1. Introduction

The stock market is the most critical market among the capital market. It can be said that if we can predict the stock price, we can get control of the finance. The stock price has a distinct feature: how it will change has no connection with whether it has ascended or descended before, which is why it is hard to forecast. The way of predicting stock has been available for no more than one hundred years. There are many different ways to indicate this. Basically, can be divided into three directions: the forecast model based on statistical methods, machine learning predictive models, and the prediction model based on the random procedure. The Markov model is precisely one that can identify the trend and ascertain the future state of a sure thing by studying its initial state. It can appropriately predict the stock price by giving the probability of how it will change. The Markov chain mainly indicates the product market share， sales status, and expected profit. This article uses the C-K equation and n step transition probability to show the Shanghai Securities.

2. The Model

The price of a stock, also the volume of trading, can be seen as a function of time. To a specific t it only has one particular value; its state is limited. Then we can build a model as follows. Set the closing number in the day n as Xn(n=1，2，3，…+∞）; we can divide (0,+∞）into k intervals of equal distance. The interval n is [Xn-1,Xn) (n∈[1，k]).Then Xn must exist in a specific break. Let the Xn(m)=i; this means when m∈[Xn-1,Xn], the state of the stock price is i. Then let \( {P _{ij }} \) represent the probability of this state turning from i to j. \( {{P^{(n)}}_{ij}} \) means the likelihood of bending from i to j in n steps. [1-4]

Then we can make such a probability transfer chart:

P= \( {P_{（ij）}} \) = \( [\begin{matrix}{P_{11}} & {P_{12}} & ⋯ & {P_{1K}} \\ {P_{21}} & {P_{22}} & … & {P_{2K}} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ {P_{K1}} & {P_{K2}} & ⋯ & {P_{KK}} \\ \end{matrix}] \)

The chart above is called “the transition matrix”. We can know that \( {P_{ij}} \) >0 since it represents the possibility—also \( \sum {P_{ij}} \) =1.The numbers 1, 2, 3, represent the three possible stock price trends.1means ascend, 2means flat, 3means descend. According to the C-K equation, \( {P_{(n)}} \) =Pn describes the possible distribution of the stock price transfer from one state to another. So we can predict the possible state that the stock price may transfer in n days by comparing the numerical value of i.

3. Empirical Analysis

Here we select 100 sets of the latest statistics per item to ensure timeliness. Then we build a chart about its closing price. The first object we choose is the 'RESTORE':

Table 1 The stock price of BESTORE [5].

Further, let’s subtract two by two to know its state in different periods:

Table 2 The price difference data of the STORE.

Since two adjacent data can't be identical, we need an interval to define "flat." Here we choose 0.5 as a limit. If the difference exceeds 0.5, it would be classified as "ascend." Likely, if it is fewer than -0.5, it would be "descend," of course.

After that, we can obtain another table:

Table 3 The change situation of the data.

Finally, we can calculate the transfer probability of each part. Among them, \( {P_{11}} \) means the probability of state 1 to state 1, ascend to ascend, is0.08. The same way, \( {P_{22}} \) =0.2, \( {P_{33}} \) =0.07, \( {P_{12}} \) =0.13, \( {P_{21}} \) =0.14, \( {P_{13}} \) =0.06, \( {P_{31}} \) =0.06, \( {P_{23}} \) =0.14, \( {P_{32}} \) =0.12

Then we can get a state transition matrix:

P = \( [\begin{matrix}{P_{11}} & {P_{12}} & {P_{13}} \\ {P_{21}} & {P_{22}} & {P_{23}} \\ {P_{31}} & {P_{32}} & {P_{33}} \\ \end{matrix}] \) = \( [\begin{matrix}0.8 & 0.13 & 0.06 \\ 0.14 & 0.2 & 0.14 \\ 0.06 & 0.12 & 0.07 \\ \end{matrix}] \)

Since the latest day we chose is the last statistic, we can set it as an initial vector. The final day lies in the state of "1", then we put the vector \( {π_{(0)}} \) = (1, 0, 0). Then we can calculate the probability of the next day in the future:

\( {π_{(1)}} \) = \( {π_{(0)}} \) P = (1, 0, 0) \( [\begin{matrix}0.8 & 0.13 & 0.06 \\ 0.14 & 0.2 & 0.14 \\ 0.06 & 0.12 & 0.07 \\ \end{matrix}] \) = (0.8, 0.13, 0.06).

We can see its state is more probably lies in \( {i_{1}} \) , which means its closing price is more likely to ascend the next day. We can even analyze \( {π_{(2)}} \) , \( {π_{(3)}} \) , etc.

But, such calculation is too complex and lacks efficiency. Also, according to a large sum of analysis, the state probability will tend toward a stable price, and this regular price has nothing to do with the initial cost. Based on the sound condition of the Markov chain:

\( \begin{cases} \begin{array}{c} πP=π \\ \sum _{i=1}^{n}{x_{i}}=1,π= \end{array} \end{cases}({x_{1}}{,x_{2}}{,x_{3}}) \)

Among them, \( {x_{1}}, {x_{2}},{x_{3}} \) represent three states (states 1, 2, and 3). Then let's juggle the equations we have:

\( {({x_{1}}{,x_{2}}{,x_{3}})=[\begin{matrix}0.8 & 0.13 & 0.06 \\ 0.14 & 0.2 & 0.14 \\ 0.06 & 0.12 & 0.07 \\ \end{matrix}](x_{1}},{x_{2}},{x_{3}}) \)

\( {x_{1}}+{x_{2}}+{x_{3}}=1 \)

\( {x_{1}}≈{0.5773,x_{2}}≈ \) 0.3695, \( {x_{3}}≈ \) 0.0532

That means it has a probability of 57.73 \( \% \) ascending, \( 36.95\% \) fat, \( 5.32\% \) descending.

The result is similar to the one we had before, and they corroborate each other, showing a feasible way to make a prediction. The next object is the China Union:

Table 4: The stock price of the China Union [5].

As you can see, the data is here. So we can likewise do the same. Unlike the former, the data here is quite close, so we elected a smaller interval: 0.3 as a limit constant. Here is an ultimate table:

Table 5 The change situation of the data2.

Then comes the probability; after a series of processes, the chart tells us that \( {P_{11}} \) =0.09, \( {P_{12}} \) =0.07, \( {P_{22}} \) =0.3, \( {P_{33}} \) =0.07, \( {P_{23}} \) =0.11, \( {P_{32}} \) =0.12, \( {P_{13}} \) =0.08, \( {P_{31}} \) 0.07, \( {P_{21}} \) =0.09.

Here the initial vector \( {π_{(0)}} \) = (1, 0, 0), and

P = \( [\begin{matrix}{P_{11}} & {P_{12}} & {P_{13}} \\ {P_{21}} & {P_{22}} & {P_{23}} \\ {P_{31}} & {P_{32}} & {P_{33}} \\ \end{matrix}] \) = \( [\begin{matrix}0.09 & 0.07 & 0.08 \\ 0.09 & 0.3 & 0.11 \\ 0.07 & 0.12 & 0.07 \\ \end{matrix}] \)

\( {({x_{1}}{,x_{2}}{,x_{3}})=[\begin{matrix}0.09 & 0.07 & 0.08 \\ 0.09 & 0.3 & 0.11 \\ 0.07 & 0.12 & 0.07 \\ \end{matrix}](x_{1}},{x_{2}},{x_{3}}) \)

\( {x_{1}}+{x_{2}}+{x_{3}}=1 \)

Figure out \( {x_{1}}{,x_{2}}{,x_{3}} \) in the same way： \( {x_{1}}=0.4340 {x_{2}}=0.1199 {x_{3}} \) =0.4461. That means it is more probably descend. So we'd better buy some tomorrow. It fell from 4.8 (12.25) to 4.78 the next day (12.26).

4. Results

This way, we can estimate the probability of falling into three states (ascend, flat, and descend). After such analysis, we can obtain more profit in the stock market. It's convenient and with real significance. Through this model, we can see how the closing price will change in an obvious way. We can sell them all tomorrow if it is more likely to ascend the next day. We can even predict the day after tomorrow to earn more…If it is flat, we can just take a break. If it is going to descend, stockholders may sell them as soon as possible to prevent a more significant loss. Despite the randomness of the stock market, as long as we know where it will move, we can always be the winners.

5. Conclusion

Generally, the Markov chain could adequately predict the stock price. With the help of this model, we can forecast the change situation just by inserting a set of data. This work emphasized building a stock predicting model and briefly introduced how it works. After doing this, we have proved its practicability through plenty of current statistics. Shortly, we may try to promote the way to predict stock prices in different ways--not only the Markov chain.