University of Ljubljana

Faculty of Mathematics and Physics

Department of Physics

Study and Modeling of Price Variations inFinancial Markets

Seminar

Nika Oman

Mentor: prof. dr. Rudolf Podgornik

January, 2009

Abstract

In this seminar, I present Mandelbrot's Levy-stable hypothesis, which was a majorbreakthrough in �nancial market statistics in the past century. The recent avail-ability of large amounts of high frequency data has made it much easier to analizeand study the �nancial markets' statistical properties. I present a few of these anal-yses which show a slight, but important disagreement with Mandelbrot's proposalin the distribution's tail behavior. In the �nal part of the seminar, a percolationmodel of stock market dynamics is presented, where the most important empiricalcharacteristics of stock market price dynamics are captured.

Contents

1 Introduction 1

2 Heavy-tailed statistics 2

2.1 Levy-stable distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The parameters of Levy-stable distributions . . . . . . . . . . . . . . . . . 42.3 Properties of Levy-stable distributions . . . . . . . . . . . . . . . . . . . . 52.4 Implications of Levy-stable hypothesis . . . . . . . . . . . . . . . . . . . . 6

3 The state of the evidence 7

4 Self-Organized Percolation model of stock market �uctuations 10

4.1 Percolation theory of networks . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Percolation model of stock market prices . . . . . . . . . . . . . . . . . . . 114.3 Connectivity evolving with time . . . . . . . . . . . . . . . . . . . . . . . . 134.4 Size-dependent activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 Nonlinear price change dependence . . . . . . . . . . . . . . . . . . . . . . 14

5 Conclusion 16

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1 IntroductionHalf a decade ago, a word �econophysics� started to circulate in the community of physi-cists. The name was coined to describe applications of methods of statistical physicsto economy in general. Although the name entered the scienti�c language only halfa decade ago, connections and interplay between physics and economy are more thanhundred years old.

In 1900 Bachelier proposed what is now called �Brownian motion� in �nancial spec-ulation, generating Gaussian distribution of price variations [1]. This discovery cameyears before physicists discovered the Brownian motion of small particles. For processesgoverned by Gaussian statistics there is a well elaborated understanding; a prominentexample of this category is the thermodynamics of systems in equilibrium [2].

In the recent decades, systems driven into non-equilibrium states and complex systemshave become of special interest for many physicists. It has been realized that there is alarge class of disordered systems which are governed by anomalous statistics, i.e. statis-tics which deviate from Gaussian statistics. These systems lack a characteristic scale andare described by hevy-tailed statistics (or power-laws). A power-law distribution lacksany characteristic scale. This property prevented the use of power-law distributions inthe natural sciences until the emergence of new paradigms in probability theory, thanksto the work of Paul Levy and Kolmogorov. During the last thirty years, physicistshave achieved important results in the �eld of phase transitions, statistical mechanics,nonlinear dynamics disordered and self-organized systems. In these �elds, power laws,scaling and unpredictable (stochastic or deterministic) time series are present and thecurrent interpretation of the underlying physics is often obtained using these concepts [2].

By modeling and analyzing the statistical properties of cotton prices, Benoit B. Man-delbrot (The Variation of Certain Speculative Prices,1963) [1] was the �rst to introduceheavy-tailed statistics to economics. Prior to this publication, the assumption was thatthe distribution of price changes in a speculative series is approximately Gaussian. Man-delbrot proposed power-law tails in the distribution (of logarithmic changes of prices)which exhibit the property of scale invariance. Together with the observation, that thewhole distribution is non-Gaussian, he concluded that the price changes were consis-tent with a class of stable distributions called Levy distributions. Although the newlyintroduced statistics was not accepted well in the economical community due to someimpractical properties, it is now regarded as an important breakthrough in statistics of�nancial markets.

Recent analysis of larger amounts of data (S&P500 index, Plerou et al., 2000)[3] showsa disagreement with Mandelbrot's proposal and a correction in the tails of the proba-bility distribution is needed. These issues are adressed in Sections 2 and 3 of this seminar.

Much of the recent work is focused on understanding the statistical properties of �nancialtime series. One reason for this interest is that �nancial markets are examples of complexinteracting systems for which a huge amount of data exist and it is possible that �nan-cial time series viewed from a di�erent perspective might yield new results. Attemptingto capture the complex behavior of stock market prices and of market participants, awealth of models have been introduced in the �nancial and physical community. In thisseminar, I will present a percolation model generating heavy-tailed statistics, which is a

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simple representation of the observed market characteristics.

2 Heavy-tailed statisticsIn this section, I will present typical features and the implications of heavy-tailed statis-tics of price variation in �nancial markets, following Mandelbrot's original hypothesis.

The classic Bachelier's model of the temporal variation of speculative prices assumesGaussian (normal) distribution. Let Z(t) be the price of a stock (or of a unit of acommodity) at the end of time priod t and xi the succesive di�erences of the formxi = ln (Z(t+ T )) − ln (Z(t)). The Gaussian distribution is characterized by two pra-

rameters: the mean value 〈xi〉 and the variance σ2i = 〈(xi − 〈xi〉)2〉. The succesive

di�erences in logarithm of price changes xi are thus independent Gaussian random vari-ables with zero mean and with variance proportional to the di�erencing interval T . Theevolution of the price (or in our case the logarithm of the price) is just a di�usion processwith a drift. Knowledge of the parameters of the Gaussian distribution describing pricechanges in one day can be used to predict the distribution of the relative price changeson a longer time scale. These will again be given by the Gaussian distribution, but withrescaled variance and mean [4].

For assessment of risk, we ask what will be the probability of an extreme event, whichfor example should be twice as large as an observed event. In the case of Gaussiandistribution we obtain: p(2x) = (p(x))4. Thus, for an event that has a probability of10−3, the event, which is twice as large, is 10−9 times less probable. This means thatthe probability of more extreme events becomes very unlikely [2].

The �nancial market reality is more complex than suggested by the model of independentstationary Gaussian distribution of relative price changes discussed above. Mandelbrot's1963 analysis of cotton and other prices revealed that the model is contradicted by factsin at least four ways [1]:

1) Large price changes are much more frequent than predicted by the Gaussian distri-bution; this re�ects the �excessively peaked� or �leptokurtic� character of price relatives.

2) Large practically instantaneous price changes occur often, contrary to prediction, andthey must be explained by causal rather than stochastic models.

3) Successive price changes do not appear to be independent, but rather exhibit a largenumber of recognizable patterns.

4) Price records do not appear stationary and statistical expressions such as the samplevariance take very di�erent values at di�erent times; this nonstationarity would put aprecise statistical model of price change out of the question [1]. One cannot expect thatthe prices will �uctuate according to the same law over twenty years. In this period manythings may happen which may a�ect performances of individual companies. One hasto weaken the stationarity assumption and substitute it by a sort of quasi-stationarity [4].

Mandelbrot felt that these departures from normality were su�cient to warrant a radi-cally new approach to the statistics of speculative prices. His approach makes two basic

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Figure 1: Histogram for normalized a) minutely, b) 1-hour, c) daily differences of logarithm of DAXindices. d) Magnification of the tail behavior of c). Normalized Gauss (dotted line) and best fit forLevy-stable distribution (dashed line) are also displayed for comparison. Parameters α and γ of Levy-stable fits: a) α = 1.16 and γ = 0.286; b) α = 1.40 and γ = 0.354; c) α = 1.70 and γ = 0.385.Figure from [5].

assertions: (1) the variances of empirical distributions behave as if they were in�nite and(2) the empirical distributions conform best to the non-Gaussian members of a familyof limiting distributions called Levy-stable distributions [1].

We will now examine further the theoretical content of this statistics.

2.1 Levy-stable distributionsThe derivation of most of the important properties of Levy-stable distributions is duePaul Levy (1925). The quali�cation �stable" of the distribution means it has the prop-erty of stability, which denotes the invariance under addition and will be discussed inSection 2.3. A rigorous and compact mathematical treatment of the statistical theory isfound in Gnedenko & Kolmogorov (1954) [1], a more comprehensive mathematical anddescriptive treatment was done by Mandelbrot and by Fama (both in 1963). Much work(mostly empirical) has been done since, especially in the last two decades, to test andimprove the initial Mandelbrot's proposal. In Fig. 1 we see an example of such analysisof the collection of the high frequency Deutsche Aktienindex (DAX) data [5].

As the basis for all later research of �nancial market statistics, I will, in this seminar,�rst present the original Mandelbrot's 1963 theoretical proposal and then discuss em-

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pirical �ndings and corrections to this theory. The �rst step is to examine some of theimportant statistical properties of the Levy-stable distributions. The implications ofMandelbrot's statistics for the theoretical and empirical work of the economists will bethen presented. The state of the evidence concerning the validity of the theory will bediscussed in Section 3.

2.2 The parameters of Levy-stable distributionsIn probability theory, the probability distribution of any random variable is de�ned bythe Fourier transform of its characteristic function:

P (u) =1

2π

∫ ∞−∞

f(t)e−itudt

Every probability distribution on R or on Rn has a characteristic function, because oneis integrating a bounded function over a space whose measure is �nite, and for everycharacteristic function there is exactly one probability distribution [6].

We can get the characteristic function from the probability distribution function P (u);it is the expected value of the function exp(itu):

f(t) =∫ ∞−∞

eitudP (u)

If the probability distribution is not symmetric, an imaginary component of a charac-teristic function is present.

Characteristic functions are used to �nd moments of a random variable. For the n-thmoment we have [6]:

Mn = i−n[dn

dtnf(t)

]t=0

The characteristic function for the Levy-stable family of distributions satis�es [1]:

lnf(t) = iδt− γ|t|α[1 + iβ(t/|t|)tg(απ/2)] (1)

Becouse the characteristic function is not di�erentable at t = 0, variance and highermoments are in�nite (except at α = 2). When α ≤ 1, also the mean is unde�ned.

The characteristic function tells us that Levy-stable distributions have four parameters:α, β, γ and δ. The location parameter is δ, and if α is greater than 1, δ is equal to theexpectation of mean of the distribution. The scale parameter is γ, while the parameterβ is an index of skewness, which can take any value in the interval −1 ≤ β ≤ 1. Whenβ = 0 the distribution is symmetric. When β > 0 (and 1 < α < 2), the distributionis skewed right (i.e., has a long tail to the right) and the degree of skewness increasesin the interval 0 < β ≤ 1 as β approaches 1. Similarly, when β < 0 (and 1 < α < 2),the distribution is skewed left, with the degree of skewness increasing in the interval−1 ≤ β < 0 as β approaches −1.

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Figure 2: Probability densities for Levy-stable distributions: a) varying skewness index β (α =0.5, γ = 1, δ = 0); b) varying characteristic exponent α (β = 0, γ = 1, δ = 0). We see that if β = 0,the probability density is symmetric. In general, the higher the α value the thinner the tails of thedistribution. Figure from [6].

Of the four parameters of a Levy-stable distribution, the characteristic exponent α isthe most important for the purpose of comparing the �goodness of �t� of the Gaussianand Levy-stable hypothesis. The character exponent α determines the height of, or totalprobability contained in, the extreme tails of the distribution, and can take any valuein the interval 0 < α ≤ 2. When α = 2, the relevant Levy-stable distribution is thenormal (Gaussian) distribution. The total probability in the extreme tails is increasingas α moves away from 2 and toward 0. The most imporatnt consequence of this is thatthe variance exsists (i.e., is �nite) only in the extreme case α = 2. The mean, however,exsists as long as α > 1.

Mandelbrot's Levy-stable hypothesis states that for distributions of price changes inspeculative series, α is in the interval 1 < α < 2, so that the distributions have means,but their variances are in�nite [1]. The Gaussian hypothesis, on the other hand, statesthat α is exactly equal to 2.

2.3 Properties of Levy-stable distributions1. Asymptotic scaling of the tails

The �rst important property of Levy-stable distributions is the asymptotically scalingnature of the extreme tail areas. Levy has shown that tails of these distributions forvalues of α less than 2 follow an asymptotic form of scaling ,i.e, asymptotically follow apower law which exhibits the property of scale invariance. Scaling denotes the fact thata power-law function satis�es f(cx) ∝ f(x), where c is a constant, that is, a rescalingof the function's argument changes the constant of proportionality but preserves theshape of the function itself. The point becomes even clearer when we take the logarithmof the function, which is what Mandelbrot did with the characteristic function of theLevy-stable distribution.

It can be shown [7] that the asymptotic behavior of the Levy-stable distribution (forα < 2)is described by:

Pr(u) ∼ 1π

αγα(1 + β)sin(πα/2)Γ(α)|u|1+α

,

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where Pr(u) denotes the (right) tail distribution and Γ is the Gamma function. Theasymptotic form of scaling can be simpli�ed into:

Pr(u)→ (u

C)−α as u→∞, (2)

where C is a constant, or, taking logartihms of both sides:

lnPr(u)→ −α (lnu− lnC) , u > 0. (3)

2. Stability or invariance under addition

By de�nition, a Levy-stable distribution is any distribution that is invariant under ad-dition. That is, the distribution of sums of independent, identically distributed, Levy-stable variables is itself Levy-stable and has the same form as the distribution of theindividual summands. A rigorous de�nition of Levy-stability is given by the logarithmof the characteristic function:

nlnf(t) = i(nδ)t− (nγ)|t|α[1 + iβ(t/|t|)tg(απ/2)], (4)

where n is the number of variables in the sum and lnf(t) is the logarithm of the charac-teristic function of the individual summands. The above expression is exactly the sameas Eq.(1), except that the parameters δ (location) and γ (scale) are multiplied by n.That is, the distribution of the sums is, except for origin and scale, exactly the sameas the distribution of the individual summands. The values of the parameters α and βremain constant under addition.

This property is responsible for much of the appeal of Levy-stable distributions as de-scriptions of empirical distributions of price changes. The price change in a speculativeseries for any time interval can be regarded as the sum of the changes from transactionto transaction during the interval. If the changes between transactions are independent,identically distributed, Levy-stable variables, daily, weekly and monthly changes willfollow Levy-stable distributions of exactly the same form, except for origin and scale.

3. Limiting distributions

It can be shown (Gnedenko & Kolmogorov, 1954) [1] that Levy-stable distributions arethe only possible limiting distributions for sums of inependent, identically distributedrandom variables. It is well known that if such variables have �nite variance, the lim-iting distribution will be the normal distribution (Central limit theorem). If the basicvariables have in�nite variance, however, and if their sums follow a limiting distribution,the limiting distribution must be Levy-stable with 0 < α < 2.

2.4 Implications of Levy-stable hypothesisThe Levy-stable hypothesis has many important implications. First, it implies thatthere is a larger number of abrupt changes in the economic variables that determinesequilibrium prices in speculative markets than predicted by Gaussian hypothesis. In amarket that is Levy-stable with α < 2, a large price change across a long time intervalwill more than likely be the result of a few very large changes that took place during

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Figure 3: (a) The daily records of the S&P500 index for the 35-year period 1962 − 1996 on alinear-log scale. Note the large jump which occurred during the market crash of October 19th 1987.Sequence of (b) 10-minute returns and (c) 1-month returns of the S&P500 index, normalized to unitvariance. (d) Sequence of i.i.d. Gaussian random variables with unit variance. For all 3 panels, thereare 850 events, i.e., in panel (b) 850 minutes and in panel (c) 850 months. In contrast to (b) and (c),there are no large events in (d). Figure from [3].

smaller intervals. This means that such a market is inherently more risky for the in-vestor than a Gaussian market. The variability of a given expected yield is higher andthe probability of large losses is greater.

Secondly, the distribution of changes between transactions must, at the very least, beasymptotically scaling. As these changes are the result of many bits of information, newinformation also re�ects the changes in the underlying economic conditions. Thus theunderlying economic conditions that determine equilibrium prices must themselves havean asymptotically scaling character.

Finally, the Levy-stable hypothesishas important implications for data analysis. From apurely statistical standpoint, if the population variance of the distribution of �rst di�er-ences is in�nite, the sample variance is a meaningless measure of dispersion. Moreover,if the variance is in�nite, other statistical tools (e.g., least-squares regression), which arebased on the assumption of �nite variance will, at best, be cosiderably weakened andmay in fact give very misleading answers.

3 The state of the evidenceMandelbrot based his Levy-stable hypothesis on the analysis of relatively short (about2000 data points) time series of cotton prices. The recent availability of �high frequency"data allows one to study economic time series on a wide range of time scales varying fromseconds up to a few decades. Plerou et al. [3] analyzed (a) 1-minute and daily recordsof the S&P500 index (Fig. 3), an index of New York Stock Exchange that consists of500 companies, often used as a benchmark to gauge the performance of the US stockmarkets, and (b) 4-year data for every transaction for the leading 1000 companies, thatrecords the prices, the shares traded, and the trading times.

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The S&P500 index Z(t) from 1962− 1996 has an overall upward drift - interrupted bydrastic events such as the market crash of October 19th 1987 (Fig. 3a). Plerou et al.analyzed the di�erence in logarithm of the index (the same as Mandelbrot), often calledthe return G(t) = ln (Z(t+ ∆t)) − ln (Z(t)), where ∆t is the time scale investigated(Fig. 3b). One only counts the number of minutes during the opening hours of the stockmarket. It is apparent from Fig. 3b that large events are much more likely to occur, incontrast to a sequence of Gaussian distributed random numbers of the same variance(Fig. 3d). As one analyzes returns on larger time scales, this di�erence is apparentlymuch less pronounced (Fig. 3c). In order to understand this process, one starts byanalyzing the probability distribution of returns on a given time scale ∆t, which in thisstudy, varied from 1 minute up to a few months.

The advent of improved computing capabilities has facilitated the probing of the asymp-totic behavior of the distribution. For example, Mantegna and Stanley (1995) [3] an-alyzed approximately 1 million records of the S&P500 index. They reported that thecentral part of the distribution of S&P500 returns appears to be well �t by a Levydistribution, but the asymptotic behavior of the distribution of returns shows fasterdecay than predicted by a Levy distribution. Hence, they proposed a truncated Levydistribution as a model for the distribution of returns; a Levy distribution in the centralpart followed by an approximately exponential truncation. The exponential truncationensures the existence of a �nite second moment, and hence the truncated Levy distribu-tion is not a stable distribution [3]. The truncated Levy process with i.i.d.(independentidentically distributed) random variables has slow convergence to Gaussian behavior dueto the Levy distribution in the center, which could explain the observed time scaling fora considerable range of time scales [3].

Recent studies on even larger time series using larger databases show quite di�erentasymptotic behavior for the distribution of returns. The recent analysis of Plerou et al.[3] included three di�erent databases covering securities from the three major US stockmarkets. They analyzed approximately 40 million records of stock prices sampled at 5minute intervals for the 1000 leading US stocks for the 2-year period 1994 − 1995 and35 million daily records for 16000 US stocks for the 35-year period 1962 − 1996. Theystudied the probability distribution of returns (Fig. 4a− c) for individual stocks over atime interval ∆t, where ∆t varies approximately over a factor of 104 - from 1 minute upto more than 1 month. A parallel study of the S&P500 index was also conducted [3].

The key �nding of the analysis is that the cumulative distribution of returns for bothindividual companies (Fig. 4c) and the S&P500 index (Fig. 4a) can be well describedby a power-law asymptotic behavior, characterized by an exponent α ≈ 3, well outsidethe stable Levy regime 0 < α < 2. Further, it was found that the distribution, althoughnot a stable distribution, retains its functional form for time scales up to approximately16 days for individual stocks and approximately 4 days for the S&P500 index, (Fig. 4b).For larger time scales the results are consistent with break-down of scaling behavior, i.e.,convergence to Gaussian [3].

We have demonstrated some of the problems with �tting �nancial data to stable distri-butions. Financial data are not stationary. There is some hope that the alpha parametermight be somewhat �xed with signi�cantly higher value than obtained by the assump-tion that all the parameters are stationary, but this is not easy to prove [8]. We couldmathematically model market data as a distribution of the product of a (log)normal

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Figure 4: (a) Log-log plot of the cumulative distribution of the normalized 1-minute returns forthe S&P500 index. Power-law regression fits in the region 3 ≤ g ≤ 50 yield α = 2.95 ± 0.07(positive tail), and α = 2.75 ± 0.13 (negative tail). For the region 0.5 ≤ g ≤ 3, regression fitsgive α = 1.6 ± 0.1 (positive tail), and α = 1.7 ± 0.1 (negative tail). (b) Log-log plot of thecumulative distribution of normalized returns of the S&P500 index. The positive tails are shown for∆t = 16, 32, 128, 512 minutes. Power-law regression fits yield estimates of the asymptotic power-lawexponent α = 2.69±0.04, α = 2.53±0.06, α = 2.83±0.18 and α = 3.39±0.03 for ∆t = 16, 32, 128and 512 minutes, respectively. (c) The positive and negative tails of the cumulative distribution ofthe normalized returns of the 1000 largest companies in the TAQ database for the 2-year period1994− 1995. The solid line is a power-law regression fit in the region 2 ≤ x ≤ 80. Figure from [3].

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random variable and a stable random variable. The idea is that we write the character-istic function for a mixture of distributions. This function will have �ve parameters, theusual α, β, γ, and δ of the Levy-stable distributions plus σ, representing the standarddeviation of the (log)normal distribution [8].

I continue now, in the next section of this seminar, with the pursuit of �nding a modelof �nancial market that would re�ect the empirical �ndings of the properties of thedistribution of returns, without concerning ourselves with a mathematical description ofthe characteristic function and with the non-stationarity of the parameters.

4 Self-Organized Percolation model of stock market fluctuationsThe e�orts to capture the observed characteristics of �nancial markets �uctuations haveproduced a series of models that each provide a particular window of understanding.The diversity of models re�ects our burgeoning understanding of this �eld which hasnot yet fully matured. In its broadest sense, a model (usually formulated using thelanguage of mathematics) is a mathematical representation of a condition, process, con-cept, etc., in which the variables are de�ned to represent inputs, outputs, and intrinsicstates and equations or inequalities are used to describe interactions of the variables andconstraints on the problem. In theoretical physics, models take a narrower meaning,such as in the Ising, Potts,..., percolation models. In economy and �nance, the termmodel is usually used in the broadest sense. Our percolation model, however, falls inthe second category [9].

4.1 Percolation theory of networksConsider a regular d-dimensional lattice whose edges are present with probability p andabsent with probability 1 − p. Percolation theory studies the emergence of paths thatpercolate the lattice (starting at one side and ending at the opposite side). For smallp only a few edges are present, thus only small clusters of nodes connected by edgescan form, but at a critical probability pc, called the percolation threshold, a percolatingcluster of nodes connected by edges appears (Fig. 5). This cluster is also called thein�nite cluster, because its size diverges as the size of the lattice increases. There areseveral much studied versions of percolation, the one presented in Fig. 5 being �bondpercolation�. The most known alternative is site percolation, in which all bonds arepresent and the nodes of the lattice are occupied with probability p. In a similar way asbond percolation, for small p only �nite clusters of occupied nodes are present, but forp > pc an in�nite cluster appears [10].

The main quantities of interest for our further discussion are:

The percolation probability, P , denoting the probability that a given node belongs tothe in�nite cluster and is:

P ={

0 if p < pc> 0 if p > pc

(5)

and the cluster size distribution, ns, de�ned as the probability of a node being the lefthand end of a cluster of size s,

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Figure 5: Illustration of bond percolation in 2D. The nodes are placed on a 25× 25 square lattice,and two nodes are connected by an edge with probability p. For p = 0.315 (left), which is below thepercolation threshold pc = 0.5, the connected nodes form isolated clusters. For p = 0.525 (right),which is above the percolation threshold, the largest cluster percolates. Figure from [10].

ns =1sPs, (6)

where Ps denotes the probability that the cluster at the origin has size s [10].

4.2 Percolation model of stock market pricesCont and Bouchaud introduced a percolation model which assumes that investors canbe classi�ed into groups (clusters) of the same opinion occurring with many di�erentsizes [12]. The simplest recipe to aggregate interacting or inter-in�uencing traders intogroups is to assume that the connectivity between traders de�ning the groups can beseen as a pure geometrical percolation problem with �xed occupancy on a given networktopology. Clusters are groups of neighboring occupied sites or investors. Then, randompercolation clusters make a decision to buy or sell on the stock market, for all sites(corresponding to the individual investors and units of wealth) in that cluster together.Thus, the individual investors are thought to cluster together to form companies orgroups of in�uence, which under the guidance of a single manager buy (probability a),sell (probability a), or refrain from trading (probability 1−2a) within one time interval.Fig. 6 shows an illustration of such a lattice of investors. The traded amount is pro-portional to the number s of sites in the cluster, and the logarithm of the price changesproportionally to the di�erence ∆ between demand and supply [12].

When the activity a is small, at most one cluster trades at a time. As a consequence,the distribution P (R) of relative price changes (or returns) R scales as the cluster sizedistribution ns of percolation theory. In contrast, for large activity a and without an in-�nite cluster, the relative price variation is the contribution (sum) of many clusters andthe central limit theorem implies that the distribution P (R) converges to the Gaussianlaw for large systems (except exactly at the critical point pc) [12].

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Figure 6: An illustration of a lattice representing investors that buy (grey), sell (white) or refrainfrom trading (black). Figure from [11].

For low activity a and right at the site percolation threshold p = pc, when the fractionp of lattice sites occupied by an investor in a d-dimensional lattice of linear extent Lbarely su�ces to form an �in�nite� cluster stretching from top to bottom, we observepower laws:

ns ∝ s−τ , P (R) ∝ R−τ for 1� s� LD, (7)

where D = d/(τ − 1) is the fractal dimension of the percolating cluster [12].

For larger activities, but still a� 1, scaling holds [9]: if we normalize height and widthof the return distribution to unity, the curves for various activities a at p = pc overlap,and thus still give the above power law. This scaling is no longer valid for large a ≈ 1/3where the curves become more like a Gaussian.

This model at the percolation threshold thus agrees qualitatively (but not quantitativelyespecially on the exponents as discussed below) with the observed characteristics of realmarkets [12]: the average return R is zero (if in�ation and other regular trends aresubtracted); at the percolation threshold, a simple asymptotic power law holds for smallactivities (short times) and becomes more Gaussian for large a (long times). Returndistribution P (R) decays as ∝ R−µ. The probability P>(R) of �nding a change largerthan R then varies as R1−µ.

Two problems of the model are:• Markets do not know the percolation threshold pc where the model gives power lawbehavior of returns.• The value of the exponent µ = τ = α+ 1 is empirically found to be τ ∼ 4, whereas inthis model, τ varies only from 2 (in two dimensions) to 2.5 (in six and more dimensions)[12].

We must turn to an alternative mechanism which gives power laws without the need totune p to pc and, hopefully, a better approximation for the empirical exponent α ≈ 3(µ ≈ 4).

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4.3 Connectivity evolving with timeThe �rst idea is that the parameter p controlling the connectivity between traders shouldnot be �xed. At some times, traders are following strong herding behavior and the e�ec-tive connectivity parameter p is high; at other times, investors are more individualisticand smaller values of p seem more reasonable. In order to take into account the complexdynamics of the network of interactions between traders, it thus seems reasonable torelax the hypothesis that p is �xed at a given value but rather evolves with its owndynamics [12].

The simplest version is to assume that p is taken purely random at each time step. Asa consequence, the distribution of relative price changes will be an average over thoseobtained for each sampled p. Averaging over an interval in p containing the percolationthreshold pc will give its main contribution to the number of large clusters from a nar-row region (width ∝ 1/sσ) about pc, and thus lead to an integrated cluster numbers∝ s−τ−σ, where σ varies from 0.4 (in two dimensions) 0.5 (in six and more dimensions).Now µ = τ + σ varies from 2.45 to 3, closer to reality (µ ' 4) [12].

We distribute randomly our investors on the L × L square lattice, with concentrationp. We sum up all the results obtained by varying p in steps of one percent, from 1 to59 percent where the percolation threshold pc = 0.593 is reached. For each concentra-tion, we make 1000 iterations where at each iteration one percent of the investors tryto move to a randomly selected neighbor site. For each cluster con�guration obtainedin this way, we sum over 1000 di�erent realizations of buying and selling decisions ofthe cluster, which thus allows much better averaging than in real markets where historycannot be repeated so easily. Many such simulations are averaged over to give smoothresults.

Fig. 7 shows for the square lattice (d = 2) the distribution of returns P (R). The simu-lations con�rm, for a range of about �ve orders of magnitude in P , the predicted powerlaw P (R) ∝ R−µ at intermediate R with an exponent µ ' 2.5. For the largest R, �nitesize e�ects reduce P (R) and, for small R, the probability is roughly constant. Increasingthe linear lattice size L from 31 via 101 to 201 shifts the power-law region to larger Rwithout changing the e�ective exponent.

The numerical deviation from the empirically observed µ = 4 is still large, and a di�er-ent approach is needed.

4.4 Size-dependent activityInstead of taking the activity a as a free parameter between zero and 1/2 (0.005 in Fig.7) and the same for all cluster sizes, we assume the following size dependence [12]:

a =1

2√s, (8)

thus getting rid of one free parameter. It is reasonable to consider that the big investors,such as the mutual or retirement funds with their prudent approach, their emphasis onlow risk, and their enormous inertia due to the fact that large positions move the marketunfavorably, trade less often than small professional investors who have to generate theirincome from active trading rather than from sheer mass. In this spirit, recent works havedocumented that the growth dynamics of business �rms , the economies of countries and

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Figure 7: Return distribution at constant activity a = 0.005. The axis of relative price variations isscaled such that a buying cluster of s investors produces an increase of the price by s. Finite latticesize effects reduce P (R) for large R - the smaller the lattice, the sooner the effect is present. Thiscan also be viewd from the point that a small number of investors present in a market cannot generatebig returns. Increasing the linear lattice size L from 31 via 101 to 201 shifts the power-law region tolarger R without changing the effective exponent. Figure from [12].

the university research activities depend on size [12], the smaller entities being the mostactive proportionally. Another not necessarily exclusive mechanism is that, within alarge cluster, the s investors have to agree by some majority to buy and sell, and donot trade if no such majority is reached. A random decision process then could lead tothe square-root behavior given by Eq. (8). With this modi�cation, the exponent µ ispredicted to be:

µ = τ + σ +12' 3. (9)

Fig. 8 shows the return distribution with activity decaying as ∝ 1/√s. The measured

e�ective exponent in the intermediate R range equals 3.5, larger than the asymptoticallyexpected value given by Eq.(9) and close to the empirical value near 4.

4.5 Nonlinear price change dependenceNow, we will add another correction to the previous two, in hopes that all three com-bined, will give us a better result for the characteristic exponent.

All previous results derive from the assumption that the change of (the logarithm of the)price is proportional to the di�erence between supply and demand. This assumption isoften made and can be in fact derived rigorously from the two assumptions that it is notpossible to make pro�ts by repeatedly trading through a circuit and that the ratio ofprices before and after a transaction is a function of the di�erence ∆ between demandand supply alone [12].

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Figure 8: Return distribution with activity decaying as ∝ 1/√s. Again, finite lattice size effects are

present for very large returns. Figure from [12].

However, many recent empirical studies suggest that the relationship between the changeof the logarithm of price and ∆ is highly nonlinear, especially for large orders [12]. As-suming that the time needed to complete a trade of size s is proportional to s and thatthe unobservable price �uctuations obey a di�usion process during that time, Zhangderived the relationship that the change of the logarithm of the price is proportional tothe square root of the di�erence ∆ between demand and supply [12], i.e. to the squareroot of s in our present formulation. This modi�es all previous results as follows.

The result (7) for the �pure� percolation model becomes:

ns ∝ s−τ , P (R) ∝ R−µ for 1� s� LD, (10)

where now,

R ∝√s, and

ds

dR∝ R. (11)

With numerical estimates in two dimensions, this gives us the exponent µ = 2τ−1 ≈ 3.1[12]. This is still smaller than the empirical value close to 4.

The result obtained by the �sweeping� of the connectivity parameter p transforms µ fromµ = τ + σ into µ = 2τ − 1 + σ, giving a value µ = 3.5. Next, incorporating the sizedependence of the activity (Eq.(9)) leads to the prediction µ = 2τ + σ = 4.5, in roughagreement with the empirical value 4.

We may even omit the size-dependent activity and use only this nonlinear price changedependence and 0 < p < pc. Then the data of Fig. 7 are transformed, without anyadditional simulations, into those of Fig. 9 which give an e�ective µ ' 3.9 in betteragreement with the real µ ' 4 than the theoretical prediction µ = 2τ − 1 + σ ' 3.5.

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Figure 9: Data from Fig. 7 replotted by assuming a price change proportional to the square root ofthe (absolute value of the) difference ∆ between demand and supply (and with sign opposite to thatof ∆). Note that, in this figure, the price change only ranges to 100 and P (R) is in the power-lawrange. Figure from [12].

5 ConclusionIn this seminar, I have presented Mandelbrot's Levy-stable hypothesis, which was a ma-jor breakthrough in �nancial market statistics in the past century. Recent advances ofcomputer technologies and the availability of large amounts of high frequency data hasmade it much easier to analize and study the �nancial markets' statistical properties.These analysis, a few presented in Section 2 of the seminar, show a slight, but importantdisagreement with Mandelbrot's proposal, especially in the distribution's tail behavior.

In the �nal part of the seminar, I have presented a simple and robust model of stockmarket dynamics without tunable parameters that selforganizes into a regime where themost important empirical characteristics of stock market price dynamics are captured.After some changes to the �pure� percolation model are introduced, a good agreementwith empirical data is obtained.

There are, of course, many issues in the �nancial market statistics which have not beenadressed in this seminar, the most obvious being the issue of large �nancial marketcrashes. These events are not accounted for even in the heavy-tailed statistics and theydemand a di�erent approach of studying. It is the focus of many e�orts in the pastdecade to describe and even predict such events.

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References

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[2] J. Peinke, F. Bottcher, St. Barth, Anomalous statistics in turbulence, �nancialmarkets and other complex systems, Ann. Phys. (Leipzig) 13, 450 - 460 (2004)

[3] V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, H. E. Stanley,Econophysics: �nancial time series from a statistical physics point of view, Physica A279, 443 - 456 (2000)

[4] Z. Burda, J. Jurkiewicz, M. A. Nowak, Is Econophysics a Solid Science?,arXiv : cond−mat/0301096v1 [cond-mat.stat-mech](2003)

[5] A. Z. Gorski, S. Droºdº, J. Speth, Financial multifractality and its subtleties: anexample of DAX, arXiv : cond−mat/0205482v1 [cond-mat.stat-mech](2002)

[6] http://en.wikipedia.org/wiki/Levy skew alpha-stable distribution, (12/2008)

[7] J.P. Nolan,Stable Distributions: Models for Heavy Tailed data,http : //academic2.american.edu/ jpnolan/stable/chap1.pdf ,(2008)

[8] http://www.mathestate.com/tools/Financial/s09/LogNormalStable.html, (01/2009)

[9] D. Sornette, D. Stauffer, H. Takayasu, Market Fluctuations II: multiplicativeand percolation models, size e�ects and predictions, arXiv : cond − mat/9909439v1[cond-mat.stat-mech](1999)

[10] R. Albert, A.-L. Barabasi, Statistical mechanics of complex networks,arXiv : cond−mat/0106096v1 [cond-mat.stat-mech](2001)

[11] http://www.cs.princeton.edu/introcs/24percolation, (01/2009)

[12] D. Stauffer, D. Sornette, Self-Organized Percolation Model for Stock MarketFluctuations, arXiv : cond−mat/9906434v1 [cond-mat.stat-mech](1999)

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