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Systems Engineering Procedia 00 (2011) 000–000

Systems Engineering Procedia 3 (2012) 110 – 118

Systems Engineering Procedia www.elsevier.com/locate/procedia

The 2nd International Conference on Complexity Science & Information Engineering

Modeling Complicated Behavior of Stock Prices Using Discrete Self-Excited Multifractal Process Jian Zhong a,b,*, Xin Zhaob a

Chongqing University, Chongqing 400030,China

b

Southwest University, Chongqing 400715,China

Abstract It is usually a challenge for finance and financial engineering to quantitatively describe the complicated behavior of stock prices because the behavior is caused by the adaptability and reflexivity of investors, driven by the endogenous interaction and perturbed by exogenous news. This paper models the behavior of stock prices basing on Self-excited multifractal(SEMF) process and deduces that probability distributions of the model are exponentially tempered Pareto distributions(CGMY model).The maximum likelihood estimator can be used to estimate parameters of CGMY and SEMF. The empirical results demonstrate that the SEMF process have a very good ability to quantitatively describe the complicated behavior of stock prices.

©©2011 Ltd. Selection Selectionand andpeer-review peer-reviewunder underresponsibility responsibility Desheng Dash 2011Published Published by by Elsevier Elsevier Ltd. of of Desheng Dash WuWu. Open access under CC BY-NC-ND license. Keywords: Behavior of stock prices; Self-excited multifractal process; Generalized tempered stable distributions; Exponentially tempered Pareto distributions; Maximum likelihood estimator

1. Introduction The term "behavior of stock prices" was first defined by Fama in 1965[1]. Its connotation actually involves two separate aspects: (1) successive price changes of stock conform to some stochastic process, and (2) the changes conform to some probability distribution[1]. Essence of the term is that behavior of stock prices should be described quantificationally by mathematic model in a scientific fashion. The scientific modeling for the behavior of stock prices is due to Bachelier in 1900[2]. Bachelier assumed that successive price changes of stock are independent and conform to Random Walk or Brownian Motion. So, price changes conform to normal or Gaussian probability distributions[2].Unfortunately Bachelier’s work did not receive much attention from economists until the 1960s. It is well known that the modern classic finance and the modern classic financial engineering originated from the portfolio selection theory presented by Harry Markowitz in 1952[3] and the option pricing theory founded by Fisher Black, Myron Scholes, and Robert Merton in 1973[4].The portfolio selection theory and the option pricing theory are based on two hypotheses, same as Bachelier's ideas: (1)price changes are independent, and (2) the changes conform to Gaussian distributions. The assumptions * Corresponding author. Tel.: 008602368367073; fax: 008602368253297. E-mail address: [email protected].

2211-3819 © 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of Desheng Dash Wu. Open access under CC BY-NC-ND license. doi:10.1016/j.sepro.2011.11.015

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are mathematical elegancy, well-motivated and the theories work reasonably well in practice when the time period is not less than a couple of days. Nevertheless, since at least the early 1960s there have been serious studies beginning to criticize the viewpoint that price changes are independent and conform to Gaussian distributions with observed data, especially daily and intraday price changes, exhibit leptokurtosis, thick tails, skewness, volatility clustering and multifractality, etc.[1][5-12].These statistic properties which deviate from independent and Gaussian distributions are called "stylized facts" or "stylized empirical facts"[11]. How to fit all of stylized facts in a model is of vital importance in finance and financial engineering, because portfolio selection, asset pricing and risk management are based on the model. There are three categories of models for this purpose: (1) stochastic volatility(SV) family models, (2)generalized autoregressive conditional heteroscedasticity(ARCH/GARCH) family models, and (3)multifractal volatility(MV) family models. The SV family models[13][14] or the ARCH/GARCH family models [15][16] can reproduce almost all of stylized empirical facts listed by Cont(2001)[11] except for multifractality. Because the dynamics of financial returns varies with the return period (sampling interval or time scale), from high-frequency data to daily, quarterly or annual data. Empirical studies have pointed out that returns with different time periods differ in their probability distributions. But the ARCH/GARCH and SV family models do not consider the influence of sampling intervals (time scale) in the volatility process. Additionally, the absolute moments of returns have been found to vary as a power law in the return period (Sattarhoff 2011)[17]. MV family models[9][18] can reproduce almost all of stylized empirical facts, and have better ability to forecast future volatility, Value at Risk and trading volume than models of classical finance, including the ARCH/GARCH family and SV family. These results indicate multifractal models are practically relevant. Moreover, the MV family models are of mathematical elegancy and computational parsimony. However, almost all of these MV models considered only external innovations without explicit dependence of future values on the past history of the process. This crucial trait makes them fundamentally unsuitable to model the behavior of stock prices, whose multifractal fluctuations are weakly coupled to external news, seem mostly be driven by reflexivity [19][20]. Filimonov and Sornette (2010) proposed a novel self-excited multifractal (SEMF) model which combined the microscopic origin of the emergent endogenous self-organization properties with macroscopic external innovations. So, the SEMF model has all the stylized facts found in financial time series, which are robust to the specification of the parameters and the shape of the memory kernel[19]. But, there are no any literatures using SEMF process to model the behavior of stock prices. The objectives of this paper are: (1) to discuss the complexity of stock price changes and probe into the cause of the complexity, and (2) to model the behavior of stock prices using the SEMF process. The paper is organized as follows: Section 2 discusses the complexity of the behavior of stock prices and probes into the cause of the complexity. Section 3 deduces that the probability distributions of SEMF process should follow generalized tempered stable distributions (GTSD). Section 4 models the behavior of stock prices using SEMF. Section 5 empirically tests the model on Shanghai composite index and Hang Seng Index (HSI) daily return series. Section 6 presents the conclusions and discusses future work. 2. Complexity of Stock Price Changes and Cause 2.1. Complexity of Stock Price Changes Along with the dissemination of computer and the coming of information epoch, the availability of huge amounts of financial data, especially high frequency data and ultra-high frequency data become possible. The applications of computer-intensive methods for analyzing statistic properties of these data have opened new horizons and contributed to the consolidation of a data-based approach for researchers in empirical finance and financial engineering. There are more and more empirical evidences of stock price changes showing that stock price changes are not independent and Gaussian distributions[1][5-12].

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These empirical evidences of stock price changes do share some quite nontrivial statistical properties, and these properties are consistent in wide range of instruments, markets and time periods. So, these properties are called “stylized facts” or “stylized empirical facts” (Cont 2001)[11]. Nowadays, stylized facts which generally accepted by economists include absence of autocorrelations(If time scales longer than 20 minutes that autocorrelations of signed stock returns are often insignificant), thick tails(The distribution of asset returns seems to display a power-law or Paretolike tail and tail index higher than 2 and less than 5 for most data sets studied), time reversal asymmetry(Large drawdowns in stock prices and stock index values not equally large upward movements), aggregational Gaussianity(As one increases the time scale the asset returns, the distribution looks more and more like a normal distribution. But, the shape of the distribution is not the same at different time scales), intermittency and volatility clustering(At any time scale ,the variability in stock return time series are irregular bursts and different measures of volatility display a positive autocorrelation over several days), slow decay of autocorrelation in absolute returns(Absolute returns are autocorrelations and it decays slowly as a power law with an exponent β∈ [0.2, 0.4]), multifractality(A global scaling property in moment of the stochastic process or measure for stock returns), mean reversion(Most stock returns are negative correlation in long-range time scale), leverage effect(Most measures of volatility of an asset are negatively correlated with the returns of that asset), volume/volatility correlation(Trading volume is correlated with all measures of volatility) and asymmetry in time scales(Coarse-grained measures of volatility predict fine-scale volatility better than the other way round) [9][11].These stylized facts reveal that the behavior of stock prices is nonlinear, dynamic and complicated. So, these stylized facts are representative of the complexity of stock price changes. 2.2. Cause of the Complexity In practice, there are a lot of frictions in the real stock market, such as transaction costs, information costs, barrier mechanism, etc. The investors in the real stock market are multitudinous. They have different endowment of resources, predilection and capacity. So, these investors are not completely rational, have adaptive learning ability, incomplete information and adopt heterogeneous interpretation and decision rules. Moreover, there are a large number of interacting elements in the stock market leading to complex interactions which influence the observable behavior of stock prices or stock indices. These properties and features of stock market are of accordance with complex adaptive system(Holland 1995)[21].Therefore, we can consider the stock market as a complex adaptive system to understand its behavior (Mauboussin 2002)[22].The coherent market hypothesis(CMH)(Vaga 1990)[23] and fractal market hypothesis(FMH)(Peters 1994)[24] have no solid economic theory foundation as EMH, and they reckon without interacting the elements of investors in stock market. On the other hand, the complexity of modeling the CMH and FMH has prevented them from being widely accepted by economists. Lo (2004)[25] proposed Adaptive Markets Hypothesis (AMH) which is based on some well-known principles of evolutionary biology: competition, mutation, reproduction, and natural selection. The AMH can reconcile the EMH with behavioral biases in a consistent and intellectually satisfying manner. Moreover, the AMH has the ability to embody ideas of reflexivity theory which promoted by billionaire investor George Soros (1988)[20]. The AMH has solid economic theory foundation, strong ability to explain the complexity of stock price behavior and is easy to be modeled. So the AMH has great potential to replace the EMH in future. But, the AMH has completely overlooked the role of cooperation in adaptation and how dominant species can suddenly implode. Our opinion is that the stock market is a complex adaptive system. The investors are of bounded rationality, heterogeneous, but self-similar. With adaptive learning ability, they interact with each other as well as economic environment. Competition for existence drives adaptation and innovation, but overadapted or over-innovated investors will become extinct species. These factors led to the complex behavior of stock prices. So, the behavior of stock prices is primarily driven by the endogenous self-

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excited nature and only impacted by external news. The forming mechanism of stock price behavior is indicated in figure 1. Government

Investors

Price

News

Listed companies Fig. 1. Forming mechanism of stock price behavior

3. Self-Excited Multifractal Process and Generalized Tempered Stable Distributions 3.1. Self-Excited Multifractal Process The multifractal volatility models are based on an assumption that financial data share statistical properties with turbulent intermittent velocity fields which areas of rapid and violent activities alternate with more peaceful ones, and this phenomenon repeats itself at any time-scale in the ”same” way(Bacry et al. 2010)[26]. Multifractality of stock prices behavior was defined as a global scaling property in moment of the stochastic process or measure for stock returns(Mandelbrot et al. 1997)[9].Today, a few of multifractal volatility models had been proposed to fit financial assets returns, such as, Multifractal Model of Asset Returns (MMAR)[9],Poisson Multifractal Model (PMM)[27], Multifractal Random Walk(MRW) [18], Discrete-time Skewed Multifractal Random Walk (DSMRW) Model[28], Markov Switching Multifractal(MSM)[29][30],Quasi-multifractal(QMF) Model[31][32], Continuous Skewed Multifractal Processes(CSMP)[26]. The results of empirical tests (Sattarhoff 2011)[17] demonstrate that these models can reproduce almost all of stylized empirical facts ,and have better ability to forecast future volatility, Value at Risk and trading volume than models of classical finance, including the ARCH/GARCH family and SV family. So, the multifractal models are of practical relevance. Moreover, these models are of mathematical elegancy and computational parsimony. However, almost all of these models use only external innovations without taking endogenous self-excited nature into consideration. This crucial trait makes them fundamentally unsuitable to model the behavior of stock prices. Filimonov and Sornette(2010)[19] proposed a novel self-excited multifractal (SEMF) model which combined the microscopic origin of the emergent endogenous self-organization properties with macroscopic external innovations. The SEMF model considers that the amplitudes of the increments of the process are expressed as exponentials of a long memory of past increments. The principal novel feature of the model lies in the self-excitation mechanism combined with exponential nonlinearity. Let discrete t=0,1,2,…,n,…, The SEMF model in discrete time is as follows: i −1

X n = ∑ i = 0 σξi exp{− n

∑X j =0

h

j i − j −1

(1) } σ where ξi represent an external noise which can be taken i.i.d. Gaussian with zero-mean and unit variance. The parameter σ sets the impact amplitude of the external noise, as well as the dimension and scale of X n .Its value determines the time scale. The sum in the exponential expresses the fact that the amplitude of the next increment of the SEMF process is strongly determined by its past realizations, weighted by the

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memory

kernel

hi ≥ 0,= i 0,..., n − 1 , hi

is

power-law

memory

kernel

2 , hn h0 exp(−φ n) , hn ≡ h0 ,where ϕ , φ and h0 are positive constants. hn = h0 n −ϕ −1=

function,

such

as

The SEMF process could be regarded as the result of combination of multifractal stochastic volatility models, such as QMF, and an autoregressive (ARCH-family) models. The self-excitation of SEMF captures the microscopic origin of the emergent endogenous self-organization properties, e.g. the “reflexive” interactions of financial markets. The SEMF process has all the standard stylized facts found in financial time series. Moreover, the SEMF is robust to the specification of the parameters and the shape of the memory kernel. 3.2. Generalized Tempered Stable Distributions Filimonov and Sornette(2010)[19] proved that the SEMF process can reproduce all the standard stylized facts found in financial time series, such as thick tails, aggregational Gaussianity, volatility clustering, multifractality, leverage effect, etc. The “Aggregational Gaussianity” means that if we increases the time scale of the asset returns, the probability distribution will more and more incline towards a normal distribution, but, the shape of the probability distribution is not the same at different time scales(Cont 2001)[11]. This distribution characteristic is consistent with generalized tempered stable distributions (Sinclair 2009)[33]. On the other hand, the empirical results of probability density function (PDF) for every power-law memory kernel function of SEMF have three regimes:(1) a plateau for small d, 1 1 (2) f d (d )  γ when 2σ  d  20σ , where 2  γ  6 , and (3) f d (d )  α , where α  1 and d log β d d

β  2 (Filimonov and Sornette 2010)[19]. These PDF regimes imply that the SEMF process seems to follow a generalized tempered stable distribution. Integrating the reason of the two aspects, the SEMF process should follow generalized tempered stable distributions. The generalized tempered stable distributions are distributions which encompass variations on tempered stable distributions[34], such as exponentially tempered Pareto distributions (the CGMY model)[35], KR tempered stable distribution[36], etc. A tempered stable distribution combines both the α − stable and Gaussian trends. In a short time frame it is close to a α − stable distribution while in a long time frame it approximates a Gaussian distribution[37]. The α − stable distribution, or Lévy stable distribution for financial asset returns was pioneered by Mandelbrot in 1963[6] and then tested and verified by Fama in 1965[1].The empirical results of Fama[1] demonstrated that stock prices seem to follow stable Paretian distributions with characteristic exponent α < 2 .Since then, the α − stable laws are widely used in economics. But α − stable distribution does not have a p − th moment for p ≥ α .Moreover, α − stable probability density functions (PDF) and cumulative distribution functions (CDF) do not have closed form expressions except for few special cases(Gaussian,Cauchy and Lévy process).Unlike α − stable distributions, generalized tempered stable distributions have all the moments finite and "semi-heavy tails", but their PDF and CDF don't have closed form expressions and the problem of parameter estimation for generalized tempered stable laws are remains except for few special cases. The exponentially tempered Pareto distributions are special cases of generalized tempered stable distributions. They have closed form expressions of PDF and CDF. Their parameter can be estimated by maximum likelihood estimator(MLE), and they have a good ability to fit stock returns(Meerschaert,Roy and Shao 2010)[38]. 4. Modeling Behavior of Stock Prices Filimonov and Sornette(2010)[19] consider the SEMF process can be used in modeling for hydrodynamic turbulence(velocity increments and energy dissipation), seismicity(stress field and earthquake triggering), biology (healthy human heart-beat rhythm), hydrology(river runoffs) and financial systems(asset returns). But, they did not give an explicit model for financial systems or stock market.

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Let {P(t )} is a price time-series of stock, r (t ) = log P (t ) − log P (t − 1) as logarithmic stock returns or logarithmic increments of the price. The discrete time (t = 1,2,...,n,...) Self-Exciting Multifractal (SEMF) model of stock price behavior has the following expression: P(t ) 1 t −1 (2) r (t ) log P (t ) − log P (= t − 1) log = σξ (t ) exp(− ∑ r (τ )h(t − τ − 1)) = P (t − 1) σ τ =0 where ξ (t ) are random variables which represent an external flow of news. ξ (t ) are i.i.d. Gaussian with zero-mean and unit variance and either positive or negative, it controls the signs of the returns. σ is a scaling parameter whose value determines the time scale. The amplitude (or volatility) of stock returns are then determined by all past returns with a decaying weight as a function of their distance to the present. The decaying weight function is power-law memory kernel function such as hn ≡ h0 2 , hn h0 exp(−ϕ n) , where ϕ , ϕ and h0 are positive constants. hn = h0 n −ϕ −1 = In this paper, we use only hn = h0 n −ϕ −1 2 as power-law memory kernel function to model stock price behavior. Other power-law memory kernel functions will be researched and compared elsewhere. So, the full parameter vector of stock price behavior in SEMF model is ψ ≡ (σ ,ϕ , h0 ) ∈ R+3 . Filimonov and Sornette (2010)[19] considered the maximum likelihood approach was appropriate to estimate these parameters. But, they had not presented specific method to estimate these parameters because they did not give an explicit probability density function for SEMF. As analyzed above, it is appropriate to take the exponentially tempered Pareto distributions as probability distribution of SEMF. Suppose r (0), r (1), r (2), , r (T ) are logarithmic stock returns and follow the tempered Pareto distribution. The CDF of logarithmic stock returns should be as follows: (3) F r (t ) (r ;θ= ) P{r (t ) > = r} γ r − a e − β r , r ≥ r (0) where γ is a scale parameter, a controls the power law tail, and β governs the exponential truncation. The corresponding density function is as follows: (4) f r ( r ;θ ) = r ≥ r (0) γ r − a −1e − β r (a + r β ) , The parameter vector θ ≡ (α , β , γ ) ∈ R+3 can be estimated by conditional MLE (Meerschaert, Roy and Shao 2010)[38].The parameter vector of stock price behavior in SEMF model is ψ ≡ (σ ,ϕ , h0 ) ∈ R+3 .The estimated value of θ ≡ (α , β , γ ) can be used to estimate the parameter ψ of SEMF by MLE and the loglikelihood function is: T

T

T

t =1

t =1

t =1

)  r (t ) + ln( a + r (t ) β ln L(r (1), , r (T );ψ= ) T ln γ − (a + 1)∑ lnr (t ) − β ∑ ∑

where = r (t ) σξ (t ) exp(−

1

t −1

∑ r (τ )h(t − τ − 1)) .Because στ

(5)

r (t ) have self-excited mechanism, we program in

=0

MATLAB to estimate the parameter ψ . 5. Empirical Test We employ Shanghai composite index daily return series of Chinese stock market and Hang Seng Index (HSI) daily return series of Hong Kong to empirically test the model. All data got from RESSET Financial Research Database (http://www.resset.cn/).Shanghai composite index daily return series range from January 4, 1993 to April 1, 2011, including 4,451 observations (weekends and holidays are excluded). HSI Index daily return series starts on January 2, 1987 and ends on April 1, 2011. There are 6,026 observations (weekends and holidays are excluded).We only empirically test the power-law kerne hn = h0 n −ϕ −1 2 function. The empirical test and comparison of other power-law kernel function will be reported elsewhere. The empirical results can be seen in Figure 2 to Figure 5. The results demonstrate that the SEMF process has very good ability to reproduce all standard stylized facts of stock returns.

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Shanghai Composite Index Daily Return

0.5 0.4

Logarithmic Return

0.3 0.2 0.1 0 -0.1 -0.2

0

500

1500

1000

2000

2500

3000

3500

3000

3500

4000

4500

Fig. 2. Daily returns of Shanghai composite index(1/4/1993-4/1/2011) Reproductive Return by SEMF Process

0.5

0.4

Logarithmic Return

0.3

0.2

0.1

0

-0.1

-0.2

0

500

1000

1500

2000

2500

4000

4500

Fig. 3. Reproductive return of Shanghai composite index by SEMF model from 1/4/1993 to 4/1/2011. Using powerare σ 0.012, law kernel hn = h0 n −ϕ −1 2 and estimated parameters values by the method= h0 0.046 . = ϕ 0.009, = Hang Seng Index(HSI) Daily Return

0.3 0.2

Logarithmic Return

0.1 0 -0.1 -0.2 -0.3 -0.4

0

1000

2000

3000

4000

5000

6000

7000

5000

6000

7000

Fig. 4. Daily returns of HIS (1/ 2/1987-4/1/2011). Reproductive Return by SEMF Process

0.3

0.2

Logarithmic Return

0.1

0

-0.1

-0.2

-0.3

-0.4

0

1000

2000

3000

4000

Fig. 5. Reproductive return of HSI by SEMF model from 1/2/1987 to 4/1/2011. Using power-law kernel are σ 0.0098, ϕ 0.012, h0 0.051 . = = = hn = h0 n −ϕ −1 2 and estimated parameters values by the method

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6. Conclusion Modeling behavior of stock prices is of vital importance in finance and financial engineering, because portfolio selection, asset pricing and risk management are based on the model. However, the stock market is a complex adaptive system. The investors are of bounded rationality, heterogeneous, but self-similar, have adaptive learning ability, they interact with each other as well as the economic environment. These factors make the behavior of stock prices nonlinear, dynamic and complicated. Therefore, to quantitatively describe the behavior of stock prices is a challenge in finance and financial engineering. A good model of stock prices behavior must have solid economic theory foundation and be practically relevant, namely, the model can reproduce all stylized facts, can explain the economic cause of formation, and can guide the practical application. Moreover, the model should be mathematically elegant and computational parsimony. The self-excited multifractal model of stock prices behavior could be regarded as the result of combination of multifractal stochastic volatility models and autoregressive (ARCH-family) models. The model considers fully endogenous driving factors and exogenous disturbance factors of stock price changes. The probability distribution of the model is tempered stable distributions. Moreover, the model is robust to the specification of the parameters and the shape of the memory kernel. These parameters of the model can be estimated by maximum likelihood estimator easily. The empirical results demonstrate that self-excited multifractal model of stock prices behavior has outstanding ability to reproduce all standard stylized facts of stock returns. So, the self-excited multifractal model of stock prices behavior and it’s extend models have extraordinary advantage and glamour to economists. This paper is only the seminal study about modeling behavior of stock prices basing on SEMF process. A lot of aspects, such as more detailed statistic properties, more accurate probability distributions, more extensive empirical test, more effective parameters estimator and methods of practical application, should be intensively studied in future. Acknowledgements The authors would like to thank the anonymous reviewers for helpful suggestions. This research was partially supported by the National Social Science Foundation of China (the id of projects are 08AJY007 and 08AJY030). References [1] Fama, E.F. The behaviour of stock market prices.Journal of Business;1965,38, pp.34-105. [2] Bachelier, L.Théorie de la Spéculation. Thesis (PhD). English translation in Cootner;1964. [3] Harry M. Markowitz.Portfolio Selection.The Journal of Finance;March, 1952, pp. 77 -91. [4] Black, Fischer and Myron Scholes.The Pricing of Options and Corporate Liabilities.Journal of Political Economy, Vol. 81, No. 3;May/June 1973, pp. 637-654. [5] Kendall, M.G. The Analysis of Economic Time-Series. Journal of the Royal Statistical Society; ,1953,96, pp.11-25. [6] Mandelbrot, B. The Variation of Certain Speculative Prices. Journal of Business;1963, 36,pp.307-332. [7] Lo, A. W., MacKinlay, A.C.Stock market prices do not follow random walks: Evidence from a simple specification test.Review of Financial Studies;1987,1, pp.41-66. [8] Su, D., Fleisher, B.M.,Risk.Return and Regulation in Chinese Stock Markets.Journal of Economics and Business;1998, 50 (3), pp.239-256. [9] Mandelbrot, B.B., Calvet, L., Fisher, A. A Multifractal Model of Asset Returns.Working Paper,Yale University. Cowles Foundation Discussion Paper #1164,Available from:http://users.math.yale.edu/~bbm3/web_pdfs/Cowles1164.pdf;1997. [10] Plerou, V. Scaling of the distribution of price fluctuations of individual companies.Physical Review ; 1999, E 60: 65196529.

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