Frontiers in Business, Economics and Management 
ISSN: 2766-824X | Vol. 17, No. 1, 2024 

 

230 

Stock Price Prediction Based on BP Neural Network and 
ARIMA Model 

Ruiqi Zhao 

Nanjing University of Posts and Telecommunications, Nanjing, China 

 

Abstract: With the progress and development of society, stock investment has become an important part of people's daily lives. 
High risk and high return are the characteristics of stock investment. Individual investors and institutional investors alway s pay 

attention to stock market trends, analyze financial data, and try to predict the development trend of stocks. Studying the future 

trend of stock prices of listed companies and predicting the stock price of a certain company not only has extremely attracti ve 

application value, but also has significant theoretical significance, attracting the attention of investors and academia. The stock 

market is unpredictable and an extremely complex system with too many influencing factors. It will be very difficult to 

thoroughly understand the mechanism of stock market changes theoretically. This study investigates the application of time series 

prediction technology in stock price prediction, proposes stock price prediction models based on ARIMA and BP neural network, 

and verifies the feasibility and effectiveness of the model prediction through prediction. 

Keywords: ARIMA, BP neural network, Stock price prediction. 

 

1. Introduction 

1.1. Purpose and Significance of Stock Price 
Prediction 

The stock market holds an important position in the field of 

financial investment. With the development of social 

economy, people are paying more and more attention to 

financial management and investment. The stock market has 

gradually become an equally important investment channel as 

banks and insurance, and is one of the main ways for many 

families and individuals to manage their finances. In recent 

years, with the continuous rise of the stock market, the 

number of investors buying into the Shanghai and Shenzhen 

stock markets has been increasing. According to statistics 

from a Chinese securities company, at the beginning of each 

year, nearly 9000 new users are added in a single day. At 

present, the total number of users in the Shanghai and 

Shenzhen stock markets far exceeds 75 million. When the 

market is at a low point, some relatively rational institutional 

investors with large amounts of funds are often able to seize 

the opportunity to enter the market early and build positions 

calmly. However, by the time the stock market is at its craziest, 

they are already busy picking the fruits. In this regard, retail 

investors should learn to analyze and invest in the stock 

market with a longer-term perspective. However, due to 

factors such as inaccurate information, differences in investor 

perceptions, the complexity of various analytical techniques, 

and the randomness of stock price changes, actual 

investments often fail to achieve, or even the opposite, 

expected results, resulting in the loss of investors' capital, 

known as stock market risk. The purpose of price prediction 

is to try to find a method that can achieve maximum returns 

with minimal risk. Since the emergence of stocks, there have 

been countless people trying to predict prices. It can be said 

that every investor is looking for a method to predict stock 

prices. Many scholars at home and abroad are dedicated to 

studying the changing trends of the stock market and 

establishing corresponding prediction models, providing 

prediction methods, striving to avoid large stock market 

fluctuations, reduce investment risks, and maintain economic 

prosperity and stability. However, although there are 

countless people studying and predicting stock prices, almost 

no one can find a perfect and foolproof method to predict 

stock prices. As a foreign economist said, stock market 

changes are like walking a drunken man, difficult to predict. 

Artificial neural networks are a rapidly developing 

interdisciplinary field widely used in signal processing, 

adaptive control, and financial prediction. 

1.2. Current Methods for Predicting Stock 
Prices 

1.2.1. Decision Tree Model 
Decision tree is also known as decision tree. In the decision 

tree method, the decision tree is first constructed from the 

instance set, which is a guided learning approach. This 

method first forms a decision tree based on the training set 

data. A decision tree represents the tree structure of a decision 

set. The final result is a tree, with leaf nodes being class names 

and middle nodes being attributes with branches 

corresponding to a possible value of the attribute. The core of 

decision tree technology in discovering data patterns and rules 

is the inductive algorithm. Inductive reasoning derives a 

regular conclusion from the characteristics, features, or 

properties represented by several facts through comparison, 

summarization, and generalization. Inductive reasoning 

attempts to obtain a complete and correct description from 

specific observations of a part or the whole of an object, that 

is, to draw universal conclusions from specific facts. 

1.2.2. Neural Network Model 
Neural networks are data models that simulate the structure 

of the human brain; therefore, neural networks are systems 

with self-learning capabilities. Just like the brain, neural 

networks learn from a set of input data and adjust model 

parameters based on this new cognition to discover patterns 

in the data. Therefore, the working process of neural networks 

can be divided into two stages. 

Learning stage: Training the network, mainly adjusting the 

connection weights and connections between neurons in the 

network. The information processing capability of neural 



 

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networks is mainly determined by the connection method and 

connection weights. Because the information processing 

capability of neural networks is mainly acquired through the 

learning stage and affects the entire working stage, the 

connection method and connection weights between neurons 

in neural networks are also known as long-term memory. 

Although the time required for different learning modes and 

algorithms in neural networks varies, generally speaking, the 

training time of neural networks is longer and much longer 

than the processing time of individual data. 

Working stage: The trained network can be used for 

practical work. At this time, the connection weights and 

connection parties of the network are fixed and unchanged. 

The working process is manifested as the mapping and 

change of input data in the state space. The final stable state 

of the neural network is the working output. Compared to the 

time cost of the learning phase, the speed of the work phase 

is relatively fast. Because the state of neurons often changes 

during the working stage, the state of neurons in a single stage 

is called short-term memory. 

1.3. Application of Neural Network Theory in 
Stock Price Prediction 

Due to the high degree of uncertainty and unpredictability 

in the stock market, as well as the strong non-linear prediction 

ability of neural network theory, many people are trying to 

use neural network theory to predict stock prices in order to 

grasp the laws of the stock market, and have achieved 

satisfactory results in different directions. Below are some 

methods of using neural network theory for prediction. 

The significance and practical application value of this 

research project lie in: 

(1) Explore new stock investment risk analysis and 

evaluation techniques, enrich and improve the system of stock 

investment risk analysis and evaluation methods. 

(2) Introducing data mining techniques into stock 

investment risk analysis and evaluation is beneficial for 

promoting the development of methods for stock investment 

risk analysis and evaluation. 

(3) Providing a quantitative analysis technique for stock 

investment risk based on multiple factors and applying 

multiple evaluation and prediction index indicators is 

beneficial for improving the accuracy of evaluations. 

(4) Provide new ideas and practical methods for the 

investment activities and decisions of individual and 

institutional investors in the stock market. 

2. Stock Price Prediction Based on 
ARIMA Model 

2.1. Main Technical Introduction 

2.1.1. Mean Square Error of MSE 
MSE is used to measure the deviation between predicted 

and true values, and its formula is as follows:      

1

𝑚
∑ (𝑦𝑖 − �̂�𝑖)

2𝑚
𝑖=1                    (1) 

In equation (1), is the predicted value, is the true value, and 

m is the data size. 

2.1.2. ADF Unit Root Inspection 
Unit root test refers to checking whether there are unit roots 

in a sequence, as the existence of unit roots is a non-stationary 

time series. Unit root refers to the unit root process, which can 

prove that the presence of unit root processes in a sequence is 

non-stationary and can lead to spurious regression in 

regression analysis. Stability is a necessary condition for the 

autoregressive model ARMA, so for time series, the first step 

is to ensure that the n-order difference sequence applied to 

autoregression is stationary. 

2.1.3. ACF, PACF autocorrelation coefficient and partial 
autocorrelation coefficient 

In time series analysis, autocorrelation function (ACF) and 

partial autocorrelation function (PACF) are commonly used 

to determine the coefficients and order of ARMA (p, q) 

models. The autocorrelation function (ACF) describes the 

linear correlation between time series observations and their 

past observations. Partial autocorrelation function (PACF) 

describes the linear correlation between time series 

observations and their past observations given intermediate 

observations. The calculation formula is as follows:   

ACF(k) = 𝜌𝑘 =
𝐶𝑜𝑣(𝑦𝑡,𝑦𝑡−𝑘)

𝑉𝑎𝑟(𝑦𝑡)
               (2) 

2.1.4. AIC Criteria 
AIC: Akaike Information Criterion (AIC) is a commonly 

used method for evaluating the quality of ARIMA models. Its 

calculation formula is as follows: 

AIC = 2k − 2ln⁡(L)                 (3) 

Among them, k is the number of model parameters, n is the 

number of samples, and L is the likelihood function. 

2.1.5. ARIMA Model 
(1) Autoregressive model AR 

The autoregressive model first needs to determine an order 

p, which represents the number of historical values used to 

predict the current value. The formula for the p-order 

autoregressive model is defined as: 

y𝑡 = 𝜇 + ∑ 𝛾𝑖𝑦𝑡−𝑖 +∈𝑡
𝑝
𝑖=1               (4) 

In the above equation, yt is the current value, u is a constant 

term, p is the order, γ i is the autocorrelation coefficient, and 

Et is the error. 

(2) Moving Average Model MA 

The moving average model focuses on the accumulation of 

error terms in the autoregressive model. The formula for the 

q-order autoregressive process is defined as follows: 

y𝑡 = 𝜇 + 𝜖𝑡 + ∑ 𝜃𝑖𝜖𝑡−𝑖
𝑞
𝑖=1               (5) 

The moving average method can effectively eliminate 

random fluctuations in predictions. 

(3) Autoregressive Moving Average Model ARMA 

By combining the autoregressive model AR with the 

moving average model MA, we obtain the autoregressive 

moving average model ARMA (p, q), which is calculated 

using the following formula: 

y𝑡 = 𝜇 + ∑ 𝛾𝑖𝑦𝑖−1 + 𝜖𝑡 + ∑ 𝜃𝑖𝜖𝑡−𝑖
𝑞
𝑖=1

𝑝
𝑖=1        (6) 

2.2. Establishment, Analysis, and Validation of 
Time Series Data Models 

The ARIMA model is used in this article, and the selected 

data includes the closing, opening, and fluctuations of a listed 

company from March 26, 2020 to March 26, 2021.  

Using ARIMA (p, d, q) model to predict stock prices can 

help us better understand financial related knowledge. 

The stock prices of listed companies can be predicted using 

the basic ARIMA (Automatic Regression Integral Moving 

Average) model. 



 

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2.3. Implementation Steps 

1) Load data and extract its stable data. If the data is not 

stable, differential operations can be performed until the data 

becomes stable. 

2) Conduct hypothesis testing, and if successful, continue 

with (h=logic 1) 

3) Generate autocorrelation and partial autocorrelation that 

help find AR and MA ranges 

4) Construct algorithms to find the best ARIMA model 

5) Use the selected ARIMA model to generate prediction 

curves and compare them with the original data 

6) Predict the future stock price and compare it with the 

actual value. 

Organize the raw data and create a trend chart as shown 

below, which provides a preliminary understanding of the 

stock's upward trend from March 2020 to March 2021. 

 
Figure 1. Original Data 

Stability analysis was conducted on the data. As shown in 

Figure 2, the sequence fluctuated randomly around 0, with 

stable fluctuations and no obvious trend changes. The data is 

a stationary time series. 

 
Figure 2. Stability Analysis 

 
Figure 3. Autocorrelation Coefficient 

 

 
Figure 4. Partial autocorrelation coefficient 

Draw autocorrelation and partial autocorrelation graphs. 

From Figures 4 and 5, it can be seen that the 1st and 2nd order 

partial autocorrelation coefficients of the PAC sequence 

exceed ± 2 times the estimated standard deviation. After the 

2nd order, the partial autocorrelation coefficient is within ± 2 

times the estimated standard deviation and rapidly decreases 

to 0, indicating that the partial autocorrelation function is 

truncated after the 2nd order; Similarly, if the correlation 

coefficient of the PAC sequence exceeds 5% and falls outside 

± 2 times the estimated standard deviation, that is, the 

autocorrelation function is rounded off, and combined with 

the table, it can be preliminarily determined that p=1 or 2 and 

q=0. Using the AIC criterion to traverse the minimum AIC 

value yields p=7 and q=5. 

 



 

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Figure 5. Predicted and Original Values 

 

 
Figure 6. Prediction 

Finally, the predicted data output by Matlab is shown below: 

 
Figure 7. Predicted Data 

3. Stock Data Prediction Model Based 
on BP Network 

3.1. Algorithm Introduction 

BP (Back Propagation) neural network was proposed in 

1986 by a team of scientists led by Rumelhart and McColland. 

It is a multi-layer feedforward network trained using error 

backpropagation algorithm. BP network can learn and store a 

large number of input-output pattern mapping relationships 

without revealing the mathematical equations that describe 

these mapping relationships in advance. Its learning rule is to 

use the steepest descent method to continuously adjust the 

weights and thresholds of the network through 

backpropagation, in order to minimize the sum of squared 

errors of the network. It is currently one of the most widely 

used neural network models. Therefore, understanding the 

structure of BP networks and weight adjustment algorithms 

plays an important role in learning other neural networks. BP 

neural network is mainly applied in the following aspects. 

Function approximation: Train a network to approximate a 

function using input vectors and corresponding output vectors. 

Pattern recognition: Connect it to the input vector using a 

specific output vector. 

Classification: Classify the appropriate way defined by the 

input vector. 

Data compression: Reduce the dimensionality of output 

vectors for transmission or storage. 

3.2. Specific Steps of BP Neural Network 

3.2.1. Determination of network layers and ANN input 
nodes 

This article adopts a three-layer network model. The input 

layer serves as a buffer memory, accepting external input data, 

so the number of nodes depends on the dimensionality of the 

input vector. The stock price trend follows a wave like pattern 

and exhibits periodicity. In technical analysis, the appropriate 

choice of analysis cycle has a direct impact on the prediction 

results. Due to time, task volume, and ease of selection, we 

have chosen 1 day, which means date=1. 

3.2.2. Data preprocessing 
Data preprocessing is the process of converting data 

obtained from the stock market into input data that can be 

recognized by ANN. Starting from day n, let ANN predict the 

rise and fall of stock prices. The raw data sequence of the most 

recently traded stocks ( )x t , as well as the processed data 

sequence  ( )x t   input to ANN, should be processed as 
follows: 

1 0! ( 1) , ( ) ( ) 0.5x date i x n date x t x t x x    = − = − = − +   
3.2.3. Number of nodes in the hidden layer 

The number of hidden layer nodes is directly related to the 

required number of input-output units for solving the problem. 

For BP networks, the following formula can be used for 

design: 

0i
n n n a= + +

                
In the formula, n  represents the number of hidden layer 

nodes; in  is the number of input nodes; 0n  is the number 

of output nodes; a  is a constant between 1 and 10. 

3.3. Output Node Definition 

The output node number of the artificial neural network is 

1, and the threshold a=0 for the trend of stock price changes 

is selected as the average value of the rise and fall over a 

certain period of time; Prediction step size k=1. The output 

node Y (t) can be defined as follows: 

When Y (t)=0, it is predicted that the daily increase or 

decrease of the stock price starting from t+1 will not exceed 

a, indicating that it is expected to be in a downward trend; 



 

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When Y (t)=1, it is expected that the stock price will be on an 

upward trend in the single day starting from t+1, with a 

cumulative increase exceeding a. 

3.4. Determination of Other Network 
Parameters 

The number of hidden layer nodes is set to 6, and the node 

action function is an S function, that is, f (s)=11+e-s; The 

learning coefficient is 0.7. When learning, it is required that 

the output error d be ≤ 0.1. When predicting, if the output of 

the output layer is greater than 0.5, it is judged as 1; Less than 

or equal to 0.5 is judged as 0. 

The data selected for this project is the closing, opening, 

and price fluctuations of Swire Group A from January 3, 2017 

to December 31, 2019. Four sets of data, including opening, 

highest, lowest, and closing, are selected to predict the closing 

data of the next day using the five sets of data from that day, 

thus establishing an equivalent stock data prediction model. 

Adopting a three-layer BP network structure consisting of an 

input layer, a hidden layer, and an output layer, the input layer 

contains four neurons, the hidden layer contains three neurons, 

and the output layer is one neuron. Among them, the 

activation function of the hidden layer neurons adopts an 

asymmetric Sigmoid function, and the function expression is:. 

The activation function of the output layer neurons adopts a 

linear function. The 738 sets of data are divided into three 

equal parts, of which two parts are used as training samples 

to train and learn the network; Another sample is used as a 

test to evaluate the generalization ability of the trained 

network. The BP algorithm is used to modify the weights of 

the hidden layer and output layer, in order to minimize the 

difference between the calculated output and the actual 

sample output, and ultimately achieve more accurate 

prediction. 

Results and Analysis 

Figure 8 shows the display content of the MATLAB 

command window, Figure 9 shows the error curve of the 

training sample set, Figure 10 shows the comparison between 

the calculated and actual outputs of the training sample set, 

and Figure 11 shows the comparison between the predicted 

and actual data of the test sample set. 

 
Figure 8. MATLAB Command Window Display Content 

 

 
Figure 9. Error Curve of Training Sample Set 

 

 
Figure 10. Comparison between Calculated Output and Actual 

Output of Training Sample Set 

 

 
Figure 11. Comparison between predicted data and actual data in 

the test sample set 

From Figures 8 and 9, it can be seen that there is a relatively 

large error between the calculated output and the actual output 

at the beginning, and the last set of data has the smallest error. 

The weight matrices of the trained network's hidden layer and 

output layer are W1 and W2, respectively. After inputting the 

test sample set into the network, the error is 2.2473, which is 

slightly larger than the error of the training sample set. 

From Figure 10, it can be seen that for the 492 sets of 



 

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training samples, the overall fit between the calculated output 

and the actual sample output is relatively good. Although 

there were several sets of data with significant fluctuations in 

the middle, the algorithm adjusted the weights in a timely 

manner. It can be seen that the last few sets of sample data fit 

relatively well, indicating that the error eventually converged. 

From Figure 11, it can be seen that the error between the 

calculated output and the actual sample output obtained from 

246 sets of test samples is larger than that of the training 

sample set, indicating that the designed network has poor 

generalization ability. After analysis, one reason is that there 

are too few training samples, and the other is that the 

prediction model is too simple, and the closing price is also 

related to other data. 

4. Summary 

This system first collects data on stocks during a certain 

period of time, including various data that can reflect the 

nature of stocks, such as stock indices, opening prices, closing 

prices, trading volumes, and some statistical data that can be 

used for macroeconomic analysis. 

Subsequently, this article applies the BP neural network 

algorithm in MATLAB to simulate and predict the future 

trend of stocks, in order to judge the quality of stocks and the 

buying and selling strategies. Due to the fact that the stock 

market is a highly complex nonlinear dynamic system, the BP 

neural network method can achieve a highly nonlinear 

mapping from R space to nR  space using only samples. It 

has good adaptability to atypical data and obvious superiority 

in dealing with missing values and nonlinear problems. 

However, due to the complexity of stock data, a single 

prediction method is limited. How to use multiple methods to 

comprehensively make qualitative and quantitative 

predictions is also the focus of future research on stock 

prediction. 

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