Financial product investment decision model based on ultimate benefit

. With the vigorous development of the financial market, more and more people enter the financial market for investment, it is very necessary to build a quantitative analysis and decision model based on data for ordinary traders. This paper selects two time series prediction models GM(1,1) and ARIMA for comparison. For GM(1,1), we improve it to obtain unbiased GM(1,1) and unbiased metabolic GM(1,1). Unbiased metabolism GM(1,1) was selected from the three grey prediction, and three combinations were formed by ARIMA and grey prediction respectively.These three results are used separately for the decision model, which prediction model to choose is determined by the final return. The investment decision model established in this paper is helpful for financial practitioners to improve investment profit and has certain application value.


Introduction
With the thriving of the financial market and the continuous prosperity of people's life, more and more people will put their spare money into the financial market for investment [1], in order to maximize the value of the existing assets. For this purpose, many people invest in volatile financial assets like gold and bitcoin with the mentality of a gambler, hoping to take advantage of sharp shortterm fluctuation and profit from price differences. Unfortunately, it is hard for ordinary people to predict the price changes of financial products accurately [2], so most people end up losing everything. Therefore, it is very necessary to construct a quantitative analysis and decision model based on data for common traders [3].

GM(1,1) Model
The best strategy is defined as maximizing daily returns. Returns are determined by commodity prices after that day, so here we need to build a predictive model to estimate future prices [4]. However, since the forecasting model usually needs to rely on a certain amount of original data, by looking for a large number of literatures, in order to ensure the good forecasting effect of the model in the short term, we consider two models [5]: GM(1,1) model and ARIMA(p,d,q) model.

Traditional GM(1,1) Model
Let the original data sequence be a strict exponential sequence.
After one accumulated generating operation, the sequence is as follows: Where (0) ( ) is the kth value of the original sequence (1) ( ) is the kth value of the new sequence, , are the parameters to be estimated, and is the number of data in the sequence [6]. According to the traditional GM(1,1) model, we construct the following two matrices [7]: Then the estimated relation between parameters and can be represented: The final fitting results are: By comparing (1) with (6), it can be seen that (1) is different from (6), that is, the traditional GM(1,1) model fails when the growth rate of the original data series is large. Therefore, it is necessary to improve it [8]. The estimated parameters ̂ and ̂ can be used to present and .Suppose that the model established for the exponential series is where ̂′ = Then (7) is the unbiased estimated GM(1,1) model of (1). However, as the data are always updating and changing, the information reflected by the old data become meaningless. To get rid of useless data, we can remove the old data. This leads to a more accurate model [9].

Comparison between the GM(1,1) price prediction results
It is known that GM(1,1) is suitable when the dataset is small. The small dataset always contains seven to ten data. Therefore, we select eight gold price data, nine gold price data and ten gold price data from September 12 th 2016 respectively as the known information, use the three GM(1,1) models to predict the gold price in the following three days and calculate the Mean Squared Error. When Mean Squared Error is large, it means the model may not be suitable. MSE(dollar/troy ounce) can be represented as follows: After using MATLAB to calculate MSE, the results are in the following table1. It is shown that Unbiased Metabolic GM(1,1) model would be the best and the model is most suitable when using 9 past prices to predict 3 future prices. Therefore, in the following session, we use Unbiased Metabolic GM(1,1) model to predict prices in the short term.

Introduction
If a time series { } is represented as follows: Then (9) is an Autoregressive Moving Average Model, which is denoted as ARMA(p, q). If a time series { } is d-differenced, where = ∇ and is ARMA(p, q), that is Then (10) is an Autoregressive Integrated Moving Average Model, which is denoted as ARIMA(p, d, q).

Basic step of ARIMA(p, d, q)
(1) Use the law of the ADF unit root to test the variance, trend and seasonal variation and identify whether the series is stationary or not, according to the scatter diagram, autocorrelation function and partial autocorrelation function diagram of the time series,.
(3) Establish the corresponding model according to the identification rules of the time series model. (4) Estimate the parameters, and then test them for significance to determine whether they are statistically significant.
(5) Perform hypothesis test to diagnose whether the residual sequence is white noise. Justify whether the model is reasonable or not. (6) Use the fitted ARIMA model for predictive analysis.

Unbiased Metabolic GM(1,1) model and ARIMA model Result Analysis
Using two models at the same time, under the condition of known gold price data volume of 30 days, 50 days, 70 days, 90 days and 100 days respectively, predict the gold prices of the next three days after the current day. The mean square errors of two models with different dataset are shown in the following table2. The table shows that as the amount of data used for prediction increases, the mean square error of the model increases gradually, indicating that the applicability of gray prediction decreases with the increase of sample size, which verifies that the model is suitable for prediction when the amount of data is small (7~10 data).
Because using the ARIMA model in MATLAB requires more than 10 known data volumes, the applicability test for this model starts with a data volume of 30. As the amount of sample data increases, the mean square error of the ARIMA model first decreases and then increases, and when the amount of data is 70, the mean square error is the smallest.
In general, the ARIMA model is more suitable when the amount of data is large, while the GM(1,1) model is better when the amount of data is small. Therefore, in the subsequent use of the predictive model, we select different combinations of the two models according to different sample data volumes.

The construction of the price prediction models
After comparing the differences between GM(1,1) model and ARIMA model, and knowing the best application scope of each model, we can construct three different price prediction models and predict the prices of commodities in the following three days.
Prediction Model 1: From the analysis before, it is found that ARIMA model is most suitable when the number of known dataset is 70. Therefore, the first model is only using ARIMA model and assumes traders do not make trading decision during the first 70 days.
Prediction Model 2:The second model is the combination of Unbiased Metabolic GM(1,1) model and ARIMA model. Because Unbiased Metabolic GM(1,1) model is suitable for prediction when the dataset is small, especially when the number of known dataset is nine. Therefore, the second model is that from day 10 to day 70, Unbiased Metabolic GM(1,1) model is used for price prediction and after day 70, ARIMA is used for prediction.
Prediction Model 3:The third model is that from day 10, the future prices are predicted by the Unbiased Metabolic GM(1,1).
The algorithms of the above three models are similar. Here the algorithm of model 2 is shown below because it is the combination of model 1 and model 3.

Strategy Decision Model
To select the best trading strategy each day, we solve it using a nonlinear programming model with the goal of maximizing the daily and total return.

Decision variable
Set 1 as the amount of gold bought, 2 as sold, 3 as the number of Bitcoins bought and 4 as sold. They should be all integers. [ 1 , 2 , 3 , 4 ] is the decision variable vector.

Objective function
In order to construct the objective function, this paper creates some indicators and sets some parameters by using the forecast price. The relevant symbols, calculation formulas and explanations are shown in Table 4 and Table 5.  In real market, the minimum transaction volume of Bitcoin is 0.01 coin, while it is 100 troy ounces for gold. Therefore, the expected daily return per day is The assumption is that traders will immediately deposit the cash they get from selling Bitcoin and gold at the bank to earn interest.By searching online, the interest rate of checking accounts in the United States, , is 0.03%. The interest is computed as follows.
Consider that traders would take future short-term returns into account, so the expected return 3 days later is also added in the objective function but has a smaller weight showing that it has a smaller effect on daily decisions.
Set as the expected short-term return, then the objective function is

Constraints
This paper assumes that traders can either buy or sell at one day. Therefore, this paper set the following constraints.
The amount of money spent on transaction that day should be less than the remaining money after yesterday's decision.

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• 100 + 3 • • 0.01 ≤ −1 Besides, if traders want to sell gold or Bitcoin, they must ensure that the amount of gold or Bitcoin they hold after yesterday decision should be greater than the amount they want to sell today.
In conclusion, the nonlinear programming problem is Since gold can be only traded on weekdays, in the subsequent process, the algorithm sets 1 , 2 equal to zero at weekend, that is, the trading volume of gold is equal to 0 at weekends.
Considering the risk of losing money, if the returns of buying gold or Bitcoins cannot cover the costs, then traders would not buy any gold or Bitcoin.

Model Results
Assume that traders initially have $1,000, and make daily decisions for 5 years, we calculate the total assets, using three prediction models and daily decision model. The final assets of the three models are 238006 $,268006 $, and 243494 $ respectively. The second model have the highest value of asset which means the combination of Unbiased Metabolic GM (1,1) and ARIMA model has the best return.

Model Evaluation
This paper considers problems from the perspective of traders in the real life. Models and algorithms embody the logical thinking process of ordinary people when making investment decisions, and abstract them into models and code languages for implementation and optimization.
When making price forecasts, this paper is not limited to a single forecasting model, but fully considers the amount of data, and uses the ARIMA model and the Unbiased Metabolic GM(1,1) model to jointly carry out price prediction. ARIMA model is for long-term prediction and improved GM(1,1) is for short-term prediction. Therefore, the prediction results are more accurate.
The assumptions of the model are closer to actual market conditions, and only a reasonable simplification of reality.The results of the price prediction models are extremely accurate.