Research Article Statistical Behavior of a Financial Model by Lattice Fractal Sierpinski Carpet Percolation

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1 Joural of Applied Mathematics Volume 2012, Article ID , 12 pages doi: /2012/ Research Article Statistical Behavior of a Fiacial Model by Lattice Fractal Sierpiski Carpet Percolatio Xu Wag ad Ju Wag Departmet of Mathematics, Key Laboratory of Commuicatio ad Iformatio System, Beijig Jiaotog Uiversity, Beijig , Chia Correspodece should be addressed to Ju Wag, wagjubjtu@yeah.et Received 5 September 2011; Accepted 10 November 2011 Academic Editor: Chei-Sha Liu Copyright q 2012 X. Wag ad J. Wag. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. The lattice fractal Sierpiski carpet ad the percolatio theory are applied to develop a ew radom stock price for the fiacial market. Percolatio theory is usually used to describe the behavior of coected clusters i a radom graph, ad Sierpiski carpet is a ifiitely ramified fractal. I this paper, we cosider percolatio o the Sierpiski carpet lattice, ad the correspodig fiacial price model is give ad ivestigated. The, we aalyze the statistical behaviors of the Hog Kog Hag Seg Idex ad the simulative data derived from the fiacial model by compariso. 1. Itroductio Fiacial fluctuatio system is oe of complex systems, ad the statistical behavior of fluctuatio of stock price chages has log bee a focus of fiacial research. With the flourishig research of complex systems, it becomes more ad more attractive to fid uiversal rules ad priciples of these systems ad further to aswer the origiatio of fiacial complex system. Recet research is o loger restricted to the traditioal areas but cocetrated o the more comprehesive domais, leadig to the birth of may burgeoig disciplies through the iteractio ad amalgamatio of mathematics ad other fields such as fiace, biology, ad sociology. For example, the theory of stochastic iteractig particle systems see 1 6 recetly has bee applied to study the behaviors of market fluctuatios, see Ad the study of fiacial market prices has bee foud to exhibit some uiversal properties similar to those observed i iteractig particle systems with a large umber of iteractig uits. Percolatio theory, as a model i iteractig particle systems for a disordered medium, has brought ew uderstadig ad techiques to a broad rage of topics i

2 2 Joural of Applied Mathematics ature ad society. First we cosider the bod percolatio o Z d,thatis,forx, y Z d,the distace δ x, y from x to y is defied by δ x, y d i 1 x i y i, where x x 1,...,x d ad y y 1,...,y d. By addig edges or bods betwee all pairs x, y of poits of Z d with δ x, y 1, we establish the d-dimesioal lattice L d Z d, E d, ad we write E d for the set of the edges. Suppose that each bod of lattice L d is either ope occupied with probability p or closed empty with probability 1 p, the coected compoets of this graph are called ope clusters. Let C x deote the ope cluster cotaiig the vertex x, ad θ p P C 0 be the probability that the origi belogs to a ifiite ope cluster. Whe the itesity p icreases from zero to oe, at some sharp percolatio threshold or critical poit p c, for the first time, oe ifiite cluster appears; for all p>p c we have exactly oe ifiite cluster, for all p<p c we have o ifiite cluster, ad at critical value p p c the icipiet ifiite clusters are supposed to be fractal. A lattice fractal is a graph which correspods to a fractal, all of them have a selfsimilarity, but most of them have o traslatio ivariace, see 1, The Sierpiski gasket ad the Sierpiski carpet are well-kow examples of fractals. The former is a fiitely ramified fractal i.e., it ca be discoected by removig a fiite umber of poits ad the latter is a ifiitely ramified fractal. Fractals also have close relatios to fiacial markets 17, electrical coductivity, supercoductivity, ad mechaical properties of percolatig systems, ad so forth. I 1, it shows that the Isig model o the lattice Sierpiski carpet does exhibit the phase trasitio i ay dimesio, but the Isig model o the lattice Sierpiski gasket has o phase trasitio i ay dimesio because of the character of the fiitely ramified fractal. Similar results of phase trasitios ca be obtaied for percolatio o the lattice Sierpiski carpet ad o the lattice Sierpiski gasket, see 18. I the preset paper, a ew method is itroduced to model ad describe the fluctuatios of market prices, amely, we use the lattice fractal Sierpiski carpet percolatio to establish a ew radom market price i a fiacial market. I this fiacial model, the local iteractio or ifluece amog traders i oe stock market is costructed, ad a cluster of percolatio is used to defie the cluster of traders sharig the same opiio about the market. For the compariso, we also cosider the most importat idex of Hog Kog fiacial market, the Hog Kog Hag Seg Idex. We aalyze the statistical properties of Hog Kog Hag Seg Idex ad the simulative data derived from the price model by compariso, which icludig the sharp peak ad the fat-tail distributio for the price chages, the distributio of returs decays with power law i the tails, the price fluctuatios are ot ivariat agaist time reversal i.e., they show a forward-backward asymmetry, ad so forth. Moreover, the behaviors of log memory ad log-rage correlatio i volatility series of market returs are exhibited. 2. Descriptio of Price Model o Lattice Sierpiski Carpet Percolatio First we give a brief descriptio of percolatio o the lattice Sierpiski carpet S d for d 2, which is defied as follows: cosider Z 2 as a graph i the usual sese ad set S 2 0 Z 2 0, 3 2, S 2 1 i 1,i 2 {0,1,2} i 1,i 2 / 1,1 {( i 1 3 1,i 2 3 1) } S 2, 2.1

3 Joural of Applied Mathematics 3 o Figure 1: Lattice percolatio o lattice Sierpiski carpet. where u S 2 S 2 {u v : v S 2 }. To make the graph more symmetric, let S 2 be the uio of ad its reflectios i every coordiate hyperplae. The we defie the lattice Sierpiski carpet as S 2 S Similarly to Sectio 1, we defie the correspodig edges set of S 2 as E S 2. Next we cosider radom graph bod percolatio o the lattice L S 2 S 2, E S 2, see Figure 1. Letp the itesity value satisfies 0 p 1, each edge of L S 2 is declared to be ope with probability p ad closed with probability 1 p idepedetly. We deote the product probability by P p or P, addefieθ p P C 0, where C 0 is the ope cluster cotaiig the origi o L S 2,ad C 0 is the umber of vertices i C 0. Let p c S 2 if{p : θ p > 0}, the percolatio o the Sierpiski carpet S 2 exhibits the existece of a phase trasitio, that is, θ p > 0forp>p c S 2, for details see 1, 18. Next we cosider a price model of auctios for a stock i a stock market. Assume that each trader ca trade the stock several times at each day t {1, 2,...,T}, but at most oe uit umber of the stock at each time. Let S t deote the daily closig price of tth tradig day. Ad let Λ be a subset of S 2, where Λ { x 1,x 2 S 2 : 3 x 1 3, 3 x 2 3 } 2.3 ad C t 0 be a radom ope cluster o Λ. Suppose that this stock cosists of Λ is large eough ivestors, who are located i Λ lattice. Ad C t 0 is a radom set of the selected traders who receive the iformatio. At the begiig of tradig i each day, suppose that the ivestors receive some ews. We defie a radom variable ζ t for these ivestors, suppose that these ivestors takig buyig positios ζ t 1 sellig positios ζ t 1, or eutral positios ζ t 0 with probability q 1,q 1 or 1 q 1 q 1 q 1,q 2 > 0,q 1 q 2 1, respectively. The these ivestors sed bullish, bearish or eutral sigal to the market. Accordig to bod percolatio

4 4 Joural of Applied Mathematics o S 2, ivestors ca affect each other or the ews ca be spread, which is assumed as the mai factor of price fluctuatios. For a fixed t {1, 2,...,T}, let B t ζ t C t Λ From the above defiitios ad mathematical fiace theory 20 24, we defie the stock price at tth tradig day as S t e α t B t S t 1, 2.5 where S 0 is the iitial stock price at time 0, ad α t >0 represets the depth fuctio of the market at tradig day t. The we have { t S t S 0 exp α t B k }, t {1, 2,...,T}. 2.6 k 1 The formula of the sigle-period stock logarithmic returs from t to t 1 is give as follows: r t l S t 1 l S t, t {1, 2,...,T} Experimet Aalysis of Market Retur Distributio I order to make empirical research o the fiacial price model ad a actual stock market by compariso, we select the daily closig prices of Hag Seg Idex i the 20-year period from September 3, 1990 to September 3, 2010, the total umber of observed data is about Recet research shows that returs o fiacial markets are ot Gaussia, but exhibit excess kurtosis ad fatter tails tha the ormal distributio, which is usually called the fat-tail pheomeo, see 21, The geeral explaatio for this pheomeo is thought to be the herd effect of ivestors i the market. The time series of returs by simulatig the price model which is developed o the Sierpiski carpet percolatio is plotted i Figure 2 a. The returs distributios of Hag Seg Idex ad the fiacial model are plotted i Figure 2 b, the part where the probability is above the 75th or below 25th percetiles of the samples deviates from the dash lie. This implies that the probability distributios of returs deviate from the correspodig ormal distributios at the tail parts. For further aalyzig the character of returs distributios for the simulative data ad Hag Seg Idex, we make the sigle-sample Kolmogorov-Smirov test by the statistical method, the basic statistics of the correspodig returs is displayed i Table 1. The value of two-tail test P is 0.000, thus the hypothesis is deied that the distributio of returs follows the Gaussia distributio.

5 Joural of Applied Mathematics Daily retur rates plot of simulatio idex Normal probability plot Retur rates Probability Data Data a The returs time series of the price model The real value Simulatio value b The plot of returs distributios Figure 2: a The returs time series of simulatio data for the price model with the itesity p b The compariso of returs distributios for 20-year period Hag Seg idex ad the simulative data with p Table 1: The Kolmogorov-Smirov test. The fiacial model Hag Seg Idex Capability The H value 1 1 The P value of double tail K-S statistics to measure The CV value I this part, we study the properties of skewess ad kurtosis o the returs for the simulative data ad Hag Seg Idex. Next we give the defiitios of skewess ad kurtosis as follows: Skewess Kurtosis r i u r 3 1 δ, 3 r i u r 4 1 δ, 4 where r i deotes the retur of ith tradig day, u r is the mea of r, is the total umber of tradig dates, ad δ is the correspodig stadard variace. Kurtosis shows the cetrality of data, ad the skewess shows the symmetry of the data; it is a measure of the peakedess of the probability distributio of a real-valued radom variable, ad the ifrequet extreme deviatios lead higher kurtosis. Skewess is importat because kurtosis is ot idepedet of skewess, ad the latter may iduce the former. It is kow that the skewess of stadard ormal distributio is 0 ad the kurtosis is 3. Next we ivestigate the statistical behaviors of the returs for differet itesity values p, where the value p chages from 0.39 to 0.55 with the iterval legth i 1 i 1 3.1

6 6 Joural of Applied Mathematics Table 2: The aalysis of the price model for differet itesity values p. p Kurtosis Skewess Mea Variace Mi Max E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E Table 2 gives a descriptio of the statistics for 17 group data of the price model. This shows that the distributio of the returs deviates from the Gaussia distributio with the itesity values p icreasig, ad the kurtosis distributio of the returs has a sharper peak, loger ad fatter tails for larger p. From the defiitios i Sectio 2, p is the itesity for the Sierpiski carpet lattice percolatio ad represets the stregth of iformatio spread i the price model. The wider the iformatio spread, the larger the value of p is. I the followig, we hope to exhibit that the umerical characteristics of simulatio results for some itesity p are very close to those of the real data. We aalyze the probability distributios of the logarithmic returs ad the cumulative distributios of the ormalized returs for these data i Figure 3, where the itesity values of the model are p 0.485, p 0.49, ad p 0.495, respectively. 4. Log Memory Test of the Model ad Hag Seg Idex We aalyze the log memory of the returs by usig Lo s modified rescaled rage statistic 31. The log memory is measured by the Hurst expoet H, calculated by Lo s modified rescaled rage statistic. For 0.5 <H<1, the series exhibits the log-term persistece, with the maiteace of tedecy; for 0 <H<0.5, the series is the atipersistet, presetig reversio to the mea; ad for H 0.5, the series correspods to a radom walk. We cosider a sample of series X 1,X 2,...,X ad let X deote the sample mea. The the modified rescaled rage statistic, deoted by Q, is defied by Q 1 k ) k ( ) ( ) max (X j X mi X j X, σ q 1 k 1 k j 1 j 1 4.1

7 Joural of Applied Mathematics P(r) 4 P(r) P(r) r a r< r log(p( r >x)) r b r> log( r ) Hag Seg idex p = p = 0.49 p = c Hag Seg idex p = p = 0.49 p = Figure 3: The plots a, b,ad c are the probability distributios of the logarithmic returs, ad the plot d is the cumulative distributios of the ormalized price returs. The data is selected from Hag Seg Idex ad from the simulatio data with the differet values p, p 0.485, p 0.49 ad p d where σ 2 ( ) 1 ) 2 2 q ( ) q (X j X ω j q )( ) (X i X X i j X ω j ( q ) 1 j 1 q σ 2 X 2 ( ω j q ) γj, j 1 j ( ) q< q 1 j 1 i j σ 2 X ad γ j deote the sample variace ad the autocovariace estimators of X.

8 8 Joural of Applied Mathematics Table 3: Statistics of returs for V ad H. V statistic results Hurst idex results Retur series First-order autocorrelatio q V Itercept c H The fiacial model Hag Seg Idex I order to make the statistical iferece for the above-modified rescaled rage statistics, Lo derived that V q 1/2 Q coverges weakly to a radom variable V, where V is the rage of a Browia bridge o the uit iterval. The the correspodig distributio fuctio of V is give by ( F v k 2 v 2) e 2 kv k 1 Form this fuctio F v, we ca get test threshold for ay level of sigificace by examiig sigificat of V q, this reflects the log memory behavior for the time series. It is importat to select the widow wide q; we take the experiece value q ( ) 3T 1/3 ( ) 2/3 2 ρ1, ρ 2 1 where ρ 1 is the estimatio of first-order autocorrelatio coefficiet of the time series. The the Hurst expoet H is defied as the limit of the ratio log Q / log. At the same time, it shows a liear growth tred betwee modified R/S statistic ad sample size, by usig regressio l Q l c H l. 4.5 With some optimal q value, the statistics of returs by the modified R/S statistic is give i Table 3 ad Figure 4. Figure 4 also shows the fluctuatios of expoet H of returs for the price model ad Hag Seg Idex. 5. Log-Rage Correlatio of the Model ad Hag Seg Idex I this sectio, detreded fluctuatio aalysis DFA method is applied o the lattice Sierpiski carpet percolatio. The DFA is a techique used to estimate a scalig expoet from the behavior of the average fluctuatio of a radom variable aroud its local tred, for the details see 26. The cumulative deviatio of time series {x t,t 1,...,N} is give by i Y i x k x, for i 1,...,N. 5.1 k 1

9 Joural of Applied Mathematics 9 log(r/s) log() The value of H log(r/s) The atural logarithm of R/S expects Fitted lie V statistic log() The atural logarithm of R/S expects Fitted lie V statistic a b The value of H Figure 4: a The plots of modified R/S statistics ad the fluctuatio of expoet H for the price model. b The correspodig plots for the actual data from Hag Seg Idex. We divide Y i ito itervals of ooverlappig ad equal legth of time. The the root mea square fluctuatio for all such legth iterval is defied as F 1 N Y i Y i 2, N i 1 5.2

10 10 Joural of Applied Mathematics 10 1 F() 10 1 F() F() = F() = The simulatio returs The fitted lie a The Hag Seg Idex returs The fitted lie b Figure 5: a DFA aalysis of the returs for the simulatio data with the itesity p b DFA aalysis of the returs for Hag Seg Idex. where Y i is the fittig polyomial of the iterval. The above defiitio is repeated for all the divided itervals. There is a power-law relatio betwee F ad, amely, F α. 5.3 The parameter α is the scalig expoet or the correlatio expoet, which exhibits the lograge correlatio of the time series. For α 0.5, it idicates that the time series is ucorrelated white oise ; for the value 0 <α<0.5, it idicates the aticorrelatios; for 0.5 <α<1, the time series has the persistet log-rage correlatio. Accordig to DFA method ad computer simulatio, the scalig expoets of the returs of the price model ad Hag Seg Idex are ad , respectively, i Figure 5. Although both the expoet values are larger tha 0.5, they are very close to 0.5. This shows that there is some strog idicatio of lograge correlatios for the returs. 6. Coclusio A ew radom stock price model is developed by the lattice Sierpiski carpet percolatio i the preset paper, ad a cluster of carpet percolatio is applied to describe the cluster of traders sharig the same opiio about the market. The statistical properties of the returs are ivestigated ad aalyzed for differet itesity values, ad the behaviors of log memory ad log-rage correlatio i volatility series are exhibited. Further, Hag Seg Idex is also itroduced ad ivestigated by compariso; the empirical results show that the price model is accord with the real market to some degree.

11 Joural of Applied Mathematics 11 Ackowledgmet The authors were supported i part by Natioal Natural Sciece Foudatio of Chia Grats o ad o , BJTU Foudatio o. S11M Refereces 1 M.-F. Che, From Markov Chais to No-Equilibrium Particle Systems, World Scietific Publishig, River Edge, NJ, USA, R. Durrett, Lecture Notes o Particle Systems ad Percolatio, Wadsworth & Brooks, G. Grimmett, Percolatio, vol. 321, Spriger, Berli, Germay, 2d editio, T. M. Liggett, Stochastic Iteractig Systems: Cotact, Voter ad Exclusio Processes, vol. 324, Spriger, New York, NY, USA, T. M. Liggett, Iteractig Particle Systems, vol. 276, Spriger, New York, NY, USA, D. Stauffer ad A. Aharoy, Itroductio to Percolatio Theory, Taylor & Fracis, Lodo, UK, P. Bak, M. Paczuski, ad M. Shubik, Price variatios i a stock market with may agets, Physica A, vol. 246, o. 3-4, pp , R. Cot ad J. P. Bouchaud, Herd behavior ad aggregate fluctuatios i fiacial markets, Macroecoomic Dyamics, vol. 4, o. 2, pp , G. Iori, Avalache dyamics ad tradig frictio effects o stock market returs, Iteratioal Joural of Moder Physics C, vol. 10, o. 6, pp , T. Kaizoji, S. Borholdt, ad Y. Fujiwara, Dyamics of price ad tradig volume i a spi model of stock markets with heterogeeous agets, Physica A, vol. 316, o. 1 4, pp , T. Lux ad M. Marchesi, Scalig ad criticality i a stochastic multi-aget model of a fiacial market, Nature, vol. 397, o. 6719, pp , D. Stauffer ad D. Sorette, Self-orgaized percolatio model for stock market fluctuatios, Physica A, vol. 271, o. 3-4, pp , H. Taaka, A percolatio model of stock price fluctuatios, Mathematical Ecoomics, o. 1264, 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, o. 2, pp , J. Wag, Q. Wag, ad J. Shao, Fluctuatios of stock price model by statistical physics systems, Mathematical ad Computer Modellig, vol. 51, o. 5-6, pp , B. B. Madelbrot, The Fractal Geometry of Nature, W. H. Freema, Sa Fracisco, Calif, USA, E. E. Peters, Fractal Market Aalysis: Applyig Chaos Theory to Ivestmet ad Ecoomics, Joh Wiley & Sos, New York, NY, USA, M. Shioda, Existece of phase trasitio of percolatio o Sierpiński carpet lattices, Joural of Applied Probability, vol. 39, o. 1, pp. 1 10, J. Wag, Supercritical isig model o the lattice fractal the Sierpiski carpet, Moder Physics Letters B, vol. 20, o. 8, pp , R. Gaylord ad P. Welli, Computer Simulatios with Mathematica: Exploratios i the Physical, Biological ad Social Sciece, Spriger, New York, NY, USA, K. Iliski, Physics of Fiace: Gauge Modelig i No-Equilibrium Pricig, Joh Wiley, New York, NY, USA, D. Lamberto ad B. Lapeyre, Itroductio to Stochastic Calculus Applied to Fiace, Chapma & Hall, Lodo, UK, S. M. Ross, A Itroductio to Mathematical Fiace, Cambridge Uiversity Press, Cambridge, UK, J. Wag, Stochastic Process ad Its Applicatio i Fiace, Tsighua Uiversity Press ad Beijig Jiaotog Uiversity Press, Beijig, Chia, F. Black ad M. Scholes, The pricig of optios ad corporate liabilities, Joural of Political Ecoomy, vol. 81, pp , P. Grau-Carles, Log-rage power-law correlatios i stock returs, Physica A, vol. 299, o. 3-4, pp , Y. Guo ad J. Wag, Simulatio ad statistical aalysis of market retur fluctuatio by Zipf method, Mathematical Problems i Egieerig, vol. 2011, Article ID , 13 pages, 2011.

12 12 Joural of Applied Mathematics 28 M. L evy ad S. Solomo, Microscopic Simulatio of Fiacial Markets, Academic Press, New York, NY, USA, D. Pirio, Jump detectio ad log rage depedece, Physica A, vol. 388, o. 7, pp , Z. Zheg, Matlab Programmig ad the Applicatios, Chia Railway Publishig House, Beijig, Chia, E. Zivot ad J. H. Wag, Modelig Fiacial Time Series with S-PLUS, Spriger, New York, NY, USA, 2006.

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