakrabartiCopyright c© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, WeinheimISBN: 3-527-XXXXX-X

1

1THE RANDOM WALKA. Chakraborti

TOPICS TO BE COVERED IN THIS CHAPTER:

• What is a Random Walk?

– The random walk formalism

– Bio-box on Carl Freidrich Gauss and L Bachelier

– The Gaussian distribution

– Wiener process– Langevin equation and Brownian motion

• Do markets follow a random walk (From Bachelier to Eu-gene Fama & beyond)

– “Stylized” facts

– ARCH/GARCH processes– Efficient Market Hypothesis (EMH)

• Power spectral density (PSD)

– Spectral density : Energy and Power

– Relation of PSD to auto-correlation– Long-time correlations : Hurst exponent and DFA ex-ponent

2 1 THE RANDOM WALK

1.1What is a Random Walk?

1.1.1Definition of Random walk

The mathematical formalization of a trajectory that consists of taking succes-sive “random” (e.g. decided by the flips of an unbiased coin) steps, is knownas a random walk.A particularly simple random walk would be that on the integers, which

starts at time zero, S0 = 0 and at each step moves by 1 or −1 with equal prob-ability (e.g. decided by the flips of an unbiased coin). To define this walkformally, take independent random variables xi, each of which is 1 with prob-ability 1/2 and −1 with probability 1/2, and set Sn = Σn

i=1xi.This sequenceSn is called the simple random walk on integers.This walk can be illustrated (see 1.1) as follows: Say you flip an unbiased

coin. If it lands on heads H, you move one to the right on the number line,and if it lands on tails T, then you move one to the left. So after five flips,you have the possibility of landing on 1,−1, 3,−3, 5,−5. You can land on 1 byflipping three heads and two tails in any order. There are 10 possible ways oflanding on 1. Similarly, there are 10 ways of landing on -1 (by flipping threetails and two heads), 5 ways of landing on 3 (by flipping four heads and onetail), 5 ways of landing on -3 (by flipping four tails and one head), 1 way oflanding on 5 (by flipping five heads), and 1 way of landing on -5 (by flippingfive tails). These results are directly related to the properties of Pascal’s triangle.The number of different walks of n steps where each step is +1 or -1 is clearly2n. For the simple random walk, each of these walks are equally likely. Inorder for Sn to be equal to a number k, it is necessary and sufficient that thenumber of +1 in the walk exceeds those of -1 by k. Thus, the number of walkswhich satisfy Sn = k is precisely the number of ways of choosing (n + k)/2elements from an n element set (for this to be non-zero, it is necessary thatn + k be an even number), which is an entry in Pascal’s triangle denoted bynC(n+k)/2. Therefore, the probability that Sn = k is equal to 2−n nC(n+k)/2.This relation with Pascal’s triangle (see 1.2) is easily demonstrated for small

values of n. At zero turns, the only possibility will be to remain at zero. How-ever, at one turn, you can move either to the left or the right of zero, meaningthere is one chance of landing on -1 or one chance of landing on 1. At twoturns, you examine the turns from before. If you had been at 1, you couldmove to 2 or back to zero. If you had been at -1, you could move to -2 or backto zero. So there is one chance of landing on -2, two chances of landing onzero, and one chance of landing on 2. We shall study more interesting aspectsof the random walk later in this chapter.

1.1 What is a Random Walk? 3

Table 1.1 Random coin flips. If there is a head H we move right on the number line (add +1),and if there is a tail T we move left on the number line (add -1).

Table 1.2 Pascal’s triangle.

The results of random walk analysis is central in physics, chemistry, eco-nomics and a number of other fields as a fundamental model for random(stochastic) processes in time. There are many systems for which at smallerscales, the interactions with the environment and their influence are in theform of random fluctuations, as in the case of “Brownian motion” 1. If the mo-

1) The motion of the particle is called Brownian Motion, in honor tothe botanist Robert Brown who observed it for the first time in his

4 1 THE RANDOM WALK

−100 −80 −60 −40 −20 0 20−50

−40

−30

−20

−10

0

10

20

30

x

y

Table 1.3 Simulated Brownian motion (5000 time steps).

tion of a pollen grain in a fluid like water is observed under a microscope, itwould look somewhat like what is shown in the figure 1.3. It is interesting tonote that the path traced by the pollen grain as it travels in a liquid (observedby R. Brown and studied first by A. Einstein), and the price of a fluctuatingstock (studied first by L. Bachelier), can both be modeled as random walks(theory of stochastic processes). It is noteworthy that the formulation of therandom walk model — as well as of a stochastic process — was first done inthe framework of the economic study by L. Bachelier [31, 32], even five yearsprior to the work of A. Einstein!There are of course other systems, that present unpredictable “chaotic” be-

havior, this time due to dynamically generated internal “noise”. Noisy pro-cesses in general, either truly stochastic or chaotic in nature, represent the rulerather than the exception. In this chapter, we will concentrate only on theformer theory of random or stochastic processes.

******************************************************************************* BIO-BOX ON JOHANN CARL FRIEDRICH GAUSS (from wikipedia)

Johann Carl Friedrich Gauss (30 April 1777 âAS 23 February 1855) wasa German mathematician and scientist who contributed significantly tomany fields, including number theory, statistics, analysis, differential ge-ometry, geodesy, geophysics, electrostatics, astronomy and optics. Some-times known as the Princeps mathematicorum (the Prince of Mathemati-cians) greatest mathematician since antiquity, Gauss had a remarkable in-fluence in many fields of mathematics and science and is ranked as one ofhistory’s most influential mathematicians.

studies of pollen. In 1828 he wrote “the pollen become dispersed inwater in a great number of small particles which were perceived tohave an irregular swarming motion”. The theory of such motion,however, was derived by A. Einstein in 1905 when he wrote: “Inthis paper it will be shown that ... bodies of microscopically visiblesize suspended in a liquid perform movements of such magnitudethat they can be easily observed in a microscope on account of themolecular motions of heat ...”


Gauss was born on April 30, 1777 in Braunschweig, in the Electorate ofBrunswick-LÃijneburg, now part of Lower Saxony, Germany, as the sec-ond son of poor working-class parents. There are several stories of hisearly genius. According to one, his gifts became very apparent at the ageof three when he corrected, mentally and without fault in his calculations,an error his father had made on paper while calculating finances.Gauss attended the Collegium Carolinum (now Technische UniversitätBraunschweig from 1792 to 1795, and subsequently he moved to the Uni-versity of Göttingen from 1795 to 1798. His breakthrough occurred in 1796when he was able to show that any regular polygon with a number ofsides which is a Fermat prime (and, consequently, those polygons withany number of sides which is the product of distinct Fermat primes and apower of 2) can be constructed by compass and straightedge. This was amajor discovery in an important field of mathematics; construction prob-lems had occupied mathematicians since the days of the Ancient Greeks,and the discovery ultimately led Gauss to choose mathematics instead ofphilology as a career.In his 1799 doctorate in absentia, Gauss proved the fundamental theoremof algebra which states that every non-constant single-variable polyno-mial over the complex numbers has at least one root. Gauss also made im-portant contributions to number theory with his 1801 book DisquisitionesArithmeticae, which contained a clear presentation of modular arithmeticand the first proof of the law of quadratic reciprocity.In that same year, Italian astronomerGiuseppe Piazzi discovered the dwarfplanet Ceres, but could only watch it for a few days. Gauss predicted cor-rectly the position at which it could be found again, and it was rediscov-ered by Franz Xaver von Zach on 31 December 1801 in Gotha, and oneday later by Heinrich Olbers in Bremen. In 1807, Gauss was appointedProfessor of Astronomy and Director of the astronomical observatory inGÃuttingen, a post he held for the remainder of his life.The discovery of Ceres by Piazzi on 1 January 1801 led Gauss to his workon a theory of the motion of planetoids disturbed by large planets, eventu-ally published in 1809. It introduced the Gaussian gravitational constant,and contained an influential treatment of the method of least squares, aprocedure used in all sciences to this day to minimize the impact of mea-surement error. Gauss was able to prove the method in 1809 under theassumption of normally distributed errors.The Gaussian distribution is one ofmany things named after Carl FriedrichGauss, who used it to analyze astronomical data, and determined the for-mula for its probability density function. However, Gauss was not the firstto study this distribution or the formula for its density functionâATthathad been done earlier by Abraham de Moivre (in 1733). His result was ex-tended by Laplace in his book Analytical theory of probabilities (1812), andis now called the theorem of deMoivreâASLaplace. Laplace used the nor-mal distribution in the analysis of errors of experiments. The importantmethod of least squares was introduced by Legendre in 1805. AlthoughGauss justified the method rigorously only in 1809, he had been using itsince 1794.In 1831 Gauss developed a fruitful collaboration with the physics profes-sor Wilhelm Weber, leading to new knowledge in magnetism (includingfinding a representation for the unit of magnetism in terms of mass, lengthand time) and the discovery of Kirchhoff’s circuit laws in electricity. Hedeveloped amethod of measuring the horizontal intensity of the magnetic

6 1 THE RANDOM WALK

field which has been in use well into the second half of the 20th centuryand worked out the mathematical theory for separating the inner (coreand crust) and outer (magnetospheric) sources of Earth’s magnetic field.Gauss died in 1855 at Göttingen.*******************************************************************************BIO-BOX ON LOUIS BACHELIERLouis Jean-Baptiste Alphonse Bachelier (March 11, 1870 - April 28, 1946)was a French mathematician. In his PhD thesis “The Theory of Specula-tion” that was published in 1900, he discussed the use of Brownianmotionto evaluate stock options. It is historically the first paper to use advancedmathematics in the study of finance. Thus, Bachelier is considered a pio-neer in the study of financial mathematics and stochastic processes. It isnotable that Bachelier’s work on random walks predated Einstein’s cel-ebrated study of Brownian motion by five years. His instructor, HenriPoincare, is recorded to have given some positive feedback. The thesis re-ceived a note of honorable, and was accepted for publication in the presti-gious Annales Scientifiques de l’Ecole Normale Superieure. After his suc-cessful thesis defence, Bachelier, further developed the theory of diffusionprocesses, which was published in prestigious journals. In 1909 he becamea “free professor” at the Sorbonne. In 1914, he published a book, “Le Jeu,la Chance, et le Hasard” (Games, Chance, and Risk). With the supportof the Council of the University of Paris, Bachelier was given a perma-nent professorship at the Sorbonne. However, World War I intervenedand Bachelier was drafted into the French army. After the completion ofthe war, he found a position in Besancon, replacing a regular professor onleave. When the professor returned in 1922, Bachelier replaced anotherprofessor at Dijon. He moved to Rennes in 1925, but was finally awardeda permanent professorship in 1927 at Besancon, where he worked for an-other 10 years.*******************************************************************************

1.1.2The random walk formalism and derivation of the Gaussian dis tribution

The original statement of the random walk problem was posed by Pearson in1905. If a drunkard begins at a lamp post and takes N steps of equal lengthin random directions, how far will the drunkard be from the lamp post? Wewill consider an idealized example of a random walk for which the steps ofthe walker are restricted to a line (a one-dimensional randomwalk). Each stepis of equal length a, and at each interval of time, the walker either takes a stepto the right with probability p or a step to the left with probability q = 1− p.The direction of each step is independent of the preceding one. Let n be thenumber of steps to the right, and m the number of steps to the left. The totalnumber of steps N = n + m. What is the probability that a random walker inone dimension has taken three steps to the right out of four steps?Instead of the above example, had we considered the probability distribu-

tions of non-interacting magnetic moments or the flips of a coin we wouldarrive at identical results (and hence we will use the terms interchangebly).All these examples have two characteristics in common. First, in each trial


there are only two outcomes, for example, up or down, heads or tails, andright or left. Second, the result of each trial is independent of all previoustrials, for example, the drunken sailor has no memory of his or her previoussteps. This type of process is called a Bernoulli process (after the mathematicianJacob Bernoulli, 1654-1705). We will cast our discussion of Bernoulli processesin terms of the random walk. The main quantity of interest is the probabilityPN(n) which we now calculate for arbitrary N and n. We know that a partic-ular outcome with n right steps and m left steps occurs with probability pnqm.We write the probability PN(n) as

PN(n) = WN(n,m)pnqm, (1.1)

where m = N − n andWN(n,m) is the number of distinct configurations ofN steps with n right steps and m left steps.From our earlier discussion of random coin flips, we will be able to deduce

easily the first several values ofWN(n,m). We can determine the general formof WN(n,m) by obtaining a recursion relation between WN and WN − 1. Atotal of n right steps and m left steps out of N total steps can be found byadding one step to N − 1 steps. The additional step is either (a) right if thereare (n− 1) right steps and m left steps, or (b) left if there are n right steps andm left steps. Because there are WN(n− 1,m) ways of reaching the first caseandWN(n,m− 1) ways in the second case, we obtain the recursion relation

WN(n,m) = WN − 1(n− 1,m) +WN − 1(n,m− 1). (1.2)

If we begin with the known values W0(0, 0) = 1,W1(1, 0) = W1(0, 1) = 1, wecan use the recursion relation to construct WN(n,m) for any desired N. Forexample,

W2(2, 0) = W1(1, 0) +W1(2,−1) = 1+ 0 = 1.

W2(1, 1) = W1(0, 1) +W1(1, 0) = 1+ 1 = 2.

W2(0, 2) = W1(−1, 2) +W1(0, 1) = 0+ 1.

Thus we identify that WN(n,m) forms the Pascal’s triangle. It is straightfor-ward to show by induction that the expression

WN(n,m) =N!n!m!

=N!

n!(N − n)!(1.3)

satisfies the relation Eq 1.2, since by definition 0! = 1. We can combine Eqs 1.1and 1.3 to find the desired result

PN(n) =N!

n!(N − n)!pnqN−n. (1.4)

8 1 THE RANDOM WALK

The form Eq 1.4 is called the “binomial distribution”. Note that for p = q =1/2, such as in the case of an unbiased coin, PN(n) reduces to

PN(n) =N!

n!(N − n)!2−N. (1.5)

The reason that Eq 1.4 is called the binomial distribution is that its form rep-resents a typical term in the expansion of (p + q)N . By the binomial theoremwe have

(p+ q)N =N

∑n=0

N!n!(N − n)!

pnqN−n. (1.6)

We use Eq 1.4 and write

N

∑n=0

PN(n) =N

∑n=0

N!n!(N − n)!

pnqN−n = (p+ q)N = 1N = 1, (1.7)

where we have used Eq 1.6 and the fact that p+ q = 1.Frequently we need to evaluate lnN! for N ≫ 1. A simple approximation

for lnN! known as Stirling’s approximation is

lnN! ≈ N lnN − N. (1.8)

A more accurate approximation is given by

lnN! ≈ NlnN− N +12ln(2πN).

We note some properties of the Binomial distribution. Suppose first that wehave exactly one Bernoulli trial. We have two possible outcomes, 1 and 0, withthe first having probability p and the second having probability q = 1 − p.Then mean µ = p.1+ q.0 = p and variance σ2 = (1− p)2p+ (0− p)2q = pq.Now, for N such independent trials, we have

(i) mean µ = Np

(ii) variance σ2 = Npq

We also note that for large N, the binomial distribution has a well-definedmaximum at n = pN and can be approximated by a smooth, continuous func-tion even though only integer values of n are physically possible. We nowfind the form of this function of n. The first step is to realize that for N ≫ 1,PN(n) is a rapidly varying function of n near n = pN, and for this reason wedo not want to approximate PN(n) directly. Because the logarithm of PN(n)is a slowly varying function, we expect that the power series expansion ofln PN(n) to converge. Hence, we expand ln PN(n) in a Taylor series about the


value of n = n at which ln PN(n) reaches its maximum value. We will writep(n) instead of PN(n) because we will treat n as a continuous variable andhence p(n) is a probability density. We find

ln p(n) = ln p(n = n)+ (n− n)d ln p(n)

dn|n=n +

12(n− n)2

d2 ln p(n)

dn2|n=n + . . .

(1.9)Because we have assumed that the expansion Eq 1.9 is about the maximumn = n, the first derivative d ln p(n)/dn |n=n must be zero. For the same reasonthe second derivative d2 ln p(n)/d2n |n=n must be negative. We assume thatthe higher terms in Eq 1.9 can be neglected, and define

ln A = ln p(n = n),

and

B = −d2 ln p(n)

dn2|n=n .

The approximation Eq 1.9 and the definitions above, allow us to write

ln p(n) ≈ ln A− 12B(n− n)2, (1.10)

or

p(n) ≈ A exp(

−12B(n− n)2

)

. (1.11)

We next use Stirling’s approximation to evaluate the first two derivatives ofln p(n) and the value of ln p(n) at its maximum to find the parameters A, B,and n. We write

ln p(n) = lnN!− ln n!− ln(N − n)! + n ln p+ (N− n) ln q. (1.12)

It is straightforward to obtain

d(ln p(n))

dn= − ln n + ln(N− n) + ln p− ln q. (1.13)

The most probable value of n is found by finding the value of n that satisfies

the condition d ln pdn = 0. We find

N − n

n=

q

p, (1.14)

or (N − n)p = nq. If we use the relation p+ q = 1, we obtain

n = pN. (1.15)

10 1 THE RANDOM WALK

Note that n = n, that is, the value of n for which p(n) is a maximum is also themean value of n. The second derivative can be found from Eq 1.13. We have

d2(ln p(n))

dn2= − 1

n− 1

N − n. (1.16)

Hence, the coefficient B defined earlier is given by

B = −d2 ln p(n)

dn2|n=n=

1n

+1

N − n=

1Npq

. (1.17)

From the properties of the Binomial distribution we see that

B =1

σ2

where σ2 is the variance.If we use the simple form of Stirling’s approximation to find the normal-

ization constant A from the relation ln A = lnp(n = n), we would find thatln A = 0. Instead, we have to use the more accurate form of Stirling’s approx-imation. The result is

A =1

(2Npq)1/2=

1√2πσ2

.

If we substitute our results for n, B, and A into Eq 1.11, we find the distribu-tion

p(n) =1√2πσ2

exp(

− (n− n)2 /2σ2)

, (1.18)

which is called the “Gaussian distrbution”.From our derivation we see that Eq 1.18 is valid for large values of N and

for values of n near n. Even for relatively small values of N, the Gaussianapproximation is a good approximation for most values of n.The most important feature of the Gaussian distribution is that its relative

width, σn/n, decreases as N−1/2. Of course, the binomial distribution alsoshares this feature. We deal it in the next subsection.

1.1.3The Gaussian or Normal distribution

The Gaussian distribution, also called the Normal distribution, is perhaps themost important family of continuous probability distributions, applicable inmany fields including physics and economics. Carl Friedrich Gauss becameassociated with this set of distributions when he analyzed astronomical datausing them, and defined the equation of its probability density function. It


is often called the “bell curve” because the graph of its probability densityresembles a bell2.The importance of the normal distribution as a model of quantitative phe-

nomena in the natural and behavioral sciences is due in part to the “centrallimit theorem”. Under certain conditions (such as being independent andidentically-distributed with finite variance), the sum of a large number of ran-dom variables is approximately normally distributed– this is the central limittheorem. The practical importance of the central limit theorem is that the nor-mal cumulative distribution function can be used as an approximation to someother well-known cumulative distribution functions, for example: A binomialdistribution with parameters N and p is approximately normal for large N,and p not too close to 1 or 0. The approximating normal distribution has pa-rameters µ = Np, σ2 = Np(1− p) = Npq. It is noteworthy that a binomialdistribution with parameter λ = Np for large n and p → 0 such that λ = Np

is constant, gives another well-known distribution, known as the “Poissondistribution”, with parameters µ = σ2 = λ.There are various ways to characterize a probability distribution. The most

customary is perhaps the probability density function (PDF); the other equiv-alent ways of expressing them are with the cumulative distribution function,the moments, the cumulants, the characteristic function, etc. The continuousprobability density function of the normal distribution is:

P(x) =1√2πσ

exp(

−(x− µ)2/2σ2)

,

where σ > 0 is the standard deviation, the real parameter µ is the mean orexpected value. Each member of the Gaussian PDF family (see Fig. 1.4) maybe defined by two important parameters, location and scale: the mean µ andvariance (standard deviation squared) σ2, respectively. The standard normaldistribution is the normal distribution with a mean µ = 0 and a varianceσ2 = 1:

P(x) =1√2π

exp(

−x2/2)

.

2) HISTORICAL DIGRESSION: The Normal distribution was first in-troduced by Abraham de Moivre in an article in 1733, which waslater reprinted in the second edition of his The Doctrine of Chances,(1738) in the context of studying binomial distributions. The resultwas extended by Laplace in his book Analytical Theory of Probabil-ities (1812) where he used the Normal distribution in the analysisof errors of experiments. Carl Friedrich Gauss in 1809, assumedin his analyses a Normal distribution of the errors. The name "bellcurve" goes back to E. Jouffret who first used the term "bell surface"in 1872 for a “bivariate normal” with independent components. Itis also known that the name "Normal distribution" was coined in-dependently by Charles S. Peirce, Francis Galton and Wilhelm Lexisaround 1875.


−10 −8 −6 −4 −2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ=0 σ2=1

µ=0 σ2=4

µ=0 σ2=6

µ=5 σ2=1/3

µ=5 σ2=2/3

Table 1.4 Gaussian PDFs.

As a Gaussian function with the denominator of the exponent equal to 2, thestandard normal density function is an eigenfunction of the Fourier transform.The probability density function has the following notable properties, amongothers:

• symmetry about its mean µ

• the mode and median both equal the mean µ

• the inflection points of the curve occur one standard deviation awayfrom the mean, i.e. at µ − σ and µ + σ.

1.1.4Wiener process

In mathematics, the Wiener process is a continuous-time stochastic processnamed in honor of Norbert Wiener. Norbert Wiener (1923) had ultimatelyproved the existence of Brownian motion and made significant contributionsto related mathematical theories, so Brownian motion is often called a Wienerprocess, although this is strictly speaking a confusion of a model with thephenomenon being modeled.A Wiener process is the scaling limit (a term applied to the behaviour of a

lattice model in the limit of the lattice spacing going to zero) of random walkin one-dimension, which means that if you take a random walk with verysmall steps you get an approximation to aWiener process. To be more precise,if the step size is ǫ, one needs to take a walk of length L/ǫ2 to approximate aWiener process walk of length L. As the step size tends to 0 (and the number


of steps increased comparatively) randomwalk converges to aWiener processin an appropriate sense.A Wiener process in multi-dimensions is the scaling limit of random walk

in the same number of dimensions. A random walk is a discrete “fractal”, buta Wiener process trajectory is a true fractal, and there is a connection betweenthe two: take a random walk until it hits a circle of radius r times the steplength. The average number of steps it performs is r2. This fact is the dis-crete version of the fact that a Wiener process walk is a fractal of “Hausdorffdimension” 2. In two dimensions, the average number of points the same ran-dom walk has on the boundary of its trajectory is r4/3. This corresponds tothe fact that the boundary of the trajectory of a Wiener process is a fractal ofdimension 4/3, a fact predicted by Mandelbrot using simulations.It is one of the best known Levy processes (stochastic processeswith station-

ary independent increments) and occurs frequently in mathematics (the studyof continuous time martingales, stochastic calculus, diffusion processes), eco-nomics (mathematical theory of finance, in particular the Black-Scholes optionpricing model) and physics (study Brownian motion, the diffusion of minuteparticles suspended in fluid, and other types of diffusion via the Fokker-Planckand Langevin equations, see next section).

1.1.5

Langevin Equation and Brownian motion

In this subsection, we shall study the basics of the Langevin equation in thelanguage of colloidal suspensions (Brownian motion). Consider a sufficientlysmall colloidal particle of mass m suspended in a liquid at absolute temper-ature T. On its path through the liquid it will continuously collide with theliquid molecules and follow a random path exhibiting Brownian motion. Inphysics, this can serve as a prototype problem whose solutions provide con-siderable insight into the mechansisms responsible for the existence of statis-tical fluctuations in a system in thermal equilibrium and “dissipation of en-ergy”. Moreover, such fluctuations constitute a background noise, which im-poses limitations on the possible accuracy of delicate physical measurements.Again for simplicity, we consider the motion restricted to one dimensions.

We consider a sufficiently small particle of massmwhose mass is described bythe position coordinate x(t) at any time t, and whose corresponding velocityis v(t) = dx/dt.It would be very complex to describe in details, the interaction of the small

particle with motion of the molecules in the surrounding liquid (other degreesof freedom). However, all such degrees of freedom can be regarded as consti-tuting a heat reservoir at the temperature T, and their interaction describedby some net force F(t). In addition, the particle may also interact with some


external system, such as gravity or electromagnetic fields, through a force de-noted by Fext(t). The velocity v of the particle may be appreciably differentfrom its mean value in equilibrium. The Newton’s equation of motion thenreads

mdv

dt= F(t) + Fext(t). (1.19)

Very little is known about the nature of the force F(t), except that it is somerapidly fluctuating function of the time t and varies in a higly irregular orrandom fashion. To make progress, one has to formulate the problem in sta-tistical terms and therefore, must consider an ensemble of verymany similarlyprepared systems, each of them consisting of a particle and the surroundingliquid. For each of these the force F(t) is some random function of time t. Wealso assume that the correlation time characterizing the force F(t) is small on amacroscopic time scale, and there is no preferred direction in space (if the par-ticle is imagined to be clamped to be stationary). Then the ensemble averageF(t) vanishes.Since F(t) is a rapidly fluctuating function of time, vmust be also fluctuating

in time. But superimposed upon these fluctuations, the time dependence of vmay also exhibit a more slowly varying trend, and thus we assume:

v = v + v (1.20)

where v denotes the part of rapidly fluctuating part of v, and whose meanvalue vanishes. The slowly varying part v is very significant (even though itis small) because it detemnines the behaviour of the particle over long periodsof time.We must consider the fact that the interaction force F(t) must be actually

affected by the motion of the particle in such a way that F itself also containsa slowly varying part F tending to restore the particle to equilbrium.Hence similar to above equation for v, we must also write that

F = F + F (1.21)

where F denotes the part of rapidly fluctuating part of F, and whose meanvalue vanishes. The slowly varying part F must be some function of v whichis such that F(v) = 0 in equlibrium when v = 0.If v is not too large, F(v) can be expanded ina power series in v, whose first

non-vanishing term must be linear in v. Thus F must have the general form

F = −γv (1.22)

where γ is some positive “frictional” constant and where the negative signindicates explicitly that the force F acts in such a direction that it tends to


reduce the v to zero as time increases. Thus we have the slowly varying part

mdv

dt= F + Fext = −γv+ Fext, (1.23)

and if one includes the rapidly fluctuating parts v and F, then we have

mdv

dt= −γv + Fext + F(t), (1.24)

assuming γv ≈ γv (since the rapidly fluctuating part γv can be neglectedcompared to the predominantly fluctuating part F(t)). This is the Langevinequation, and describes in this way the behaviour of the colloidal particle atall later times if the initial conditions are specified. We note that since theLangevin equation contains the frictional force −γv, it implies the existenceof processes whereby the energy associated with the particle is dissipated indue course of time to the other degrees of freedom (molecules of the liquidsurrounding the collodial particle).Now, while describing Brownian motion in absence of external forces Fext,

we have

mdv

dt= −γv + F(t). (1.25)

The Stokes’s law of hydrodynamics gives:

γ = 6πηa, (1.26)

where η is the viscosity of the liquid and a is the radius of the colloidal particle(assumed to be spherical).Let the system be in thermal equilibrium. Clearly the mean displacement x

of the particle vanishes by symmetry, since there is no preferred direction inspace. Our aim is to calculate the mean-square displacement 〈x2〉 = x2 of theparticle in a time interval t.We have v = x and dv/dt = dx/dt, so that multiplying the Langevin equa-

tion by x throughout, we get

mxdx

dt= m

{

d

dt(xx) − x2

}

= −γxx + xF(t). (1.27)

Now we take the ensemble average of the above equation, and use the factthat irrespective of the value of x or v, we have 〈xF〉 = 〈x〉〈F〉 = 0. Using the“equipartition theorem” of classical statistical mechanics, we have 1

2m〈x2〉 =12kBT, such that

m〈 ddt

(xx)〉 = md

dt〈xx〉 = kBT − γ〈xx〉. (1.28)


This is a simple differential equation which can be solved to get the valueof the quantity 〈xx〉. Thus, one gets

〈xx〉 = C exp (−αt) +kBT

γ, (1.29)

where C is a constant of integration and α = γ/m is the characteristic timeconstant of the system. Assuming that each particle in the ensemble starts outat time t = 0 and the position x = 0, so that xmeasures the displacement fromthe initial position, the constant C satisfies 0 = C + kBT

γ . Hence, we have

〈xx〉 =12d

dt〈x2〉 =

kBT

γ(1− exp(−αt)) , (1.30)

or

〈x2〉 =2kBT

γ

(

t− α−1(1− exp(−αt))

. (1.31)

For us the case t ≫ α−1 when exp(−αt) → 0, is relevant and gives rise tothe interesting equation

〈x2〉 =2kBT

γt. (1.32)

The particle then behaves like a diffusing particle executing a random walkso that 〈x2〉 ∝ t. Indeed, the diffusion equation in physicsfor random walksgives a relation 〈x2〉 = 2Dt, where D is the diffusion constant, and comparingthese two we get

D =kBT

γ, (1.33)

which is known as the “Einstein relation”. Using the Stokes’s law, we alsohave

〈x2〉 =kBT

3πηat. (1.34)

1.2Do markets follow a random walk?

Prices of assets in a financial market, produce what is called a “financial time-series”. Different kinds of financial time-series have been recorded and stud-ied for decades, all over the world. Nowadays, all transactions on a financialmarket are recorded leading to huge amount of data available either commer-cially or for free in the Internet. And financial time-series analysis has been

1.2 Do markets follow a random walk? 17

of great interest to not only the practitioners (an empirical discipline) but alsothe theoreticians for making inferences and predictions. The inherent uncer-tainty in the financial time-series and its theory makes it specially interestingto economists, statisticians and physicists [40]. It is a formidable task to makean exhaustive review on this topic but we try to give a flavour of some of theaspects here.

1.2.1What if the time-series were similar to a random walk?

The answer is simple: It would not be possible to predict future price move-ments using the past pricemovements or trends. Louis Bachelier, who was thefirst one to investigate such studies in 1900 [31], had come to the conclusionthat “The mathematical expectation of the speculator is zero” and he had de-scribed this condition as a “fair game.” Let us discuss this issue in a bit moredetails.In economics, if P(t) is the price of a stock or commodity at time t, then the

“log-return” is defined as: rτ(t) = ln P(t+ τ)− ln P(t), where τ is the intervalof time. The definition of daily log-return is illustrated in Fig. 1.5, using theprice time-series for the General Electric.We generate a random time-series using random numbers from a Normal

distribution with zero mean and unit standard deviation, in order to comparewith the real empirical returns, and plot them in Fig. 1.6.If we divide the time-interval τ into N sub-intervals (of width ∆t), the total

log-return rτ(t) is by definition the sum of the log-returns in each sub-interval.If the price changes in each sub-interval are independent (for the data shownin Fig. 1.6a) and identically distributed with a finite variance, according to thecentral limit theorem the cumulative distribution function F(rτ) would con-verge to a Gaussian (Normal) distribution for large τ. The Gaussian (Normal)distribution has the properties (a) when the average is taken, the most prob-able change is zero, (b) the probability of large fluctuations is very low, sincethe curve falls rapidly at extreme values and (c) it is a stable distribution. Thedistribution of returns were first modelled for “bonds” by Bachelier [31], as aNormal distribution,

P(r) =1√2πσ

exp(−r2/2σ2),

where σ2 is the variance (second moment) of the distribution.The classical financial theories had always assumed this Normality, until

Mandelbrot [33] and Fama [98] pointed out that the empirical return distri-butions are fundamentally different– they are “fat-tailed” and more peakedcompared to the Normal distribution. Based on daily prices in different mar-kets, Mandelbrot and Fama found that F(rτ) was a stable Levy distribution


Table 1.5 Price (USD), log-price and log-return plotted with time for General Electric during theperiod 1982-2000.

Table 1.6 Time-series. (a) Random time-series (3000 time steps), (b) Return time-series of theS&P500 stock index (8938 time steps).

whose tail decays with an exponent α ≃ 1.7, a result that suggested that short-


term price changes were not well-behaved since most statistical properties arenot defined when the variance does not exist.For the probability density function of a stochastic process P(x), the char-

acteristic function G(y) is given by the Fourier transform of the probabilitydensity function:

G(y) =∫ +∞

−∞P(x) exp(iyx) dx,

and by performing the inverse Fourier transform we obtain the probabilitydensity function:

P(x) =12π

∫ +∞

−∞G(y) exp(−iyx) dy.

Levy and Khintchine [10, ?, ?] determined the entire class of stable distribu-tions described by the most general form of a characteristic function:

lnG(y) = iµy− γ |y|α[

1− iβy

|y| tan(π

2α)

]

[α 6= 1] ,

and

lnG(y) = iµy− γ |y|[

1+ iβy

|y|2πln |y|

]

[α = 1] ,

where 0 < α ≤ 2, γ is a positive scaling factor, µ is any real number and β isan asymmetry parameter between −1 and 1. The analytical form of the Levystable distribution is known only for a few values of α and β. For symmet-ric stable distributions, β = 0 and if the distributions have zero mean (firstmoment), µ = 0. The characteristic function for the Gaussian distribution, aspecial case of Levy stable distribution with α = 2, β = 0 and µ = 0 is thus

G(y) = exp(−γ |y|2),

where γ ≡ σ2/2 is the positive scale factor. The symmetric stable Levy dis-tribution with zero mean, of index α and scale factor γ is the inverse Fouriertransform:

PLevy(x) =1π

∫ ∞

0exp(−γ |y|α) cos(yx)dy.

If we assume that γ = 1, and look at the asymptotic approximation validfor large values of |x|:

PLevy(|x|) ∼Γ(1+ α) sin(πα/2)

π |x|1+α∼ |x|−(1+α) ,

we find that it has a power-law behaviour. We also find that⟨

|x|q⟩

diverge forq ≥ α when α < 2. It follows, in particular, that all Levy stable processes withα < 2 have infinite variance, as mentioned earlier.


It is now known, using more extensive data, that the decay of the distribu-tion is fast enough to provide finite second moment. With time, several otherinteresting features of the financial data were unearthed. A point worth men-tioning is that the physicists have been analysing financial data with the mo-tive of finding common or “universal” regularities in the complex time-series,which is very different from those of the economists who are traditionally ex-perts in statistical analysis of financial data. The results of the empirical stud-ies on asset price series by the physicists show that the apparently randomvariations of asset prices share some statistical properties which are interest-ing, non-trivial and common for various assets, markets and time periods.These are called “stylized empirical facts”. This brings to our next question.

1.2.2What are the “Stylized” facts?

Stylized facts have been usually formulated using general qualitative prop-erties of asset returns and hence distinctive characteristics of the individualassets are not taken into account. Below we quote just a few from the paperby Cont [84], which reviews the several empirical studies of the returns andother relevant issues.

(i) Fat tails: large returns asymptotically follow a power law F(rτ) ∼ |r|−α,with α > 2 (with α = 3.01± 0.03 for the positive tail and α = 2.84± 0.12for the negative tail [85]). With α > 2, the second moment (the vari-ance) is well-defined, excluding stable laws with infinite variance. Therehas been various suggestions for the form of the distribution: Student’s-t, hyperbolic, normal inverse Gaussian, exponentially truncated stable,etc. but no general consensus exists on the exact form of the distributiondescribing the tails (see Fig. 1.7).

(ii) Aggregational Normality: as one increases the time scale over whichthe returns are calculated, their distribution closes to Normality. Theshape is different at different time scales. The fact that the shape of thedistribution changes with τ makes it clear that the random process un-derlying prices must have non-trivial temporal structure 3.

(iii) Absence of linear auto-correlations: the auto-correlation of log-returns,ρ(T) ∼ 〈rτ(t+ T)rτ(t)〉 is illustrated in Fig. 1.8. It normally rapidlydecays to zero within a few minutes: for τ ≥ 15 minutes, it is practi-cally zero [86]. This supports in a way the “efficient market hypothesis”.When τ is increased, weekly and monthly returns exhibit some auto-correlation but the statistical evidence varies from sample to sample.

3) Any non-gaussian iid with finite variance has this property!!! Whatis however special is slow convergence.


−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1

10−1

100

101

102

Log−returns

Pro

babi

lity

dens

ity fu

nctio

n

Normal distributionS&P log−returns

Table 1.7 S&P 500 daily log-return distribution and Normal kernel density estimate. For calcu-lating log-returns, we have used the daily closure prices from January 3, 1950 to October 29,2009, for 15054 days. The mean is -2.76E-4 and variance is 9.34E-5.

0 50 100 150 200−0.2

0

0.2

0.4

0.6

0.8

Delay time / days

Aut

ocor

rela

tion

0 50 100 150 200−0.2

0

0.2

0.4

0.6

0.8

Delay time / days

Aut

ocor

rela

tion

Absolute log−returnsLog−returns

Table 1.8 Autocorrelation functions. For calculating log-returns we have used the daily closureprices from January 3, 1950 to October 29, 2009, for 15054 days.

(iv) Volatility clustering: price fluctuations are not identically distributedand the properties of the distribution, such as the absolute return orvariance, change with time and this is called time-dependent or “clus-tered volatility” (see Fig. 1.9) . The volatility measure of absolute returnsshow a positive auto-correlation over a long period of time (see Fig. 1.8)– decays roughly as a power-law with exponent between 0.1 and 0.3[86, 89, 90]. Therefore high volatility events tend to cluster in time andlarge changes tend to be followed by large changes and so also for smallchanges.

Some of these features have been studied very well by the class of eco-nomic models called ARCH and GARCH models.


1950 1960 1970 1980 1990 2000 2010−0.2

−0.1

0

0.1

0.2

Date

Log−

retu

rns

1950 1960 1970 1980 1990 2000 2010

0

1

2

3

4x 10

−3

Date

Vol

atili

ty

Table 1.9 Returns and Volatility. For calculating log-returns we have used the daily closureprices from January 3, 1950 to October 29, 2009, for 15054 days. For volatility calculations wehave used the moving time window of 20 days.

1.2.3Short note on multiplicative stochastic processes ARCH/GA RCH

Considerable interest has been in the application of ARCH/GARCH modelsto financial time-series which exhibit periods of unusually large volatility fol-lowed by periods of relative tranquility. The assumption of constant varianceor “homoskedasticity” is inappropriate in such circumstances. A stochasticprocess with auto-regressional conditional “heteroskedasticity” (ARCH) is ac-tually a stochastic process with “non-constant variances conditional on thepast but constant unconditional variances” [58]. An ARCH(p) process is de-fined by the equation

σ2t = α0 + α1x

2t−1 + ...+ αpx

2t−p, (1.35)

where α0, α1, ...αp are positive parameters and xt is a random variable withzero mean and variance σ2

t , characterized by a conditional probability distri-bution function ft(x), which may be chosen to be Gaussian. The nature of thememory of the variance σ2

t is controlled by the parameter p.The generalized ARCH processes, called the GARCH(p, q) processes, intro-

duced by Bollerslev [59] is defined by the equation

σ2t = α0 + α1x

2t−1 + ...+ αqx

2t−q + β1σ2

t−1 + ...+ βpσ2t−p, (1.36)

where β1, ..., βp are the additional control parameters.The simplest GARCH process is the GARCH(1,1) process with Gaussian

conditional probability distribution function ft(x), and is given by

σ2t = α0 + α1x

2t−1 + β1σ2

t−1. (1.37)

It was shown in[60] that the variance is given by

σ =α0

1− α1 − β1, (1.38)


and the kurtosis is given by

κ = 3+6α2

1

1− 3α21 − 2α1β1 − β2

1

. (1.39)

The random variable xt can be written in term of σt by defining xt ≡ ηtσt,where ηt is a random Gaussian process with zero mean and unit variance.One can also rewrite Eq. 1.37 as a random multiplicative process

σ2t = α0 + (α1η2

t−1 + β1)σ2t−1. (1.40)

1.2.4

Is the market efficient?

In financial econometrics, one of the most debatable issues is whether the mar-ket is “efficient” or not; the “efficient” asset market is one in which the infor-mation contained in past prices is instantly, fully and continually reflected inthe asset’s current price. It was Eugene Fama who proposed the efficient mar-ket hypothesis (EMH) in his Ph.D. thesis work in the 1960’s. He made theargument that in an active market that includes many well-informed and in-telligent investors, securities would be fairly priced and reflect all the availableinformation. In his own words:

“An ‘efficient’ market is defined as a market where there are largenumbers of rational, profit-maximizers actively competing, witheach trying to predict futuremarket values of individual securities,and where important current information is almost freely avail-able to all participants. In an efficient market, competition amongthe many intelligent participants leads to a situation where, at anypoint in time, actual prices of individual securities already reflectthe effects of information based both on events that have alreadyoccurred and on events which, as of now, the market expects totake place in the future. In other words, in an efficient market atany point in time the actual price of a security will be a good esti-mate of its intrinsic value.”

– Eugene F. Fama, “RandomWalks in Stock Market Prices,” Finan-cial Analysts Journal, September/October 1965 (reprinted January-February 1995).

Besides, there continues to be disagreement on the degree of market effi-ciency. The three widely accepted forms of the efficient market hypothesisare:

• “Weak” form: all past market prices and data are fully reflected in secu-rities prices and hence technical analysis is of no use.


• “Semistrong” form: all publicly available information is fully reflectedin securities prices and hence fundamental analysis is of no use.

• “Strong” form: all information is fully reflected in securities prices andhence even insider information is of no use.

The efficient market hypothesis has provided the basis for much of the fi-nancial market research. In the early 1970’s, a lot of the evidence seemed tohave been consistent with the efficient market hypothesis: the prices followeda random walk and the predictable variations in returns, if any, turned out tobe statistically insignificant. While most of the studies in the 1970’s concen-trated mainly on predicting prices from past prices, studies in the 1980’s alsolooked at the possibility of forecasting based on variables such as dividendyield (e.g. Fama & French [1988]). Several later studies also looked at thingssuch as the reaction of the stockmarket to the announcement of various eventssuch as takeovers, stock splits, etc. In general, results from event studies typi-cally showed that prices seemed to adjust to new information within a day ofthe announcement of the particular event, an inference that is consistent withthe efficient market hypothesis. Studies beginning in the 1990’s started look-ing at the deficiencies of asset pricing models. The accumulating evidencesstarted suggesting that stock prices could be predicted with a fair degree ofreliability. To answer the question of whether predictability of returns repre-sented “rational” variations in expected returns or simply arose as “irrational”speculative deviations from theoretical values, further studies have been con-ducted in the recent years. Researchers have now discovered several otherstock market “anomalies” that seem to contradict the efficient market hypoth-esis. Once an anomaly is discovered, in principle, investors attempting toprofit by exploiting such an inefficiency should result in the disappearance ofthe anomaly. In fact, numerous anomalies that have been discovered via back-testing, have subsequently disappeared or proved to be impossible to exploitbecause of high transactions costs.In many cases, strong performers in one period frequently turn around and

underperformed in subsequent periods. Numerous studies have found lit-tle or no correlation between strong performers from one period to the next.And this lack of consistent out-performance among active managers can befurnished as evidence in support of the efficient market hypothesis:

“Market efficiency is a description of how prices in competitivemarkets respond to new information. The arrival of new infor-mation to a competitive market can be likened to the arrival of alamb chop to a school of flesh-eating piranha, where investors are- plausibly enough - the piranha. The instant the lamb chop hitsthe water, there is turmoil as the fish devour the meat. Very soonthe meat is gone, leaving only the worthless bone behind, and the

1.3 Power spectral density 25

water returns to normal. Similarly, when new information reachesa competitive market there is much turmoil as investors buy andsell securities in response to the news, causing prices to change.Once prices adjust, all that is left of the information is the worth-less bone. No amount of gnawing on the bone will yield any moremeat, and no further study of old information will yield any morevaluable intelligence.”

– Robert C. Higgins, Analysis for Financial Management (3rd edi-tion 1992)

Before ending the discussion, we must mention that the nature of efficientmarkets is paradoxical in the sense that if every practitioner truly believed thata market was efficient, then the market would not have been efficient since noone would have then analyzed the behaviour of the asset prices. In effect,efficient markets depend on market participants who believe the market is in-efficient and trade assets in order to make the most of the market inefficiency.

1.3Power spectral density

In statistical signal processing and physics, the concept of a spectral density–power spectral density (PSD) or energy spectral density (ESD)– is a positivereal function of a frequency variable associated with a stationary stochasticprocess, or a deterministic function of time, which has dimensions of power perHz, or energy per Hz. It is often called simply the “spectrum” of the signal.In a sense, the spectral density captures the frequency content of a stochasticprocess and helps identify periodicities.

1.3.1The spectral density

The energy spectral density describes how the energy (or variance) of a signalor a time series is distributed with frequency. If f (t) is a finite-energy (squareintegrable) signal, the spectral density Φ(ω) of the signal is the square of themagnitude of the continuous Fourier transform of the signal (where energy istaken as the integral of the square of a signal, which is the same as physicalenergy if the signal is a voltage applied to a unit-ohm load):

Φ(ω) =

∣

∣

∣

∣

1√π

∫ ∞

−∞f (t) exp(−iωt)dt

∣

∣

∣

∣

2

=F(ω)F∗(ω)

2π

where ω is the angular frequency (defined as 2π times the ordinary frequency)and F(ω) is the continuous Fourier transform of f (t), and F∗(ω) is its complexconjugate.


If the signal is discrete with values fn, over an infinite number of elementsn, we can still define an energy spectral density:

Φ(ω) =

∣

∣

∣

∣

∣

1√π

∞

∑n=−∞

fn exp(−iωn)

∣

∣

∣

∣

∣

2

=F(ω)F∗(ω)

2π

where F(ω) is simply the discrete-time Fourier transform of fn.However, if the number of defined values is finite, the sequence does not

actually have an energy spectral density. But the sequence can be treated asperiodic, using a discrete Fourier transform to make a discrete spectrum, or itcan be extended with zeros and a spectral density can be computed as in theinfinite-sequence case. Also, the continuous and discrete spectral densities areoften denoted with the same symbols, even though their dimensions and unitsdiffer: the continuous case has a time-squared factor that the discrete case doesnot have. They can be constructed to have same dimensions and units bymeasuring time in units of sample intervals or by scaling the discrete case tothe desired time units. The multiplicative factor of 1

2π is also not absolute, butdepends rather on the particular normalizing constants used in the definitionof the various Fourier transforms.Note that the above definitions of energy spectral density require that the

Fourier transforms of the signals exist, i.e., the signals are square-integrable(or square-summable). An alternative is the power spectral density (PSD),which describes the distribution of the power of a signal or time series withfrequency. Here, power considered may be the actual physical power, or forconvenience with abstract signals, may be defined as the squared value ofthe signal (the actual power if the signal was a voltage applied to a unit-ohmload). This instantaneous power (the mean or expected value of which is theaverage power) is then given by:

P = s(t)2.

Since a signal with non-zero average power is not square-integrable, the Fouriertransforms do not exist in this case. Fortunately, theWiener-Khinchin theoremprovides a simple alternative:

The power spectral density is the Fourier transform of the autocorrelation

function R(τ) of the signal, if the signal is a stationary random process.

This results in the formula:

S( f ) =∫ ∞

−∞R(τ) exp(−2πi fτ)dτ.

The power of the signal in a given frequency band can be calculated byintegrating over positive and negative frequencies,

P =∫ F2

F1S( f )d f +

∫ F1

−F2S( f )d f .

1.3 Power spectral density 27

Note that the power spectral density of a signal exists if and only if the signalis a stationary process. If the signal is not stationary, then the autocorrela-tion function must be a function of two variables, so truly no power spectraldensity exists. However, similar techniques may be used to estimate a time-varying spectral density.The power spectrum G( f ) is defined by

G( f ) =∫ f

−∞S(′ f )d′ f .

Noteworthy properties:

(i) The spectral density of f (t) and the autocorrelation function of f (t) forma Fourier transform pair.

(ii) One of the results of Fourier analysis is Parseval’s theorem which phys-ically means that the area under the energy spectral density curve isequal to the area under the square of the magnitude of the signal, thetotal energy:

∫ ∞

−∞| f (t)|2 =

∫ ∞

−∞Φ(ω)dω.

The above theorem holds true in the discrete cases as well. Anothersimilar result: the total power in a power spectral density being equal tothe corresponding mean total signal power, which is the autocorrelationfunction at zero lag.

1.3.2Are there any long-time correlations?

The random walk theory of prices assumes that the returns are uncorrelated.But are they truly uncorrelated or are there long-time correlations in the finan-cial time-series? This question has been studied especially since it may lead todeeper insights about the underlying processes that generate the time-series[41].We discuss this in the next chapter with details but here we introduce two

measures to quantify the long-time correlations, and study the strength oftrends: the R/S analysis to calculate the Hurst exponent and the detrendedfluctuation analysis [42].

1.3.2.1 Hurst Exponent from R/S AnalysisIn order to measure the strength of trends or “persistence” in different pro-cesses, the rescaled range (R/S) analysis to calculate the Hurst exponent canbe used. One studies the rate of change of the rescaled range with the changeof the length of time over which measurements are made. We divide the time-series ξt of length T into N periods of length τ, such that Nτ = T. For each


period i = 1, 2, ...,N, containing τ observations, the cumulative deviation is

X(τ) =iτ

∑t=(i−1)τ+1

(ξt − 〈ξ〉τ) , (1.41)

where 〈ξ〉τ is the mean within the time-period and is given by

〈ξ〉τ =1τ

iτ

∑t=(i−1)τ+1

ξt. (1.42)

The range in the i-th time period is given by R(τ) = maxX(τ) −minX(τ),and the standard deviation is given by

S(τ) =

1τ

iτ

∑t=(i−1)τ+1

(ξt − 〈ξ〉τ)2

12

. (1.43)

Then R(τ)/S(τ) is asymptotically given by a power-law

R(τ)/S(τ) = κτH , (1.44)

where κ is a constant and H the Hurst exponent. In general, “persistent”behavior with fractal properties is characterized by a Hurst exponent 0.5 <

H ≤ 1, random behavior by H = 0.5 and “anti-persistent” behavior by 0 ≤H < 0.5. Usually Eq. (1.44) is rewritten in terms of logarithms, log(R/S) =H log(τ) + log(κ), and the Hurst exponent is determined from the slope.

1.3.2.2 Detrended Fluctuation Analysis (DFA)In the DFA method the time-series ξt of length T is first divided into N non-overlapping periods of length τ, such that Nτ = T. In each period i =1, 2, ...,N the time-series is first fitted through a linear function zt = at + b,called the local trend. Then it is detrended by subtracting the local trend, inorder to compute the fluctuation function,

F(τ) =

1τ

iτ

∑t=(i−1)τ+1

(ξt − zt)2

12

. (1.45)

The function F(τ) is re-computed for different box sizes τ (different scales) toobtain the relationship between F(τ) and τ. A power-law relation betweenF(τ) and the box size τ, F(τ) ∼ τα, indicates the presence of scaling. Thescaling or “correlation exponent” α quantifies the correlation properties of thesignal: if α = 0.5 the signal is uncorrelated (white noise); if α > 0.5 the signalis anti-correlated; if α < 0.5, there are positive correlations in the signal.

References 29

Table 1.10 Power spectrum analysis (left) and detrended fluctuation analysis (right) of auto-correlation function of absolute returns, from Ref. [89].

1.3.2.3 DFA and PSD analyses of auto-correlation function of absolu te returnsThe analysis of financial correlations was done in 1997 by the group of H.E.Stanley [89]. The correlation function of the financial indices of the New Yorkstock exchange and the S&P 500 between January, 1984 and December, 1996were analyzed at one minute intervals. The study confirmed that the auto-correlation function of the returns fell off exponentially but the absolute valueof the returns did not. Correlations of the absolute values of the index returnscould be described through two different power laws, with crossover timet× ≈ 600 minutes, corresponding to 1.5 trading days. Results from powerspectrum analysis and DFA analysis were found to be consistent. The powerspectrum analysis of Fig. 1.10 yielded exponents β1 = 0.31 and β2 = 0.90 forf > f× and f < f×, respectively. This is consistent with the result that α =(1 + β)/2 and t× ≈ 1/ f×, as obtained from detrended fluctuation analysiswith exponents α1 = 0.66 and α2 = 0.93 for t < t× and t > t×, respectively.

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0

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30 References

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