Data Analysis and Statistical Behaviors of Stock Market Fluctuations


 Reynold Carpenter
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1 44 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Data Aalysis ad Statistical Behaviors of Stock Market Fluctuatios Ju Wag Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Bigli Fa ad Dogpig Me Departmet of Mathematics, Beijig Jiaotog Uiversity, Beijig 00044, Chia Abstract I this paper, the data of Chiese stock markets is aalyzed by the statistical methods ad computer scieces. The fluctuatios of stock prices ad trade volumes are ivestigated by the method of Zipf plot, where Zipf plot techique is frequetly used i physics sciece. I the first part of the preset paper, the data of stocks prices ad trade volumes i Shaghai Stock Exchage ad Shezhe Stock Exchage is aalyzed, the statistical behaviors of stocks prices ad trade volumes are studied. We select the daily data for Chiese stock markets durig the years , by aalyzig the data, we discuss the statistical properties of fat tails pheomea ad the power law distributios for the daily stocks prices ad trade volumes. I the secod part, we cosider the fat ails pheomea ad the power law distributios of Shaghai Stock Exchage Idex ad Shezhe Stock Exchage Idex durig the years , ad we also compare the distributios of these two idices with the correspodig distributios of the Zipf plot. Idex Terms data aalysis, statistical methods, Zipf method, statistical properties, computer simulatio, market fluctuatio I. INTRODUCTION The stock prices ad the trade volumes play a importat role i the market fluctuatios i a stock market. I this paper, we focus our attetio o the statistical properties of esembles of the stock prices ad the trade volumes. I the first part of the preset paper, usig 443 stocks traded i Shaghai Stock Exchage (SHSE) ad Shezhe Stock Exchage (SZSE) durig the years , we formed esembles of daily stock prices ad daily trade volumes. The database which used i the preset paper is from the websets of Shezhe Stock Exchage ad Shaghai Stock Exchage ( Cosiderig the history of fiacial situatio of Chiese stock markets, the daily price limit (ow 0%), the tradig rules of the two stock markets, ad the fiacial policy of Chiese govermet, we select the data of the daily closig price (for each tradig day) for each stock coverig the recet 5year period durig the year , the total umber of observed data for the stocks prices ad the trade volumes is about I the secod part of the preset paper, we cosider the returs of the fat tails pheomea ad the power law distributios of Shaghai Stock Exchage Idex ad Shezhe Stock Exchage Idex. We select the data for 5 miutes from 9:30 (the opeig time of each tradig day i Chia) at February 3, 2002 to 5:00 (the closig time of each tradig day) at April 7, The total umber of observed data is for SHSE idex ad for SZSE idex. The we study the statistical properties of returs for SHSE idex ad SZSE idex. Recetly, some research work has bee doe to ivestigate the statistical properties of fluctuatios of stock prices i a stock market, see [,2,3,5,6,7,9,0,]. Their work shows that the fluctuatios of price chages are believed to follow a Gaussia distributio for log time itervals but to deviate from it for short time steps, especially the deviatio appears at the tail part of the distributio, usually called the fattails pheomea. The empirical research has show the powerlaw tails i price fluctuatios ad i trade volume fluctuatios. The study o powerlaw scalig i fiacial markets is a active topic for physicists to uderstad the distributio of fiacial price fluctuatios. I the preset paper, Zipf plot method of statistical aalysis is itroduced to study the market fluctuatios. The techique, kow as a Zipf plot, is a plot of log of the rak vs. the log of the variable beig aalyzed. Let ( x, x2, L, x N ) be a set of N observatios o a radom variable x for which the cumulative distributio fuctio is F( x ), ad suppose that the observatios are ordered from the largest to the smallest so that the idex i is the rak of x i. The Zipf plot of the sample is the graph of l x i agaist l i. Because of the rakig, in= Fx ( i ), so l i = l[ F( x )] + l N. Thus, the log of the rak is simply a trasformatio of cumulative distributio fuctio. For example, i studyig Eglish word occurrece frequecy, it was foud that if the words have the descedig orders of frequecy, the frequecy of occurrece of each word ad i
2 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER its symbol rakig has simple iverse relatios, that is Pr () = cr b. Makig a trasformatio, the above equatio ca be coverted ito l Pr ( ) = l c bl r, where Pr () is the frequecy of the word whose rak is r. Plottig the graph by l Pr ( ) agaist l r, the graph is close to a lie with the slope of b. Zipfs law describes that oly a few Eglish words are ofte used, most of the words are rarely used. I recet years, Zipfs law has bee widely applied to the literature, computers, etworks, maagemet, oil, ad may other fields. I the preset paper, Zipf plot is applied to study tail pheomea of market fluctuatios i the Chiese stock markets, i particularly, the prices of stocks itegrated with trade volumes are studied by the statistical aalysis. II. ZIPF PLOT OF QUOTING DATA FROM CHINESE STOCKS PRICES I this sectio, we discuss esemble of 443 stocks daily prices of Chiese stock markets o September 8, The theory of statistical method ad computer simulatio is applied i the followig sectios, see [4,8,2]. Amog these 443 stocks, deote that the stocks prices i descedig order, that is, S() is the stock price with the highest price, S(2) is the stock with the secod highest price, ad S() is the th highest price, =, L,443. Figure is the plot of S() agaist o a double logarithmic scale, that is the Zipf plot of stocks prices. log(s ()) β= < β <0.56. Note that the threshold value p 0 depeds o the tradig dates. Figure 2 shows the Zipf plot of the trade volumes of Chiese stocks. I Figure 2, the Zipf plot for the trade volumes higher tha shows a straight lie ad is well described by C ( ) α with the parameter α =0.77 ( C() ), =, L,443. By the similar procedure ad aalysis, we cosider the daily trade volumes of the 443 Chiese stocks from 2002 to 2006 (totally 205 tradig days i 5 years). Similarly, we ca show that the daily trade volumes of each tradig day higher tha a certai threshold value q 0 subject to a distributio C ( ) ( () ()), where the rage α C q0 of the threshold value of the daily parameter α is 0.65<α <.2. log(c()) α= log() Figure 2. Zipf plot of daily trade volumes of 443 Chiese stocks o September 8, The horizotal axis shows logarithm of the stock order log( ), the vertical axis shows logarithm of stock trade volume log( C ( )). III. THE STATISTICAL CHARACTERS OF PARAMETERS β AND α I this sectio, for each tradig day, we discuss the properties of the parameters β ad α. The followig Figure 3 ad Figure 4 show the fluctuatios of the values β ad α log() Figure. Zipf plot of daily prices of 443 Chiese stocks o September 8, The horizotal axis shows logarithm of the stocks order log( ), the vertical axis shows logarithm of the daily price of stock log( S ( )). β I Figure, the Zipf plot for the stocks prices higher tha 2.72 RMB shows a straight lie ad is well described by S ( ) β with the parameter β = 0.47 ( S() 2.72). By the similar procedure ad aalysis, we cosider the daily prices of the 443 Chiese stocks from 2002 to 2006 (totally 205 tradig days i 5 years). We ca show that the daily prices of each tradig day higher tha some threshold value p 0 subject to a distributio S ( ) β ( S() p0 ()), where the rage of the threshold value of the daily parameter β is t Figure 3. The fluctuatios of the power law expoet β i the 5year period , correspodig to each tradig day. The vertical axis idicates the power law expoet value β, ad the horizotal axis idicates the correspodig tradig dates. Figure 3 shows the movemet of the power law expoet β with respect to the tradig dates from 2002
3 46 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 to 2006, ad shows the value rage of the parameters β. This ca reflect the activity ad the tred of Chiese stock markets i large degrees, so this will be helpful for us to uderstad the status of Chiese macroecoomic. With the reformatio ad developmet of Chiese ecoomic systems, the Chiese stock markets develop rapidly, ow the movemet of stocks prices has a strog ifluece o the ecoomic behavior of idividuals ad firms, as a result, it affects the ecoomic developmet of Chia directly. I recet years, Chiese stock markets expad quickly, from Jue 2006 to December 2006, about 65 ew compaies were listed i Shaghai Stock Exchage ad Shezhe Stock Exchage. Accordig to Zipflaw of five years from 2002 to 2006, the icreasig umber of Chiese stocks is oe of mai factors o the icreasig tred of parameters β. The other reaso of parameters β s icreasig i Figure 3 is that, the degree of the disparity amog stocks prices becomes lower as Shaghai Composite Idex ad Shezhe Composite Idex declie from 2002 to Shaghai Composite Idex ad Shezhe Composite Idex are ad respectively o Jauary 4, 2002, ad ad respectively December 3, Shaghai Composite Idex ad Shezhe Composite Idex are declied 7.52% ad 7.03% respectively i the year From Figure 3, we ca see that the value β fluctuates calmly i From , Chiese stock markets develop rapidly, Shaghai Composite Idex ad Shezhe Composite Idex rose by 30%. So i Figure 3, the macro parameters β was icreasig, i particularly, the parameter β was substatially icreasig i From above aalysis o the parameter β, this implies that β ca reflect the market fluctuatios i some scope. α t Figure 4. The fluctuatios of the power law expoet α i the 5year period , correspodig to each tradig day. The vertical axis idicates the power law expoet value α, ad the horizotal axis idicates the correspodig tradig dates. I the above Figure 4, we cosider the movemet of the parameter α of the trade volumes by Zipf method. I Chiese stock markets durig the years , the fluctuatio of the parameter values α is a relatively stable. The degree of the disparity amog stocks trade volumes is relatively stable. This implies that the trade volumes of Chiese stock markets had o otable chages from 2002 to IV. THE STATISTICAL PROPERTIES OF PARAMETERS β AND α I this sectio, we cotiue to discuss the statistical properties of the power law expoets β, α defied i above Sectio 2 ad Sectio 3. The ormalized parameters are defied as β mea( β ) β =, std( β ) α mea( α) α = std( α) where mea( β ) ad mea( α ) are the meas of β ad α respectively, std( β ) ad std( α) are the stadard deviatios of β ad α respectively. Skewess ad kurtosis are the importat statistics used to describe the data distributios. Skewess describes the distributio patters of symmetry. Whe the value of skewess equals zero, the distributio patter is symmetric; whe the value of skewess is greater tha zero, the distributio patter is positive skewess; ad whe the value of skewess is less tha zero, the distributio patter is egative skewess. The greater the absolute value of skewess is, the greater the degree of deviatio is. Kurtosis is used to describe the steep degree of distributio. If the steep degree of the data distributio is same as that of stadard ormal distributio, the value of kurtosis equals to 3. Whe the kurtosis value is bigger tha 3, the the steep degree of the distributio is bigger tha that of stadard ormal distributio, the distributio is peak. O the cotrary, the distributio is plai. The mathematical defiitios of skewess ad kurtosis of the vector X = ( x, x2, L, x ) are 3 3 Skewess = ( xi x) S i= 4 4 Kurtosis = ( xi x) S i= where x is the mea of the vector X = ( x, x2, L, x ), ad S is the stadard deviatio of the vector X = ( x, x2, L, x ). TABLE I. PROPERTIES OF THE NORMALIZED PARAMETERS Miimum maximum skewess kurtosis β α Accordig to the observed tradig data of Chiese stock markets, we have Table. From Table, the skewess value of the parameter β is 0.58, it is bigger tha zero. The distributio of the parameter β has positive skewess. The value of skewess of the parameter α is , close to zero. The values of kurtosis show that, comparig with the ormal distributio, the distributios of parameters β, α are plai. This implies that the statistical distributios of the observed data deviate from the Gaussia distributio i some parts.
4 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER V. THE STATISTICAL PROPERTIES OF PARAMETERS DIFFERENCES I this sectio, we cotiue to study the statistical properties of the parameters β ad α, i details that, the properties of the parameters differeces β ad α. We choose the same markets data i the five years from 2002 to 2006 as i Sectio 4, ad let the differeces be β = β+ β, α = α+ α, for =, 2, L,204. The the ormalized parameters are defied as β mea( β ) α mea( α) β =, α = std( β ) std( α) where mea( β ) is the mea of β, std( β ) is the stadard deviatio of β, mea( α) is the mea of α, std( α) is the stadard deviatio of α. Next, accordig to the five years data of Chiese stock markets, we study the cumulative probability distributio of β ad α, that is, P( β > x), P( α > x). By the computer simulatio, we plot the cumulative probability distributios o the double logarithmic scale, see the Figure 5. Figure 5. The plot of the cumulative probability distributios of ad P(X>x) t=2.75( β) α uder the double logarithmic scale. t=2.9( α) x β From above Figure 5, we ca see clearly that the tail distributio of the cumulative probability distributio of t β follows the distributio P( β > x) x, (that is the powerlaw distributio), where t = The tail of the cumulative probability distributio of α also t follows the powerlaw distributio P( α > x) x, where t = 2.9. The values of the expoet parameter t decide the level of the fluctuatios of the correspodig parameter differeces, the research work o the values of the parameters t is a mai part i the study of the fluctuatios of the parameters. The smaller the expoet value t is, it meas that the more fluctuatios the correspodig parameter differece ( β or α ) has. I preset paper, α represets the disparity amog the degrees of chages for the tradig volumes i each tradig day, β represets the disparity amog the degrees of chages for the stock prices i each tradig day. Figure 5 shows the properties of fluctuatios of β ad α. This work may help for us to uderstad the statistical properties of fluctuatios i Chiese stock markets. VI. THE RETURNS OF SHSE INDEX AND SZSE INDEX BY ZIPF PLOT I this sectio, accordig to 5 miutes data from 9:30 at February 3, 2002 to 5:00 at April 7, 2007, we cosider the returs of SHSE idex ad SZSE idex by Zipf plot method. Chia has two stock markets, Shaghai Stock Exchage ad Shezhe Stock Exchage, ad the idices studied i the preset paper are Shaghai Composite Idex ad Shezhe Composite Idex. These two idices play a importat role i Chiese stock markets. The database is from Shaghai Stock Exchage ad Shezhe Stock Exchage, see ad For a price time series Pt (), the retur rt () over a time scale () t is defied as the forward chage i the logarithm of () t, for t =, L, rt ( ) = l( Pt ( + t)) l( Pt ( )). Now we give a ew ormalized method for the returs of rt (). The followig sequece is ordered from the largest value to the smallest oe, that is the order statistics ((), r L,()) r. Let med() r be the media value of the vector ((), r L,()) r, ad defie Ri () = ri () medr ( ), i =, 2, L. () O a double logarithmic scale, where R ( ) deotes the returs i descedig order, R () the retur with the highest value, R (2) the retur with the secod highest value, ad so o. I the followig, we show the compariso of distributios betwee the returs of SHSE idex (SZSE idex) ad the correspodig ormal radom variable, here we suppose that the returs ad the ormal radom variable have the same mea ad variace. Figure 6. The top curve is a Zipf plot, the double logarithmic plot of returs of SHSE idex vs. rak. The bottom curve is a Zipf plot for the correspodig logormal distributio.
5 48 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER 2008 Figure 6 shows the Zipf plot of returs alog with the Zipf plot for the log ormal. The Zipf plot suggests that the log ormal fits the distributio of returs well i some parts. However, i other parts of Figure 6, the Zipf plot makes clear that the returs with the large value are bigger tha the correspodig values of log ormal, or the Zipf plot of returs lie above the Zipf plot of ormal. More specifically, with the aid of Zipf plot, the deviatios from the log ormal ca be see clearly i Figure 6. First, o the right side of Figure 6, a good fittig distributios ca be see. The mai part of deviatio is, however, that the tail parts of the distributios o the left side of the graph. This deviatio from log ormality is statistically sigificat. From Figure 6, the fat tails pheomea ca be see clearly for the returs of SHSE idex, this shows that the distributio of the returs deviates from the Gaussia distributio i the tail parts. Figure 7 shows the Zipf plot of returs for SZSE idex alog with the Zipf plot for the log ormal. The Zipf plot shows that the two curves separate obviously o the left side of the graph. Figure 7 shows the similar statistical properties of returs for SZSE idex as that of returs for SHSE idex. Figure 8. The double logarithmic Zipf plot of rak returs { R(), L, R ( )} of SHSE durig the years By the observed data of Shaghai Stock Exchage durig the years , we plot the double logarithmic Zipf plot of rak returs { R(), L, R( )} i Figure 8. The diamod lie deotes the positive returs sequece, ad the circle lie deotes the egative returs sequece. Figure 8 displays that the distributios of the positive returs ad the egative returs follow the power law distributio. For the positive returs, the expoet is with the cofidece iterval of [0.4266, ], where the sigificat level is For the egative returs, the expoet is with the cofidece iterval of [0.3767, ]. Figure 7. The top curve is a Zipf plot, the double logarithmic plot of returs of SZSE idex vs. rak. The bottom curve is a Zipf plot for the correspodig logormal distributio. VII. THE EMPIRICAL ANALYSIS OF POWER LAW DISTRIBUTION OF SHSE INDEX AND SZSE INDEX BY ZIPF PLOT I this sectio, for SHSE idex ad SZSE idex, we study the power law distributios of returs sequece { R(), L, R( )} which is obtaied by Zipf method, see the defiitio () i Sectio VI. I order to show the power law distributios of these two Chiese idices, we eed to make some adjustmet i Figure 6 ad Figure 7, that is, we exchage the horizotal axis with the vertical axis. The applyig the theory of liear regressio fuctio, we aalyze the observed data, further we estimate the coefficiet of determiatio. We are more cocered about that if this liear regressio fuctio is a good fit for the observed data, if the returs sequece { R(), L, R( )} follows the power law distributio. I the followigs, we give two figures Figure 8 (SHSE idex) ad Figure 9 (SZSE idex) to show the power distributio of returs sequece { R(), L, R( )}. Figure 9. The double logarithmic Zipf plot of rak returs { R(), L, R ( )} of SZSE durig the years Similarly to above Figure 8, by the observed data of Shezhe Stock Exchage durig the years , we plot the double logarithmic Zipf plot of rak returs { R(), L, R( )} i Figure 9. The diamod lie deotes the positive returs sequece, ad the circle lie deotes the egative returs sequece. Figure 9 displays that the distributios of the positive returs ad the egative returs follow the power law distributio. For the positive returs, the expoet is with the cofidece iterval of [0.463, 0.485], where the sigificat level is For the egative returs, the expoet is with the cofidece iterval of [0.3655, ].
6 JOURNAL OF COMPUTERS, VOL. 3, NO. 0, OCTOBER CONCLUSION The objective of this research is to ivestigate the power law behavior ad the fat tails pheomea of Chiese stock markets. Some research work has bee doe i [5,9] for Chiese stock markets. I this paper, we cotiue the research work by Zipf plot method. ACKNOWLEDGMENT The authors are supported i part by Natioal Natural Sciece Foudatio of Chia Grat No , BJTU Foudatio No.2006XM044. The authors would like to thak the support of Istitute of Fiacial Mathematics ad Fiacial Egieerig i Beijig Jiaotog Uiversity, ad thak Z. Q. Zhag ad S. Z. Zheg for their kid cooperatio o this research work. REFERENCES [] J. Elder, A. Serletis, O fractioal itegratig dyamics i the US stock market, Chaos, Solitos & Fractals, vol. 34, pp , [2] P. Gopikrisha, V. Plerou, H.E. Staley, Statistical properties of the volatility of price fluctuatios, Physical Review E, vol. 60, pp , 999. [3] B. Hog, K. Lee, J. Lee, Power law i firms bakruptcy, Physics Letters A, vol. 36, pp. 68, [4] K. Iliski, Physics Of Fiace: Gauge Modelig i Noequilibrium Pricig, Joh Wiley & Sos Ltd., 200. [5] M. F. Ji ad J. Wag, Data Aalysis ad Statistical Properties of Shezhe ad Shaghai Lad Idices, WSEAS Trasactios o Busiess ad Ecoomics, vol. 4, pp , [6] T. Kaizojia, M. Kaizoji, Power law for esembles of stock prices, Physica A, vol. 344, pp , [7] A. Krawiecki, Microscopic spi model for the stock market with attractor bubblig ad heterogeeous agets, Iteratioal Joural of Moder Physics C, vol. 6, pp , [8] D. Lamberto, B. Lapeyre, Itroductio to Stochastic Calculus Applied to Fiace, Chapma ad Hall, Lodo, [9] Q. D. Li ad J. Wag, Statistical Properties of Waitig Times ad Returs i Chiese Stock Markets, WSEAS Trasactios o Busiess ad Ecoomics, vol. 3, pp , [0] T. H. Roh, Forecastig the volatility of stock price idex, Expert Systems with Applicatios, vol. 33, pp , [] J. Wag ad S. Deg, Fluctuatios of iterface statistical physics models applied to a stock market model, Noliear Aalysis: Real World Applicatios, vol. 9, pp , [2] J. Wag, Stochastic Process ad Its Applicatio i Fiace, Tsighua Uiversity Press ad Beijig Jiaotog Uiversity Press, Beijig, 2007.
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