Effect of boundary conditions on stochastic Ising-like financial market price model
- Wen Fang^{1} and
- Jun Wang^{1}Email author
https://doi.org/10.1186/1687-2770-2012-9
© Fang and Wang; licensee Springer. 2012
Received: 20 May 2011
Accepted: 2 February 2012
Published: 2 February 2012
Abstract
Price formation in financial markets based on the 2D stochastic Ising-like spin model is proposed, with a randomized inverse temperature of each trading day. The statistical behaviors of returns of this financial model are investigated for zero boundary condition and five different classes of mixed boundary conditions. For comparison with actual financial markets, we also analyze the statistical properties of Shanghai Stock Exchange (SSE) composite Index, Shenzhen Stock Exchange (SZSE) component Index and Hushen 300 Index. Fluctuation properties, fat-tail phenomena, power-law distributions and fractal behaviors of returns for these indexes and the simulative data are studied. With the plus boundary condition, for example the boundary condition ¿_{6}, the value of market depth parameter ¿ is smaller than those of the corresponding market depth parameters ¿ with zero boundary condition ¿_{1} and weak mixed boundary conditions ¿_{2} and ¿_{3}. And the changing range of tails exponents of boundary condition ¿_{6} is much smaller than those of the other five boundary conditions.
Keywords
1 Introduction
As the stock markets are becoming deregulated worldwide, the modeling of the dynamics of the forward prices is becoming a key problem in risk management, physical assets valuation and derivatives pricing, see [1–6], and it is also important to understand the statistical properties of fluctuations of stock price in globalized securities markets, for example see [7, 8]. A complex behavior can emerge due to the interactions among smallest components of that system, see [9], and it is often a successful strategy to analyze the behavior of a complex system by studying these components. In financial markets, these components are comprised by the market participants who buy and sell assets in order to realize their trading and investment decisions. Similar to physical systems, the superimposed flow of all individual orders submitted to the exchange trading system initiated by market participants and its change in time generate a complex system with fascinating properties, see [1, 2].
Recently, the theory of stochastic interacting particle systems [10–12] has been applied to investigate the statistical behaviors of fluctuations for stock prices, and the corresponding valuation and hedging of contingent claims for these price process models are also studied, see [1, 2, 11, 12]. In the present article, we suppose that traders determine their positions at each time by observing the market information (and then evaluating the market behavior, market sentiment and their trading strategies), each trader is thought to be a subunit in the stock market, and may take positive (buying) position, negative (selling) position or neutral position, denoted by +, -, and 0, respectively. Traders with buying positions or selling positions are called market participants, and the configuration of positions for all traders is assumed as the main factor resulting in price fluctuations in this financial model. The reason that we use interacting particle systems to investigate the fluctuation of stock markets is that all of these systems consist of subunits. The Ising spin system, which can describe the mechanism of making a decision in a closed community, is the most popular ferromagnetic model of interacting particle systems. The subunits in a 2D Ising model are called spins (with the interactions between the nearest neighbors), the clusters of parallel spins in the square-lattice Ising model can be defined as groups of traders acting together on the stock market model, for example see [1, 3, 4]. The objective of this work is to study the financial phenomena of the price model developed by the stochastic Ising-like spin model. In this model, all of the spins are flipped by following Ising dynamic system [13], and the inverse temperature of each trading day is randomly chosen in a certain interval. And for different boundary conditions, the statistical behaviors of the price model are studied. Further, the empirical research in financial market fluctuations for the actual stock market and the financial model is made by comparison analysis.
2 Description of 2D Ising-like spin model
If we set ¿(u) = 0 for all u ¿ ¿^{2}, then we call the resulting boundary condition the zero or open boundary condition, if ¿(u) = +1 for all u ¿ ¿^{2}, the boundary condition is called the plus boundary condition, if ¿(u) = -1 for all u ¿ ¿^{2}, then the resulting boundary condition is called the minus boundary condition, and if there are either ¿(u) = +1 or ¿(u) = -1 for some u ¿ ¿^{2}, and ¿(u) = 0 for the others, we call this kind of condition the mixed boundary condition. The spin of the Ising model can point up (spin value +1) or point down (spin value -1), and it flips between the two orientations. At sufficiently low temperatures, the energy effect predominates and we have known that the model exhibits phase transition. Correlations are related to the phase transition and the spin fluctuations of the model. As ß increases (from 0), the correlations begin to extend, these correlations take the form of spin fluctuations, which are islands of a few spins each that mostly point in the same direction. As ß approaches the critical inverse temperature ß_{ c }from below, spin fluctuation are present at all scales of length. At ß = ß_{ c }, the correlations decay by a power law, but for ß > ß_{ c }, there are two distinct pure phases. Correlations play an important role in studying the fluctuations of the phase interfaces for the statistical physics model, see [4, 14]. In the following section, since the financial price model heavily depends on the number of spin values, we set the intensity of interaction among the market investors ß = ¿¿, where ¿ is a random variable with the uniform distribution in 0[1], and ¿ is a intensity parameter, then we obtain the Ising-like spin model.
3 Financial model and boundary conditions
Next, we consider the different boundary conditions for the model in a finite square ¿, which is defined by ¿ = {(u_{1}, u_{2}): 1 ¿ u_{1} ¿ n, 1 ¿ u_{2} ¿ n} for a large integer n, and then we have |¿_{ ext }¿| = 4n. The six classes of boundary conditions ¿_{1}, ¿_{2}, ¿_{3}, ¿_{4}, ¿_{5}, ¿_{6} of the financial model based on the Ising-like system are given as follows, where ${\xbf}_{i}\xbf{\left\{+1,0,\phantom{\rule{2.77695pt}{0ex}}-1\right\}}^{{\xbf}^{2}}$ for i = 1,..., 6.
A. Boundary Condition 1. The boundary condition ¿_{1} is defined as follows: ¿_{1}(u) = 0 for all u ¿ ¿_{ ext }¿. ¿_{1}(u) = 0 means that the site u is open or there is no spin on the site u, this boundary condition is called the zero boundary condition.
where m_{1} is a positive integer which depends on the number 4n for a fixed l. In this case, for the first connected sites of length l in {u^{ i }} from the site (1, n + 1), we assign the same spins (" + " spin) on these sites, then on the next connected sites of length l we assign the " - " spins on these sites, and so on. This is a mixed boundary condition, and suppose that l = 10 in the following parts of the present article.
In this case, the plus spins are assigned on the top side and the bottom side of the exterior boundary ¿_{ ext }¿, and the minus spins are assigned on the left side and the right side of ¿_{ ext }¿.
where m_{2} is a positive integer such that m_{2} < n/3.
E. Boundary Condition 5. The boundary condition ¿_{5} is defined as follows: Four sides boundary conditions of the exterior boundary ¿_{ ext }¿ are same as that of the right side boundary condition in ¿_{4}.
F. Boundary Condition 6. The boundary condition ¿_{6} is defined as follows: ¿(u) = +1 for all u ¿ ¿^{2}, the boundary condition is called the plus boundary condition.
In our computer simulations, the system size is n = 100, and the position probabilities p_{1} = p_{-1} = 0.5. Each step represents one trading minute, 240 steps constitute one trading day, the random variable ¿ is stochastically chosen in the uniform distribution 0[1] once for every step, where the value of inverse temperature ß = ¿¿ is defined in Section 2. We typically simulate 1200000 steps, which corresponds to 5000 trading days for each boundary condition and each fixed intensity parameter ¿. We analyze the historical data sets of Hushen 300 Index, Shanghai Stock Exchange (SSE) Composite Index and Shenzhen Stock Exchange (SZSE) Component Index, which records every trade for all the securities in the Chinese stock market during the period from January 4, 2005 to December 31, 2010, a total number of observed trading days is 1457, see http://www.sse.com.cn and http://www.szse.cn. The Hushen 300 Index consists of 300 actively traded large cap companies in the SSE (179 companies) and SZSE (121 companies), which has a good representative of the market. For these databases, the records of the daily closing price are continuous in regular open days for every week, due to the removal of all market closure times.
4 Statistical behaviors of financial model with different boundary conditions
Statistical behaviors of price fluctuations are very important to understand and model financial market dynamics, which have long been a focus of economic research. Stock price volatility is of interest to traders because it quantifies risk, optimizes the portfolio, and provides a key input of option pricing models that are based on the estimation of the volatility of the asset, see [6, 7].
4.1 The statistical properties of the model
where r_{ t } denotes the return of t th trading day, $\stackrel{\xbf}{r}$ is the mean of r, n is the total number of the data, and ¿ is the corresponding standard variance. The kurtosis shows the centrality of data and the skewness shows the symmetry of the data. It is known that the skewness of standard normal distribution is 0.
The statistical properties of Chinese stock market
Mean | Variance | Max | Min | Kurtosis | Skewness | |
---|---|---|---|---|---|---|
SSE | 0.000546 | 0.000382 | 0.090345 | -0.092561 | 5.584019 | -0.347366 |
SZSE | 0.000962 | 0.000466 | 0.091615 | -0.097501 | 4.962429 | -0.377435 |
Hushen 300 | 0.000786 | 0.000431 | 0.089310 | -0.096949 | 5.267240 | -0.412498 |
The statistical properties of price model with six classes boundary conditions
Boundary | ¿ | ¿ | Mean | Variance | Max | Min | Kurtosis | Skewness |
---|---|---|---|---|---|---|---|---|
¿ _{1} | 1 | 0.5 | 0.00003 | 0.00040 | 0.09890 | -0.09380 | 4.85178 | 0.13589 |
¿ _{1} | 0.8 | 0.58 | 0.00007 | 0.00037 | 0.09616 | -0.09512 | 4.88207 | 0.10954 |
¿ _{1} | 0.6 | 0.9 | -0.00043 | 0.00050 | 0.09792 | -0.08802 | 4.46219 | 0.13983 |
¿ _{1} | 0.5 | 0.92 | 0.00009 | 0.00039 | 0.09090 | -0.09734 | 4.35889 | -0.08853 |
¿ _{1} | 0.45 | 1.1 | -0.00028 | 0.00048 | 0.09680 | -0.09416 | 4.17186 | -0.07398 |
¿ _{1} | 0.4 | 1.5 | 0.00034 | 0.00068 | 0.09480 | -0.09810 | 3.49030 | -0.02603 |
¿ _{1} | 0.3 | 1.7 | 0.00048 | 0.00063 | 0.09554 | -0.09656 | 3.37645 | 0.05148 |
¿ _{2} | 1 | 0.5 | 0.00059 | 0.00044 | 0.09662 | -0.09870 | 4.79473 | 0.05917 |
¿ _{2} | 0.8 | 0.59 | 0.00019 | 0.00037 | 0.09853 | -0.08767 | 5.10466 | 0.06324 |
¿ _{2} | 0.6 | 0.7 | 0.00037 | 0.00031 | 0.09982 | -0.09100 | 4.93810 | 0.07865 |
¿ _{2} | 0.5 | 1.05 | 0.00049 | 0.00048 | 0.09303 | -0.09618 | 4.04856 | 0.06254 |
¿ _{2} | 0.45 | 1.1 | 0.00028 | 0.00046 | 0.09812 | -0.09108 | 4.11700 | 0.10904 |
¿ _{2} | 0.4 | 1.2 | 0.00020 | 0.00045 | 0.09552 | -0.09888 | 3.85836 | -0.02008 |
¿ _{2} | 0.3 | 1.75 | 0.00004 | 0.00067 | 0.09135 | -0.09835 | 3.32317 | -0.00956 |
¿ _{3} | 1 | 0.53 | 0.00058 | 0.00045 | 0.09932 | -0.09158 | 5.00763 | 0.12602 |
¿ _{3} | 0.8 | 0.66 | 0.00035 | 0.00048 | 0.09332 | -0.09913 | 4.61243 | -0.00751 |
¿ _{3} | 0.6 | 0.85 | 0.00010 | 0.00045 | 0.09571 | -0.09758 | 4.69311 | 0.00167 |
¿ _{3} | 0.5 | 1.05 | 0.00033 | 0.00047 | 0.09744 | -0.09891 | 4.11189 | 0.00977 |
¿ _{3} | 0.45 | 1.1 | 0.00048 | 0.00053 | 0.09724 | -0.09856 | 4.12251 | 0.04151 |
¿ _{3} | 0.4 | 1.25 | 0.00024 | 0.00049 | 0.09875 | -0.09425 | 3.79751 | -0.00912 |
¿ _{3} | 0.3 | 1.75 | 0.00007 | 0.00067 | 0.09135 | -0.09975 | 3.32325 | -0.01994 |
¿ _{4} | 1 | 0.48 | -0.00004 | 0.00039 | 0.09504 | -0.09926 | 4.76332 | -0.00688 |
¿ _{4} | 0.8 | 0.64 | 0.00035 | 0.00043 | 0.09894 | -0.09536 | 4.93876 | 0.13699 |
¿ _{4} | 0.6 | 0.7 | 0.00037 | 0.00033 | 0.09688 | -0.09898 | 5.14323 | 0.09849 |
¿ _{4} | 0.5 | 0.85 | 0.00019 | 0.00033 | 0.09367 | -0.09979 | 4.49162 | -0.04789 |
¿ _{4} | 0.45 | 1.2 | -0.00022 | 0.00057 | 0.09216 | -0.09648 | 3.84336 | -0.03486 |
¿ _{4} | 0.4 | 1.3 | 0.00014 | 0.00053 | 0.09178 | -0.09880 | 3.87194 | -0.03044 |
¿ _{4} | 0.3 | 1.75 | 0.00027 | 0.00067 | 0.09240 | -0.09765 | 3.35798 | 0.02708 |
¿ _{5} | 1 | 0.41 | 0.00010 | 0.00037 | 0.09938 | -0.09479 | 4.92090 | 0.09551 |
¿ _{5} | 0.8 | 0.46 | 0.00030 | 0.00033 | 0.09752 | -0.09504 | 5.85585 | -0.16673 |
¿ _{5} | 0.6 | 0.73 | 0.00031 | 0.00041 | 0.09928 | -0.09505 | 4.73252 | 0.16074 |
¿ _{5} | 0.5 | 0.88 | -0.00036 | 0.00042 | 0.09979 | -0.09258 | 4.88828 | 0.01417 |
¿ _{5} | 0.45 | 0.95 | 0.00016 | 0.00040 | 0.09937 | -0.09804 | 4.43247 | -0.10505 |
¿ _{5} | 0.4 | 1.2 | 0.00052 | 0.00052 | 0.09888 | -0.09264 | 3.99503 | 0.05671 |
¿ _{5} | 0.3 | 1.42 | -0.00023 | 0.00050 | 0.09883 | -0.09656 | 3.56444 | 0.00311 |
¿ _{6} | 1 | 0.3 | 0.00021 | 0.00037 | 0.09951 | -0.09877 | 4.62543 | 0.14243 |
¿ _{6} | 0.8 | 0.48 | -0.00025 | 0.00057 | 0.09168 | -0.09936 | 4.38937 | -0.09951 |
¿ _{6} | 0.6 | 0.55 | 0.00053 | 0.00038 | 0.09680 | -0.09768 | 4.62340 | 0.08621 |
¿ _{6} | 0.5 | 0.78 | -0.00011 | 0.00051 | 0.09625 | -0.09937 | 4.14862 | -0.00981 |
¿ _{6} | 0.45 | 0.84 | 0.00038 | 0.00046 | 0.09912 | -0.09442 | 4.14778 | 0.01051 |
¿ _{6} | 0.4 | 0.92 | -0.00006 | 0.00042 | 0.08777 | -0.09954 | 3.97269 | 0.07773 |
¿ _{6} | 0.3 | 1.3 | 0.00014 | 0.00051 | 0.09074 | -0.09698 | 3.81253 | -0.07299 |
In Table 2, for each fixed boundary condition ¿, as the intensity parameter ¿ decreases, the depth parameter ¿ of the model has the tend to increase. The interaction among the market investors decreases for the intensity parameter decreasing, this means that investors may pay less attention to the other people's investment attitude around them. In this situation, Table 2 shows that the market depth parameter may play a great role in the price model. And for each fixed value of intensity parameter ¿, when the boundary conditions are neither " + " nor " - " predominates the other, for example the boundary conditions ¿_{1}, ¿_{2}, and ¿_{3}, the values of depth parameters ¿ are larger than that of the corresponding depth parameter ¿ of plus boundary condition ¿_{6}, which is the plus is the overwhelming part of the boundary sites. When the interaction among the market investors increases or the external environment is dominant by one view for a long time, the value of market depth parameter ¿ may decrease. And the ranges of variances of returns with six boundary conditions ¿_{1}, ¿_{2}, ¿_{3}, ¿_{4}, ¿_{5}, ¿_{6} are [0.00037, 0.00068], [0.00031, 0.00067], [0.00045, 0.00067], [0.00033, 0.00067], [0.00033, 0.00050], [0.00037, 0.00057], respectively. Since the variances of SSE, SZSE and Hushen 300 are 0.000382, 0.00046, and 0.000431, respectively, from the view of fluctuations, the simulation of model with boundary condition ¿_{5} is closer to this period of January 4, 2005 to December 31, 2010 in real Chinese stock markets.
Power law and fractal behavior of Chinese stock markets
Stock | ¿ | H _{1} | H _{2} | h | p | cv | k |
---|---|---|---|---|---|---|---|
Hushen 300 | 2.9755 | 0.69374 | 0.64784 | 1 | 3.89E-06 | 0.0355 | 0.067 |
SSE | 2.9291 | 0.69399 | 0.6505 | 1 | 2.17E-08 | 0.0355 | 0.0791 |
SZSE | 3.0819 | 0.70723 | 0.6445 | 1 | 1.17E-05 | 0.0355 | 0.0641 |
The power law and fractal behavior of the financial price model
Boundary | ¿ | ¿ | H _{1} | H _{2} | h | p | cv | k |
---|---|---|---|---|---|---|---|---|
¿ _{1} | 1 | 2.9858 | 0.62932 | 0.52649 | 1 | 3.52E-14 | 0.0192 | 0.0562 |
¿ _{1} | 0.8 | 2.9914 | 0.62918 | 0.56231 | 1 | 1.28E-13 | 0.0192 | 0.0551 |
¿ _{1} | 0.6 | 3.1079 | 0.62977 | 0.54157 | 1 | 1.50E-10 | 0.0192 | 0.0482 |
¿ _{1} | 0.5 | 3.1914 | 0.62464 | 0.57616 | 1 | 3.48E-05 | 0.0192 | 0.0331 |
¿ _{1} | 0.45 | 3.2673 | 0.60693 | 0.49562 | 1 | 1.93E-05 | 0.0192 | 0.0339 |
¿ _{1} | 0.4 | 3.5877 | 0.61531 | 0.52876 | 1 | 0.0242 | 0.0192 | 0.021 |
¿ _{1} | 0.3 | 3.7026 | 0.62881 | 0.54277 | 0 | 0.0671 | 0.0192 | 0.0184 |
¿ _{2} | 1 | 2.9605 | 0.65835 | 0.57159 | 1 | 2.99E-14 | 0.0192 | 0.0564 |
¿ _{2} | 0.8 | 2.9962 | 0.64852 | 0.57482 | 1 | 7.42E-13 | 0.0192 | 0.0534 |
¿ _{2} | 0.6 | 3.0368 | 0.60431 | 0.57264 | 1 | 8.08E-09 | 0.0192 | 0.0439 |
¿ _{2} | 0.5 | 3.2509 | 0.61399 | 0.50458 | 1 | 1.62E-06 | 0.0192 | 0.0374 |
¿ _{2} | 0.45 | 3.2829 | 0.62111 | 0.55096 | 1 | 2.45E-05 | 0.0192 | 0.0336 |
¿ _{2} | 0.4 | 3.4894 | 0.65298 | 0.54266 | 1 | 3.58E-04 | 0.0192 | 0.0293 |
¿ _{2} | 0.3 | 3.7573 | 0.62058 | 0.512 | 0 | 0.0584 | 0.0192 | 0.0188 |
¿ _{3} | 1 | 2.9426 | 0.62758 | 0.52255 | 1 | 2.33E-14 | 0.0192 | 0.0566 |
¿ _{3} | 0.8 | 3.1303 | 0.64643 | 0.58522 | 1 | 1.95E-12 | 0.0192 | 0.0525 |
¿ _{3} | 0.6 | 3.0115 | 0.64581 | 0.5978 | 1 | 4.02E-09 | 0.0192 | 0.0447 |
¿ _{3} | 0.5 | 3.3363 | 0.60074 | 0.52125 | 1 | 5.80E-06 | 0.0192 | 0.0357 |
¿ _{3} | 0.45 | 3.235 | 0.60871 | 0.48946 | 1 | 2.68E-05 | 0.0192 | 0.0335 |
¿ _{3} | 0.4 | 3.4714 | 0.65577 | 0.54144 | 1 | 6.15E-04 | 0.0192 | 0.0284 |
¿ _{3} | 0.3 | 3.7332 | 0.61767 | 0.49896 | 0 | 0.1043 | 0.0192 | 0.0172 |
¿ _{4} | 1 | 3.0509 | 0.63533 | 0.57367 | 1 | 1.33E-13 | 0.0192 | 0.055 |
¿ _{4} | 0.8 | 2.9779 | 0.64633 | 0.53821 | 1 | 4.68E-12 | 0.0192 | 0.0517 |
¿ _{4} | 0.6 | 2.9488 | 0.63554 | 0.54294 | 1 | 2.68E-12 | 0.0192 | 0.0522 |
¿ _{4} | 0.5 | 3.2594 | 0.6237 | 0.53025 | 1 | 1.61E-05 | 0.0192 | 0.0342 |
¿ _{4} | 0.45 | 3.4414 | 0.60189 | 0.46971 | 1 | 1.88E-04 | 0.0192 | 0.0304 |
¿ _{4} | 0.4 | 3.433 | 0.62462 | 0.57237 | 1 | 0.002 | 0.0192 | 0.0263 |
¿ _{4} | 0.3 | 3.7723 | 0.63278 | 0.54339 | 0 | 0.0528 | 0.0192 | 0.019 |
¿ _{5} | 1 | 3.0892 | 0.6298 | 0.55672 | 1 | 1.64E-18 | 0.0192 | 0.0645 |
¿ _{5} | 0.8 | 2.6359 | 0.67195 | 0.56048 | 1 | 1.25E-24 | 0.0192 | 0.0746 |
¿ _{5} | 0.6 | 2.9661 | 0.62938 | 0.55315 | 1 | 1.41E-09 | 0.0192 | 0.0458 |
¿ _{5} | 0.5 | 2.9836 | 0.63758 | 0.58806 | 1 | 1.78E-09 | 0.0192 | 0.0456 |
¿ _{5} | 0.45 | 3.1455 | 0.62375 | 0.52021 | 1 | 1.82E-07 | 0.0192 | 0.0402 |
¿ _{5} | 0.4 | 3.4035 | 0.6429 | 0.56318 | 1 | 5.06E-04 | 0.0192 | 0.0287 |
¿ _{5} | 0.3 | 3.7789 | 0.61718 | 0.51631 | 0 | 0.2839 | 0.0192 | 0.0139 |
¿ _{6} | 1 | 3.2594 | 0.61485 | 0.53234 | 1 | 6.40E-14 | 0.0192 | 0.0557 |
¿ _{6} | 0.8 | 3.088 | 0.64219 | 0.54102 | 1 | 5.51E-13 | 0.0192 | 0.0537 |
¿ _{6} | 0.6 | 3.1714 | 0.64848 | 0.59233 | 1 | 1.50E-10 | 0.0192 | 0.0482 |
¿ _{6} | 0.5 | 3.2267 | 0.65057 | 0.58373 | 1 | 9.70E-10 | 0.0192 | 0.0463 |
¿ _{6} | 0.45 | 3.267 | 0.63234 | 0.56437 | 1 | 5.29E-07 | 0.0192 | 0.0389 |
¿ _{6} | 0.4 | 3.3155 | 0.6084 | 0.50981 | 1 | 3.59E-05 | 0.0192 | 0.033 |
¿ _{6} | 0.3 | 3.4349 | 0.60951 | 0.51533 | 1 | 0.0028 | 0.0192 | 0.0256 |
4.2 Power-law behavior and fractal phenomena of financial time series
for some ¿ ¿ 3. The main feature of this function is invariance of scale, in other words, the shape of the function is preserved. Power-law distributions show no typical scale or size, and in some cases they are connected with fractals, which also lack typical scales. Power-law distributions occur very often in natural and social fields. A few notable examples are Pareto's law for income distributions, behavior near a second-order phase transition and Zipf's law. They are commonly cited as examples of power laws.
where $\left[\frac{R\left(N\right)}{S\left(N\right)}\right]$ is the rescaled range, E[x] is the expected value, N is the number of data points in a time series, C is a constant.
Hurst exponent is referred to as the index of dependence, and is the relative tendency of a time series to either strongly regress to the mean or cluster in a direction. When 0 ¿ H ¿ 0.5, the analyzed series is anti-persistent, presenting reversion to the mean; if H = 0.5, the series presents random walk; and if 0.5 ¿ H ¿ 1, the series is persistent, with the maintenance of tendency. This develops a method of the long memory estimates for the volatilities series. The long memory is measured by the Hurst exponent H, calculated through the rescaled range analysis (R/S), which can be described as follows, for details, see [19].
Thus, we obtain the new time series of ¿t-returns.
where q = [N/s] and N is the number of observation.
5 Conclusion
For the financial modeling, any model aiming at understanding price fluctuations needs to define a mechanism for the formation of the price. In the present article, the financial model based on the Ising-like spin system is the contributor towards our ultimate understanding of the impact of external condition and interaction among the market investors and critical phenomena of the empirical stock markets. In this financial model, we suppose that the financial market not only depends on the perspective that the price movements are caused primarily by the external environment, but also depends on the spread of investing information which is due to the interaction among the market investors. The boundary condition ¿ may represent the information or the situation on this stock, including the estimate for this stock price, positive or negative news, trends, political event and economic policy, etc. The parameter ¿, phase transitions and critical phenomena of the model can be explained as the intensity of interaction among the market participants in the financial market.
In the model, the intensity parameter ¿ represents the strength of information spread and the depth parameter ¿ describes the strength of market fluctuation, both of them are defined in Section 3. Note that the range of daily price fluctuation is limited in Chinese stock markets, that is, the changing limits of daily returns for stock prices and stock market indexes are between -10% and 10%. In order to make the financial price model satisfy the changing limits of daily returns for Chinese stock markets, the value of ¿ is chosen dependently on the value of ¿ in Section 4. According to the empirical research of the model in Table 2, for each fixed boundary condition ¿, the value of depth parameter ¿ has the tend to decrease as the intensity parameter ¿ is increasing. And for each fixed value of intensity parameter ¿, when the boundary conditions are neither " + " nor " - " predominates the other, for example the boundary conditions ¿_{1}, ¿_{2} and ¿_{3} (which are usually called the "weak boundary conditions"), the corresponding values of depth parameters ¿ are larger than that of depth parameter ¿ of plus boundary condition ¿_{6} (when the plus (or the minus) is the overwhelming part of boundary sites, it is usually called the "strong boundary condition"). The large value of intensity parameter ¿ of the financial price model will exhibit the strong interaction among the market participants, this implies that the information or news about the market may spread far and wide among the market investors. The strong boundary condition of the model also shows that the external investing environment has a strong impact on the fluctuation of financial market. In the present article, by following the trading rules of Chinese stock markets, we find that when the interaction among the market investors increases or the external environment is dominant by one view for a long time, the value of depth parameter ¿ may decrease. This behavior may suggest that the possibility of long time continuous large volatilities of stock prices is small.
In Tables 3 and 4, the tails power law distributions and the fractal behaviors of market returns for the price model and the real stock markets are analyzed by empirical research and computer simulation. The values of exponents ¿ of returns are around the value 3 for the financial indexes of SSE, SZSE and Hushen 300. For the boundary condition ¿_{6} of the model, the largest value and the smallest value of tails exponent ¿ are 3.4349 and 3.088, respectively. Then the changing range of tails exponents of boundary condition ¿_{6} is 0.3469 (= 3.4349 - 3.088), which is much smaller than the corresponding changing ranges of tails exponents of other five boundary conditions. This shows that the intensity ¿ of the model with the boundary condition ¿_{6} has a weak impact on the tails distributions by comparing with other boundary conditions cases. This behavior is due to that the strong boundary conditions (for example the boundary condition ¿_{6}) or the strong external environment have a deep influence on the price dynamics in this work. According to the evolution of the price model which is modelled by Ising-like dynamic system, the strong external environment may make most of market participants take a similar investing strategy. So that, for the boundary condition ¿_{6}, the interaction among the market participants has a relative weak effect on the price fluctuation, and the tails distributions exhibit more stable behavior for different values of intensity ¿ (by comparison with other boundary conditions cases).
In Section 4, the empirical analysis displays that the fractal behaviors and the persistence properties exist in the real Chinese stock markets, SSE, SZSE, and Hushen 300. And we also analyze the fractal behaviors and other statistical properties of the financial model with six kinds of boundary conditions, and we also investigate the fluctuations of exponents H of absolute returns for the real markets and the price model. The empirical research shows the price financial model also exhibits fractal and persistence properties for different boundary conditions, this means that the six kinds of boundary conditions of this article can not change the existence of fractal behaviors for the absolute returns of the model. From the above summary, we think that the financial model of the present article is reasonable for the real stock market to some extent.
Declarations
Acknowledgements
The authors were supported in part by National Natural Science Foundation of China Grant No. 70771006 and Grant No. 10971010, Fundamental Research Funds for the Central Universities No. 2011YJS077, and BJTU Foundation No. S11M00010. The authors would like to thank the support of Institute of Financial Mathematics and Financial Engineering in BJTU. Thanks to the anonymous referees for their useful comments and suggestions, which helped us to improve our work.
Authors’ Affiliations
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