An Empirical Analysis of Oil and Stock Markets’ Volatility Based on the DGC-MSV-t Model

We investigate the spillover effect between crude oil future prices, crude oil spot prices, and stock index by using the multivariate stochastic volatility model.+ese tests between eachmarket show the significant Granger causes of spillover effect. More andmore evidences show that the crude oil price has been affected by other financial markets. +e oil future played an important role in the energy market. WTI and Brent oil future have more spillover effect than INE oil future. +e result shows that S&P stock market is more sensitive to the oil price than Shanghai stock market. +e cross-market spillover effect we found can give some advices for the investor of oil and stock market. DIC test shows that DGC-MSV-t is considered effective and more accurate.


Introduction
Crude oil is the blood of modern industry and the most financialized energy product. According to the report of CNPC, China's oil dependence on foreign sources reached 70.8%, with an increase of 1.2 percentage per year. It is expected that China's oil demand will continue to rise in the future, and its dependence on foreign sources will remain high for a long time. In addition, the report also believes that the United States has achieved energy independence through the development of domestic shale gas oilfields, and its control and influence on the global oil market are increasing. OPEC's oil market share and influence are constantly being squeezed and keep going down. ere are many unexpected factors in the crude oil price including geopolitical factors [1,2]. WTI and Brent Crude Oil Futures are the most important pricing benchmarks for the US and European oil markets. In 2018, Shanghai International Energy Exchange (INE) established the first Chinese crude oil future trading product. After two years, the INE has surpassed the Oman and became the world's third largest oil future product. Crude oil prices are highly related to the national economy [3,4]. A lot of research studies show that spillover effect and co-movement between stock and oil price exist in both developed market and emerging market [5,6]. Chen [7] has given sufficient evidence to prove US market has one-way spillover effect to China market due to the closed economic and trading relations. e commodity future market are highly corrlated [8], spillover effect still exists even in the Bitcoin market [9]. e three oil crises in history have proved that the oil trading market is extremely vulnerable to emergencies, such as wars, terrorist attacks, and diseases. Clean energy also brings more spillover to the crude oil price [10]. Luo and Qin [11] have proved the oil price has a significant spillover to the Chinese stock index. In addition, Boubaker and Raza [12] have found that all BRICS stock markets have spillover or subspillover from oil price. e oil-importing and oilexporting countries are both affected by the spillover of cross market [13][14][15]. A lot of work has been carried out using the OVX index to find the volatility spillover of different markets [16,17]. Risk diversification and hedging need to clarify the relationship between markets [18]. Especially in 2020, the spot oil price and oil future are crashing in the COVID-19 pandemic [19]. e stochastic volatility model has added uncertain random disturbance into the time series. e Monte Carlo method is used to estimate the random factors [20], and the degree of random disturbance is estimated on the basis of fitting historical time series [21]. e volatility models of time series mainly include GARCH [22] and SV models [23]. MSV model is effective and performs better [24]. e time-series problem can be solved by the SV model and the MCMC method [25,26]. Various SV models have been built to solve different problems. Ghosh et al. [27] add the nonlinear method to solve nonlinear SV problem, and Nugroho and Morimoto [28] add mean equation to solve mean-SV problem. Chib et al. [29] have improved the SV model with leverage. Omori et al. [30] and Zhongxian et al. [31] have changed the N-distribution to T-distribution, which is suitable for some problems. Jacobs and Li [32] have improved the simulation method of two-factor simulation. Based on the multivariate stochastic volatility model, this paper introduces dynamic correlation coefficients, t-distribution, and Granger causality to construct models. Using the Monte Carlo method, we try to find the volatility spillover effect among INE crude oil futures' price and Shengli oilfield spot price of China, WTI crude oil futures and spot price of USA, BRENT crude oil futures and spot price of UK, the Shanghai Stock Index, and the S&P Index. e article consists of four sections. Section 2 introduces the DGC-MSV-t model, MCMC method, and Gibbs sampling. Section 3 is empirical analysis and result of the data of stock indexes, oil spot price, and oil future price. Section 4 is the conclusion of this paper.

Stochastic Volatility Model.
where y t represents the historical logarithmic return of crude oil and stock prices. Equation (1) shows the basic model with the known y t and the unknown ψ t which are unobservable variables. We combined the Ganger-MSV model and the dynamic-MSV model as Yu and Meyer [33] and replace N-distribution with T-distribution: Equation (2) has multivariate time series. Taking the WTI future (AF) and Brent future (BF) for examples, y af represents the price volatility of WTI future and y af represents the price volatility of Brent future ψ � ψ afaf ψ bfaf ψ afbf ψ bfbf . ψ afbf represents the cross-market spillover from WTI future to the Brent future. ψ bfaf is the opposite. ψ afaf and ψ bfbf represent the autocorrelation of WTI future and Brent future. ρ t represents the dynamic correlation [33]. o reflects the degree of T-distribution. q t+1 , μ and ψ, and 2 Journal of Mathematics

MCMC Method and Gibbs Sampling.
We use the Markov chain as follows: erefore, the one-step transition probability is as How to determine this conditional probability is a key issue. With further research, some powerful tool has been used to solve the N-P problem. Gibbs sampling set X � (X 1 , X 2 ) to follow the M-N distribution:    Journal of Mathematics representative oil future in the world. e descriptive statistics is shown as Table 1. J-B value shows that some data are different from the normal distribution.

Parameter Estimation.
Taking America WTI crude oil future (AF) and China INE crude oil future (CF) as example, we abandon the first 10,000 iterations. en, we simulate the last 80,000 iterations to get the result as Table 2.
In Table 2, ψ cfaf represents the volatility spillover from WTI future to INE future. As proposed by [33], if ψ is greater than 0 means significant spillover effect exists. e 2.5% quantile of ψ cfaf is less than 0, but 5% quantile is greater than 0, which means the spillover from WTI to INE is significant in 95% confidence interval. e 5% quantile of ψ afcf is less than 0 and the 5% quantile of ψ afcf is greater than 0. Within 90% confidence interval which is greater than 0, we classify the spillover of ψ afcf as subsignificant. e volatility level parameter μ cf of INE future is 0.800 6. And μ af of WTI future is 0.952. e value of μ cf is lower than μ af , which means the risk of the INE future is lower than the WTI future. e volatility persistence parameter ψ cfcf of INE future is 0.744 and ψ afaf of WTI future is 0.783 9. e INE future volatility persistence is lower than WTI future. Figure 1 shows the Gelman test results of μ cf and μ af . We can see the two Markov links are lower than 1.1, and it can be considered as convergent. μ cf and μ af are convergent. Other parameters' results are also convergent. In Table 3, the DIC test result shows that the DGC-MSV-t model is better than other models. Dbar reflects the difference between the     model and the actual data, the smaller the better. pD reflects the complexity of the model; the larger the value, the more complicated. Dbar and pD jointly determine the DIC test value. e total DIC score is the lowest in the DGC-t-MSV model. Considering the adaptability and complexity comprehensively, it can be seen that the DGC-t-MSV model is the most suitable model for testing the volatility spillover. Figure 2 shows the dynamic correlation result between America WTI crude oil future (AF) and China INE crude oil future (CF).
In Table 4, we can see all the spillover from each oil futures to others. ψ bfaf , ψ bfcf , ψ bsas , ψ bsas , ψ csas , ψ bscs , ψ bsas , ψ csas , and ψ bscs in 90% confidence are lower than zero. Brent oil future has significant one-way spillover to WTI oil future. WTI future has higher spillover to INE future. Also, we can see all the spillover between spot oil prices. e Brent  oil spot price has one-way spillover effect to WTI and Shengli oilfield spot price. Unexpected, Shengli oilfield spot price has one-way spillover to WTI spot price. Using the same method, we can get the relationship of spillover between oil future and spot price in Table 4. It shows that WTI spot price and future price has significant two-way spillover effect. In Table 5, the Brent spot price has one-way spillover effect to Brent future and WTI future. Brent future has spillover to the price of WTI spot price. Shengli oilfield spot price has a subsignificant one-way spillover to Brent future. ree crude oil spot prices have volatility spillover to INE futures. Table 6 shows the spillover estimation result between stock and oil. ere only exist a few spillover relations between the stock market and the oil market. e spillover from WTI, Brent, and INE oil future to the stock market of S&P exists. However, the Shanghai stock index has a significant one-way spillover to the Shengli oilfield spot price.

Conclusion
e cross-market spillover effect we found can give some advice to correctly diversify investment and reduce risks. (1) e empirical results show that WTI and Brent oil future have more spillover to both spot price and future price. e oil future played an important role in the energy market and economic. INE oil future is a fast-growing product of China. However, INE oil future still lacks international influence.
(2) S&P stock market is more sensitive to the oil price than Shanghai stock market. After experiencing three oil crises, the investors of USA fully understand the impact of crude oil prices on the market. e investor in China does not pay attention to the volatility of crude oil price. S&P includes many oil company, which is not included in the Shanghai stock market. (3) e volatility of the Chinese and American stock markets is not highly correlated and suitable for diversified investment. DIC test shows that DGC-MSV-t is considered effective and more accurate. Our possible future studies will be focus on the difference when market volatility increases and decreases based on regime-switching method of stochastic volatility.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare there are no conflicts of interest regarding the publication of this paper.