Differential equation model of financial market stability based on Internet big data

The close connections and interactions among financial sub-markets have affected the development, changes and operation laws of each market, forming an overall linkage development trend. Existing research results fully demonstrate that the three markets of the stock market, foreign exchange market and currency market are significant. They mutually influence the relationship. Changes in one market directly or indirectly affect the changes in the other two markets. This article attempts to use the ideas and analysis methods of evolutionary finance and system dynamics to dig out the evolutionary characteristics and interaction relationships of financial markets [1]. Then we will reveal the operating laws of the financial market, the direction of market fluctuations and the nonlinear evolution mechanism. It is well known that the devaluation of the RMB in the foreign exchange market will cause capital outflows and cause the stock market to be depressed. When the interest rate is lowered, a large amount of money flows into the stock market from the money market, which enhances the attractiveness of the stock market. Then, how will the stock market change when the currency depreciates and interest rates are lowered at the same time? There are numerous such problems among the three markets. The study of such issues has important theoretical and practical significance. For the government, when an emergency occurs in one market, it can indirectly restore the abnormally volatile market to normal by adjusting the other two markets. For investors, when a market fluctuates, investors can use the nonlinear evolution structure to anticipate changes in the other two markets to avoid risk. This shows that it is particularly important to study the nonlinear evolution of the three markets.

In response to the above problems, this article selects 12 important decision-making indicators from 19 important indicators in the three markets through causality testing. On this basis, we have established a comprehensive indicator that can fully reflect the changes in the three markets [2]. Then the article is based on three overall markets as the research object established a three-market nonlinear evolution model based on ordinary differential equations. Ordinary differential equations can fully reflect the time-varying characteristics of variables, and it can more comprehensively reflect the evolutionary relationship between the three markets. Finally, the paper uses China's monthly data from January 2005 to May 2019 to simulate the specific evolution between the three markets. Form, the established nonlinear evolution model with constraints better describes the nonlinear evolution structure of the three markets and the mutual influence between the three markets.

This article starts from two aspects: theoretical analysis and references. This choice not only considers the economic significance of the indicators in the real market, but also considers the market indicators that are often used in existing studies. We select the market indicators of this article based on three factors: (1) important indicators of a single market. The stock turnover (ST) of the stock market can reflect the trading status of the stock market, and it is one of the important indicators of the stock market; (2) important indicators between markets. Money market money supply M0 and broad money (M2) can explain the stock market price changes to a certain extent, so we choose M0 and M2 as money market indicators to reflect part of the stock market information at the same time; (3) We select available and complete indicators. Based on the consideration of the above three factors, we selected 19 market indicators that can accurately reflect the characteristics of the market from the stock, currency and foreign exchange markets, as shown in Table 1.

Whether we choose market indicators to reflect the interaction between markets is very important. Since the existing research has not conducted a comprehensive test on the interaction of 19 market indicators, this paper conducts a comprehensive causality test on the indicators [3]. We screened out decision-making indicators that not only reflect the market conditions in which they are located, but are also closely related to the other two markets. This article uses the causality test process shown in Figure 1. This causality test method speeds up the calculation on one hand, and on the other, sufficient causality judgements have been made on the indicators.

Fig. 1

This study takes three overall markets as the research objectives. We need to combine multiple decision-making indicators in the stock, currency, and foreign exchange markets into corresponding market comprehensive indexes to represent each market. The construction method is similar to the foreign exchange market stress index. In the stock market, the market value of stocks and the turnover of stocks are extremely large indicators [7]. The larger the value, the more capital investors invest in the stock market, and the more active stock transactions are. Similarly, the Shanghai Composite Index (SHCI) and the Shenzhen Stock Exchange Component Index (SZCI) are also very large indicators. The larger the value the higher the investor's expectations of the stock market are. Based on this, the following stock market comprehensive index is established in this article: (9) SMI=SCC−μSCCσSCC+SCC−μSHCIσSHCI+SZCI−μSZCIσSZCI+ST−μSTσST SMI = {{SCC - \mu SCC} \over {\sigma SCC}} + {{SCC - \mu SHCI} \over {\sigma SHCI}} + {{SZCI - \mu SZCI} \over {\sigma SZCI}} + {{ST - \mu ST} \over {\sigma ST}}

The larger the value of M0, M1 and M2, the greater the money supply. The interbank lending rate (IBOR) is a very small indicator. The smaller the value, the bank can obtain funds at a lower cost. Based on this, we build the following currency market composite index: (10) MMI=M0−μM0σM0+M1−μM1σM1+M2−μM2σM2−IBOR−μIBORσIBOR MMI = {{M0 - \mu M0} \over {\sigma M0}} + {{M1 - \mu M1} \over {\sigma M1}} + {{M2 - \mu M2} \over {\sigma M2}} - {{IBOR - \mu IBOR} \over {\sigma IBOR}}

In the foreign exchange market, we used the foreign exchange market stress index to establish the following foreign exchange market comprehensive index: (11) FMI=FER−μFERσFER+IEV−μIEVσIEV+U/C−μU/CσU/C−H/C−μH/CσH/C FMI = {{FER - \mu FER} \over {\sigma FER}} + {{IEV - \mu IEV} \over {\sigma IEV}} + {{U/C - \mu U/C} \over {\sigma U/C}} - {{H/C - \mu H/C} \over {\sigma H/C}}

In order to illustrate the effectiveness of the three market composite indexes, this article gives a historical trend chart of the three market composite indexes from January 2005 to May 2019 (as shown in Figure 2).

Fig. 2

In order to discuss the effects of the above two non-constrained and constrained nonlinear evolution models, this paper uses the BP neural network model and the multiple linear regression model to model under the same data. We compare and analyse the effects of the unconstrained nonlinear evolution model and the constrained nonlinear evolution model. The data are a comprehensive index obtained through the causality test and the index construction process. We randomly select 80% of the full sample as the learning sample, and the remaining 20% as the test sample. In order to improve the accuracy of the model, the number of iterations of the COMDE algorithm is 10,000 generations, the number of populations is 200 and the remaining parameters are default values [13]. The number of iterations of the BP neural network is set to 10,000 times, and the remaining parameters are default values. The values of K₁ and K₂ are more flexible; when the values of K₁ and K₂ are large, the model has a higher learning accuracy, but the structure is relatively unclear. When the values of K₁ and K₂ are small, the model has a clear structure, but the learning accuracy is poor. This article considers two factors: K₁ = 5 and K₂ = 10. Generally, the information in the past period of time has a guiding effect on the future market.

From the perspective of model accuracy, the constrained nonlinear evolution model has the best overall performance under different values of m. It has strong learning ability for learning samples and evolutionary ability for testing samples. From the definition of model form, the constrained nonlinear evolution model and the multiple linear regression model have the simplest structure. The constrained nonlinear evolution model has only a few non-zero parameters that can clearly analyse the influence relationship between each comprehensive index. Compared with the unconstrained nonlinear evolution model the structure of is relatively unclear [14]. The BP neural network model is a black box operation. It is not difficult to find that the effect and structure of the constrained nonlinear evolution model are optimal, and it can analyse the nonlinear evolution relationship of the three markets well.

From the different values of m, the error of the constrained nonlinear evolution model is the smallest when m = 6. As m increases, the error begins to increase. It shows that the stock market, the currency market and the foreign exchange market have the most significant interactions in the short term. When the government regulates the market, it should pay attention to short-term market information. The effectiveness of market information gradually decreases over time as: (18) {ΔSMIt=0.1156−0.3096FMIt−2−0.4919FMIt−3+0.9231FMIt−4−0.1646SMIt−6 +0.0506FMIt−6+0.1185FMIt−1FMIt−2−0.1479FMIt−1FMIt−4 +0.5909SMIt−2FMIt−3−0.8077SMIt−2FMIt−5+1.0156SMIt−3FMIt−5 −0.7162SMIt−3FMIt−6−0.3323FMIt−3FMIt−6+0.3052FMIt−4FMIt−5 −1.0760SMIt−4FMIt−5+0.9167SMIt−4FMIt−6ΔMMIt=0.1205−0.5259FMIt−1−0.0366MMIt−2−0.7861FMIt−2+0.1343FMIt−3 +1.2355FMIt−4−0.2591MMIt−1FMIt−6−0.1529FMIt−1MMIt−5 −0.1968FMIt−2SMIt−4+0.7831FMIt−2FMIt−5−2.4403FMIt−3MMIt−5 +2.3672FMIt−3FMIt−5+4.2843FMIt−4MMIt−5−2.4791FMIt−4FMIt−5 −0.4506FMIt−4FMIt−6−1.6089MMIt−5FMIt−5ΔFMIt=0.0041−0.1914FMIt−1+0.2331FMIt−2+0.1648FMIt−4−0.1495FMIt−5 −0.0559FMIt−6+0.0399FMIt−1FMIt−4−0.2672FMIt−1FMIt−6 −0.5993FMIt−2MMIt−3+0.4584FMIt−2FMIt−6+0.5579MMIt−3FMIt−3 +0.0740MMIt−3FMIt−4−0.3212FMIt−32−0.0252FMIt−3SMIt−4 +0.0735SMIt−4FMIt−4+0.0155FMIt−4FMIt−6 \left\{{\matrix{{\Delta SM{I_t} = 0.1156 - 0.3096FM{I_{t - 2}} - 0.4919FM{I_{t - 3}} + 0.9231FM{I_{t - 4}} - 0.1646SM{I_{t - 6}}} \hfill\cr {\quad \quad \quad+ 0.0506FM{I_{t - 6}} + 0.1185FM{I_{t - 1}}FM{I_{t - 2}} - 0.1479FM{I_{t - 1}}FM{I_{t - 4}}} \hfill\cr {\quad \quad \quad+ 0.5909SM{I_{t - 2}}FM{I_{t - 3}} - 0.8077SM{I_{t - 2}}FM{I_{t - 5}} + 1.0156SM{I_{t - 3}}FM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 0.7162SM{I_{t - 3}}FM{I_{t - 6}} - 0.3323FM{I_{t - 3}}FM{I_{t - 6}} + 0.3052FM{I_{t - 4}}FM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 1.0760SM{I_{t - 4}}FM{I_{t - 5}} + 0.9167SM{I_{t - 4}}FM{I_{t - 6}}} \hfill\cr {\Delta MM{I_t} = 0.1205 - 0.5259FM{I_{t - 1}} - 0.0366MM{I_{t - 2}} - 0.7861FM{I_{t - 2}} + 0.1343FM{I_{t - 3}}} \hfill\cr {\quad \quad \quad+ 1.2355FM{I_{t - 4}} - 0.2591MM{I_{t - 1}}FM{I_{t - 6}} - 0.1529FM{I_{t - 1}}MM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 0.1968FM{I_{t - 2}}SM{I_{t - 4}} + 0.7831FM{I_{t - 2}}FM{I_{t - 5}} - 2.4403FM{I_{t - 3}}MM{I_{t - 5}}} \hfill\cr {\quad \quad \quad+ 2.3672FM{I_{t - 3}}FM{I_{t - 5}} + 4.2843FM{I_{t - 4}}MM{I_{t - 5}} - 2.4791FM{I_{t - 4}}FM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 0.4506FM{I_{t - 4}}FM{I_{t - 6}} - 1.6089MM{I_{t - 5}}FM{I_{t - 5}}} \hfill\cr {\Delta FM{I_t} = 0.0041 - 0.1914FM{I_{t - 1}} + 0.2331FM{I_{t - 2}} + 0.1648FM{I_{t - 4}} - 0.1495FM{I_{t - 5}}} \hfill\cr {\quad \quad \quad- 0.0559FM{I_{t - 6}} + 0.0399FM{I_{t - 1}}FM{I_{t - 4}} - 0.2672FM{I_{t - 1}}FM{I_{t - 6}}} \hfill\cr {\quad \quad \quad- 0.5993FM{I_{t - 2}}MM{I_{t - 3}} + 0.4584FM{I_{t - 2}}FM{I_{t - 6}} + 0.5579MM{I_{t - 3}}FM{I_{t - 3}}} \hfill\cr {\quad \quad \quad+ 0.0740MM{I_{t - 3}}FM{I_{t - 4}} - 0.3212FMI_{t - 3}^2 - 0.0252FM{I_{t - 3}}SM{I_{t - 4}}} \hfill\cr {\quad \quad \quad+ 0.0735SM{I_{t - 4}}FM{I_{t - 4}} + 0.0155FM{I_{t - 4}}FM{I_{t - 6}}} \hfill\cr}} \right.

In order to study the nonlinear evolution relationship between the three markets, the results of the constrained nonlinear evolution model when m is 6 are given here as shown in Eq. (18). At the same time, this is also a nonlinear evolution structure between the three markets. Among them, SMI, MMI and FMI represent the stock, currency and foreign exchange market composite indexes, respectively [15]. The relationship between the three market composite indexes is shown in Table 4.

The items in Table 4 represent the way and degree of influence of the left market on the upper market. From Table 4, the following conclusions can be drawn: (1) each market has the most significant impact on itself, and all participate in the evolution of its own market in linear and nonlinear forms [16]. It shows that the operation mechanism of China's financial sub-markets is relatively independent. Although the various sub-markets are closely connected, the independent development of each sub-market is still the main. (2) The foreign exchange market has an important influence on the stock market and the money market. The foreign exchange market participates in the evolution of the stock and money market in a linear and nonlinear form, and the degree of influence is huge. When the foreign exchange market fluctuates, the other two markets will also fluctuate in the short term. (3) The stock market has less effect on the evolution of the other two markets. This is due to the strong government regulation in the currency and foreign exchange markets, and fluctuations in the stock market are difficult to transmit to the currency and foreign exchange markets.

This paper constructs three market-comprehensive indexes of stocks, currency and foreign exchange through index selection and causality test, and establishes three nonlinear evolution models of markets driven by data. We use the COMDE algorithm to obtain the nonlinear evolution structure and evolution relationship of the stock, currency and foreign exchange markets, and break through the nonlinear evolution research based on theoretical research. The constrained nonlinear evolution model is for the nonlinear evolution problem between the three markets. They have good learning and evolution ability and they have the clearest structure. The three markets not only have independent operating mechanisms, but also have a significant influence relationship with the other two markets. In particular, the impact of the foreign exchange market on the stock and currency markets is the most obvious of the three markets.