Research on Sustainable Development of the Stock Market Based on VIX Index

sustainability

Article Research on Sustainable Development of the Stock Market Based on VIX Index

Lei Ruan School of Business, Northeast Normal University, Changchun 130117, China; [email protected]

Received: 28 September 2018; Accepted: 5 November 2018; Published: 9 November 2018

Abstract: The frequent occurrence of financial crises has made the dynamic linkage between international financial markets an important research topic. In the past, scholars mostly studied the correlation between financial markets directly, however ignored the impact of exogenous financial variables on financial markets. The stock market is an important part of the financial market and plays an important role in the overall economy. Information asymmetry is common and has a certain degree of impact on investors’ returns. However, many scholars believe that the problem of information asymmetry in China has seriously negatively impacted investors, forming an unsustainable state. At present, there are still many problems in the Chinese stock market, especially the stock market fraud, which brings great challenges to the sustainable development of the stock market. Based on the idea of the STCC model, it is assumed that the Copula parameter is affected by the exogenous variables and the time-varying dynamic Copula model-ST-VCopula model is established. Based on the model, the influence of market volatility (VIX index) on the stock market is explored and then the stock index data of several countries are empirically analyzed. The empirical results show that the VIX index has a significant impact on the linkage between stock markets. The VIX index is easy and more intuitive to obtain, providing another way for the dynamic linkage research between the market, which can provide investors with some guidance and advice when conducting financial activities such as diversification.

Keywords: VIX index; sustainable development; stock market

1. Introduction Most studies directly relate to correlation and analyze the dynamic changes of correlation [1]. There are few literatures on the correlation between stock market returns that are affected by exogenous variables, and the main determinants of this correlation are difficult to determine [2]. The stock market is an important part of the financial market; it can transfer funds from the hands of people without productive purposes to the hands of people with productive purposes. It is an important channel for enterprise financing and is one of the short- and long-term investments of investors [3–5]. Therefore, the healthy operation of the stock market plays an important role in personal wealth, enterprise development, financial market and the overall economy [6]. The original intention of designing the VIX (influence of market volatility) index is to reflect investors’ expectations for the future of the market [7]. However, with the deepening of research, people gradually found that in addition to this feature, the VIX index also has a wider range of applications. Therefore, the influence of exogenous variables on market correlation is a topic of great research significance. Relevance is one of the most important means to analyze financial problems [8]. Especially, the significant increase of correlation is considered as one of the important test methods of contagion. Information asymmetry exists in the financial market, and the stock market is no exception. Asymmetric information refers to the fact that one party of a transaction lacks sufficient knowledge of the other party and, thus, affects its decision-making [9]. The consequences and manifestations of asymmetric information are adverse

Sustainability 2018, 10, 4113; doi:10.3390/su10114113 www.mdpi.com/journal/sustainability Sustainability 2018, 10, 4113 2 of 12 selection and moral hazard. After years of empirical testing, people found that the VIX index and stock market returns have a negative correlation, which prompted people to think of using the VIX index to hedge stock portfolio risk [10]. Of course, this was for the long-term development of the stock market, to protect the interests of investors and for the company to create a good financing market environment. It also puts forward some policy suggestions to promote the healthy development of China’s stock market and improve the efficiency of the stock market in reflecting market information so as to form a more effective stock market [11,12]. Kang et al. show that the dynamic correlation between commodities (gold and oil) and BRIC markets is time-varying and sensitive to major financial and economic events [13]. Some studies considered the impact of the global financial crisis (GFC). Baur and McDermott used regression analysis data from 1979 to 2009, which used virtual variables representing extreme market conditions [14]. They believe that gold cannot effectively hedge the extreme volatility of the BRICs market, including India. Xu and Hamori used the cross correlation function method to show that the volatility link between the U.S. and BRIC stock markets has been significantly weakened after the GFC [15]. We use vector autoregression (VAR) and multivariate GARCH models, as well as linear and nonlinear causal frames. Mensi et al. used the quantile regression method to study the dependence structure between BRIC stock markets and several influential global factors from September 1997 to September 2013 [16]. Arui found evidence of a time-varying correlation between U.S. economic policy uncertainty and BRICS stock markets [17]. Ko and Lee point out that the relationship between uncertainty in international economic policy and stock prices is usually negative, however that it changes over time [18]. From January 2001 to March 2013, Nath Sahu et al. showed that there is a long-term relationship between oil and India’s stock market [19]. They also confirm that there is no checking variable between short-term causality and reveal that a positive shock to oil prices has a small but sustained positive impact on Indian stocks in the short term.

2. Game Model Suitable for China’s Current Market For the securities market, especially in China, where the transparency of information is not 100%, there is indeed an incomplete information disclosure of listed companies [20]. Market information is not reflected in the information asymmetry, information transmission leakage, information analysis distortion and other information asymmetry, and basically cannot reach the effective market situation [21]. There are various forms of counterfeiting in the stock market, such as the manipulation of accounting profits by listed companies, the black-box operation of large institutions, large investors, etc. [22]. We call the policymakers who have the tendency to fabricate fraud the counterfeiters. In the stock market, even if small and medium shareholders buy counterfeit stocks, they cannot use them for a period of time to know the authenticity of the goods, just as when they buy other goods [23]. The complete rationality hypothesis of neoclassical economics guarantees the optimal behavior of investors and the general equilibrium of the market under the condition of obtaining complete information [24]. Similarly, under this assumption, the stock market will undoubtedly achieve general equilibrium and achieve Pareto optimality [25]. However, under the real economic conditions, the complete rationality does not exist, mainly in two aspects: First, information is incomplete. The information distribution among investors in the securities market is not uniform and the adequacy of information disclosure of the listed companies often fails to meet the expectations of investors [26]. The reason why the relevant information of listed companies cannot be fully disclosed may be that the relevant information of listed companies has externality. Once disclosed, the unique strategic value of the information may be known to other companies in the same industry, thus losing the competitive advantage of the information [27]. At this point, the lack of information disclosure of listed companies has endogenous characteristics. Second, market participants have limited rationality. Bounded rationality means that decision makers are limited by incomplete information and exhibit non-optimal behavior [28]. If left laissez-faire, these incomplete rationalities can lead to the failure of many securities markets. 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[31]. aTheThe certain normal normal inhibitory state state of of effect the the securities onsecurities stock marketmarket, market, fraud such such [ 31as as]. information information The normal asymmetry, stateasymmetry, of the securitiesbounded bounded market,rationality rationality such and and as informationthethe resulting resulting asymmetry, market market failure failure bounded under under rationality realistic realistic conditions andconditions the resulting to to truly truly market reflect reflect failure the the information information under realistic and and conditions efficiency efficiency toofof trulythe the securities securities reflect the market. market. information and efficiency of the securities market. AAA share shareshare investors’ investors’investors’ investmentinvestmentinvestment incomeincomeincome ratioratioratio andandand aaa shareshareshare investors’investors’investors’ profitprofitprofit ratioratioratio are areare shown shownshown in inin FiguresFiguresFigures1 and11 andand2. The 2.2. TheTheX axis XX axis showsaxis showsshows the proportion thethe proportionproportion of the investors’ ofof thethe investors’investors’ investment investmentinvestment and the Y andaxisand the representsthe YY axisaxis therepresentsrepresents proportion the the proportion ofproportion the investors’ of of the the earnings. investors’ investors’ earnings. earnings.

FigureFigureFigure 1.1. 1. AA A shareshare share investors’investors’ investors’ returnreturn return onon on investmentinvestment investment ratio.ratio. ratio.

Figure 2. Proﬁt share of share investors. FigureFigure 2. 2. Profit Profit share share of of share share investors. investors. The actual income distribution of investors in the share market is shown in Table1 and Figure3. TheThe actual actual income income distribution distribution of of investors investors in in the the share share market market is is shown shown in in Table Table 1 1 and and Figure Figure The X axis in the chart represents the proﬁt volume index and the Y axis represents the yield ratio. 3.3. The The X X axis axis in in the the chart chart represents represents the the profit profit volume volume index index and and the the Y Y axis axis represents represents the the yield yield ratio. ratio.

Sustainability 2018, 10, x FOR PEER REVIEW 4 of 12 Sustainability 2018, 10, 4113 4 of 12 Sustainability 2018, 10, x FOR PEER REVIEW 4 of 12 Table 1. The actual income distribution of investors in the share market. TableTable 1. 1.The The actual actual income income distribution distribution of of investors investors in in the the share share market. market. Investor Classification Profit Amount (100 million yuan) InvestorNaturalInvestor person Classification Classiﬁcation investor ProfitProﬁt Amount Amount 19,874.19 (100 (100 million million yuan) yuan) Natural person investor 19,874.19 GeneralNatural legal person person investor 6871.96 19,874.19 ProfessionalGeneralGeneral legal institutions legal person person 13,541.98 6871.96 6871.96 ProfessionalProfessionalTotal institutions institutions 13,541.9840,288.13 13,541.98 TotalTotal 40,288.13 40,288.13

Figure 3. The actual income distribution of investors in the share market. Figure 3. The actual income distribution of investors in the share market. Figure 3. The actual income distribution of investors in the share market. TheThe shareholding shareholding of of investors investors in in the the share share market market is shown is shown in Figure in Figure4. The 4. TheX axis X axis represents represents the investmentthe investmentThe shareholding proportion proportion index of investors index and theand inY thetheaxis Y sh representsaxisare representsmarket the is proﬁt shownthe profit index. in Figureindex. 4. The X axis represents the investment proportion index and the Y axis represents the profit index.

FigureFigure 4. 4.A A share share investors’ investors’ investment investment ratio. ratio. Figure 4. A share investors’ investment ratio. The risk events in the financial market have good correspondence with the peak of the VIX The risk events in the financial market have good correspondence with the peak of the VIX index. index. When the risk event occurs, the investor’s panic is intensified. The VIX index reached a local WhenThe the risk risk events event in occurs, the financial the investor’s market have panic good is intensified. correspondence The VIX with index the peak reached of the a VIXlocal index. high, high, which should be accompanied by a sharp fall in stock prices. This further illustrates the close Whenwhich theshould risk eventbe accompanied occurs, the investor’sby a sharp panic fall isin intensified. stock prices. The This VIX furtherindex reached illustrates a local the high,close relationship between the VIX index and the investor’s panic, and how the investor’s panic will have a whichrelationship should between be accompanied the VIX index by anda sharp the investor’sfall in stock panic, prices. and howThis thefurther investor’s illustrates panic thewill closehave negative impact on stock prices. Therefore, it can be concluded that there is also a certain relationship relationshipa negative impactbetween on the stock VIX indexprices. and Therefore, the investor’s it can panic, be concluded and how the that investor’s there is panic also willa certain have between the VIX index and the stock price. There is a strong negative relationship between VIX index arelationship negative impact between on thestock VIX prices. index Therefore,and the stock it can pr ice.be Thereconcluded is a strongthat there negative is also relationship a certain and the stock market index. So, we can use the VIX index to create derivatives based on the VIX index, relationshipbetween VIX between index and the the VIX stock index market and index.the stock So, prweice. can There use the is VIXa strong index negative to create relationship derivatives between VIX index and the stock market index. So, we can use the VIX index to create derivatives Sustainability 2018, 10, x FOR PEER REVIEW 5 of 12 Sustainability 2018, 10, 4113 5 of 12 based on the VIX index, providing investors with a financial product against market risk. However, providingthe game model investors at this with time a financial is not a product complete against information market game risk. However,model. Therefore, the game the model incomplete at this timeinformation is not a completegame model information is not inconsistent game model. with Therefore, the efficient the market incomplete hypothesis. information The efficient game model market is nothypothesis inconsistent holds with that the the efficient efficient market market hypothesis. eliminates Theall unused efficient profit market opportunities. hypothesis holdsTherefore, that thethe efficientcurrent marketprice of eliminatessecurities fully all unused reflects profit all of opportunities. the available Therefore,information. the The current elimination price of of securities unused fullyprofit reflects opportunities all of the availableis essential information. for efficient The markets, elimination however of unused it does profit not opportunities require that isall essential market forparticipants efficient markets, have a good however grasp it doesof the not information. require that The all market counterfeiter participants can make have athe good stock grasp price of therise information.and make a huge The counterfeiter profit. Therefore, can make if its thefraudulent stock price behavior rise and is not make recogniz a hugeed profit. by the Therefore, stock market, if its fraudulentthe fraudulent behavior stock is will not use recognized its false prof by theits stockto compete market, with the the fraudulent fraudulent stock stock will once use itsthe false counterfeit profits tostock compete enters with the stock the fraudulent market. Under stock the once competitive the counterfeit pressure stock of enterscounterfeit the stock stocks, market. the share Under price the of competitivecounterfeit stocks pressure declines of counterfeit and the stocks,profits theare sharedama priceged, which of counterfeit induces stocks the counterfeiters declines and to the abandon profits arethe damaged,truth and whichfake it.induces the counterfeiters to abandon the truth and fake it.

3.3. PredictionPrediction andand AnalysisAnalysis ofof thethe StockStock MarketMarket TrendTrend Based Based on on VIX VIX Index Index ThereThere isis indeed indeed a negativea negative relationship relationship between between the VIXthe indexVIX index and the and stock the market,stock market, which meanswhich thatmeans investors that investors can arbitrage can arbitrage through through the height the of height the VIX of index.the VIX So, index. we need So, we to use need the to VIX use index the VIX to estimateindex to theestimate stock marketthe stock forecast market analysis, forecast get analysis, a more get reliable a more forecast reliable value, forecast buy lowvalue, and buy sell low high and to achievesell high the to purposeachieve ofthe earning purpose investment of earning returns. investme Sincent returns. the VIX indexSince the is an VIX investor’s index is expectation an investor’s of marketexpectation volatility, of market there shouldvolatility, be there a certain should lag orderbe a certain for the lag future order trend for ofthe stocks. future We trend use of the stocks. monthly We datause the of themonthly VIX index data andof the the VIX S&P500 index to and make the aS&P500 line chart to make as shown a line in chart Figure as5 shown. In the in ﬁgure, Figure the 5. XIn axisthe figure, represents the X the axis time represents index and the the timeY axis index represents and the theY axis investment represents index. the investment index.

Figure 5. Monthly charts of VIX index (inﬂuence of market volatility) and S&P500. Figure 5. Monthly charts of VIX index (influence of market volatility) and S&P500. As you can see from Figure5, VIX, as an emotional indicator of market collapse or crash, shows a levelAs of riskyou abovecan see the from average Figure and 5, VIX, reached as an a phasedemotional peak. indicator At the of same market time, collapse when the or crash, market shows is in gooda level condition, of risk above the VIXthe average index tends and toreached be stable a phased and has peak. a certain At the degree same time, of decline. when Inthe the market course is ofin thegood S&P500 condition, rise, thethe VIXVIX indexindex will tends fall, to while be stable in the and S&P500 has a fall,certain the degree VIX index of decline. will rise In sharply. the course In the of meantime,the S&P500 we rise, can the see VIX from index the observation will fall, while of Figure in the5 thatS&P500 when fall, the the VIX VIX index index reaches will arise high sharply. point, itIn isthe often meantime, the time we when can the see market from startsthe observation to reverse orof reboundFigure 5 when that when it reaches the theVIX bottom. index reaches The securities a high regulatorypoint, it is often authorities the time should when adopt the market institutional starts to arrangements reverse or rebound for supervision, when it reaches punishment the bottom. and informationThe securities disclosure. regulatory In orderauthorities to effectively should control adopt theinstitutional stock market arrangements counterfeiting, for thesupervision, securities regulatorypunishment authorities and information should pay disclosure. attention toIn the order institutional to effectively arrangements control for counterfeitingthe stock market and strengthencounterfeiting, supervision the securities and punishment regulatory of authorities counterfeiters. should The historicalpay attention chart to of thethe VIXinstitutional index is shownarrangements in Figure for6. counterfeiting The X axis in theand chartstrengthen represents supervision the time and trend punishment of the VIX of index counterfeiters. and the Y axis The representshistorical chart the VIX of the exponent. VIX index is shown in Figure 6. The X axis in the chart represents the time trend of the VIX index and the Y axis represents the VIX exponent. Participants in the game include securities regulators and counterfeiters. This means that the stock market trend has a certain lag order for the VIX index, which means that the stock market has a certain lag in response to the VIX index. After that, this paper takes the VIX index and S&P500 monthly data as an example to calculate the coefficients. Then, by comparing with the coefficient of Sustainability 2018, 10, x FOR PEER REVIEW 6 of 12

adding the lag period, we can get the approximate prediction of S&P500 to the lag order of the VIX Sustainability 2018, 10, 4113 6 of 12 index. The coefficients obtained are shown in Table 2.

Figure 6. Historical trend of the VIX index. Figure 6. Historical trend of the VIX index. Participants in the game include securities regulators and counterfeiters. This means that the Table 2. Coefficient table between S&P500 and the lagged VIX index. stock market trend has a certain lag order for the VIX index, which means that the stock market has a certain lag in response to the VIXLag index. Period After that, this Correlation paper takes Coefficient the VIX index and S&P500 monthly data as an example to calculate the1 coefficients. Then, by comparing0.58 with the coefficient of adding the lag period, we can get the approximate2 prediction of S&P5000.65 to the lag order of the VIX index. The coefficients obtained are shown5 in Table2. 0.45

Table 2. Coefficient table between S&P500 and the lagged VIX index. According to Table 2, when the VIX index lags 2, the coefficient between the VIX index and the S&P500 is 0.65, which is the maximuLag Periodm value in Correlationthe table. So, Coefficient there is reason to believe that the stock market’s response time to the VIX 1index is around two 0.58 months. Based on this, we can establish a model, time series analysis, and, ultimately,2 according to 0.65 the parameters of the model, the VIX index can predict the trend of the stock market.5 0.45 The presence of the unit roots of the three series that were considered was examined using the EnhancedAccording Dickery to Table Fuller2, when(ADF), the Phillip VIX indexPerron lags (PP) 2, and the Kwiatkowski–Phillips–Schmidt–Shin coefficient between the VIX index and (KPSS) the S&P500tests. The is 0.65, empty which HY is model the maximum for ADF value and inPP the testing table. is So, that there the is series reason contains to believe unit that roots the stockand is market’stherefore response not fixed. time In contrast, to the VIX the index null hypothesis is around twoof the months. KPSS test Based is that on this,the series we can is fixed. establish Duea to model,data availability time series (1446 analysis, observations), and, ultimately, the data according was compiled to the parametersdaily from 16 of March, the model, 2011 theto 9 VIX December, index can2016, predict covering the trend the implied of the stock volatility market. index of the US (VIX) and BRIC stock markets, as well as oil (ImpliedThe presence volatility of index the unit for rootsOVX) of and the gold three (GVZ). series thatThe werecross-sectional considered regression was examined analysis using of stock the Enhancedliquidity Dickerychanges Fuller that were (ADF), caused Phillip by Perron changes (PP) in and shareholding Kwiatkowski–Phillips–Schmidt–Shin structure found that institutional (KPSS) tests.investors’ The empty shareholding HY model ratio for ADFwas andnegatively PP testing correla is thatted the with series stock contains liquidity. unit Equity roots and concentration is therefore is notpositively fixed. In related contrast, to stock the null liquidity. hypothesis There of is theno significant KPSS test isrelationship that the series between is fixed. equity Due checks to data and availabilitybalances and (1446 the observations), shareholding the ratio data of wasmanagers. compiled From daily the from perspective 16 March, of 2011 ownership to 9 December, structure, 2016, this coveringpaper analyzes the implied the liquidity volatility of index private of theplacement US (VIX) stocks and BRICand finds stock that markets, the proportion as well as of oil institutional (Implied volatilityinvestors index is negatively for OVX) correlated and gold (GVZ).with the The liquidity cross-sectional of stocks. regression It is of great analysis theoretical of stock and liquidity practical changessignificance that wereto examine caused the by changeschange of in stock shareholding liquidity structurefrom the foundperspective that institutional of the change investors’ of stock shareholdingownership structure ratio was for negatively perfecting correlated the stock withownership stockliquidity. structure Equityof listed concentration companies, promoting is positively the relatedperfection to stock of corporate liquidity. Theregovernance, is no significant providing relationship valuable information between equity for relevant checks andpolicy balances authorities, and theprotecting shareholding small ratio and ofmedium managers. investors From and the perspectiveimproving stock of ownership liquidity. structure, this paper analyzes the liquidityThe initial of private phase placementis described stocks by the and availability finds that of the data proportion collected offrom institutional Thomson investorsReuters data is negativelystreams. According correlated withto the the VIX liquidity method, of the stocks. implied It is volatility of great theoretical index (IVI) and is derived practical from significance the price to of examineput and the call change options, of stockrepresenting liquidity a fromforward-looking the perspective measure of the of change the vola of stocktility ownershipto be realized structure in their forrespective perfecting asset the stockmarkets ownership over the structurenext 30 days. of listed With companies, indiavix as promoting a dependent the variable, perfection it was of corporate observed Sustainability 2018, 10, 4113 7 of 12 governance, providing valuable information for relevant policy authorities, protecting small and medium investors and improving stock liquidity. The initial phase is described by the availability of data collected from Thomson Reuters data streams. According to the VIX method, the implied volatility index (IVI) is derived from the price of put and call options, representing a forward-looking measure of the volatility to be realized in their respective asset markets over the next 30 days. With indiavix as a dependent variable, it was observed that the increase (decrease) of crudevix led to the increase (decrease) of indiavix. The latest discoveries have increased MaGyeReh et al. (2016); the implied volatility of crude oil affects the implied volatility of India. In addition, we report that the increase (decrease) in gold reserves has led to an increase (decrease) in the Indian stock market: India’s broader stock market volatility expectations are sensitive to volatility expectations in the oil and gold markets. India’s VIX has no effect on crude oil and gold VIX. The shift of investors from the stock market to the gold market led to an increase in the volatility of gold prices, which led to an increase in the implied volatility of gold prices. This shift will also lead to a loss of liquidity in the stock market, leading to an increase in implied volatility. In the post-crisis period, investors have become risk-averse and have turned to the gold market during short-term uncertainty, thus exacerbating these volatility linkages. For any function G, conditional expectation s can be expressed as the following Taylor expansion:

Kg Gg(s) = (1) 1 + Td0s

If we only extend to the ﬁrst order and move S to the left side of the equation, we can get:

S = (U, A, V, g) (2)

Considering that a bivariate joint distribution function i and j with edge distribution Q and u is an information set, then there exists a conditional Copula function which makes:

n Q(uij) = Max {gij(T)} (3) ∑ ≤ ≤ i=1 1 j m

The Copula function assumes that P is the parameter vector in the Copula function:

N M 2 T Pf −nm = pi hi,n Wi,i (4) ∑ ∑ l m l 2 i=1,i6=n l=1

The joint density function of two variables is obtained from the function expression (1) that is jointly distributed by the condition.

Kr Gr(s) = (5) 1 + Trs The parameters of Copula are dynamically modeled and then the parameters of each model are estimated by maximizing the logarithmic likelihood function values of the following dynamic Copula functions: 2k ≤ n, k ≤ a, 2k − 1 ≤ d ≤ n − r (6)

Among them, k, n and d are the parameters of the edge distribution and the Copula function, respectively, and r is the length of the time series. Sustainability 2018, 10, 4113 8 of 12

If the exogenous variable (VIX index) does not have a signiﬁcant impact on the Copula parameters, the weight G does not have an impact on the parameters. Therefore, the original hypothesis and the opposite hypothesis can be set as follows:

T m h T T Ti G = G1 , G2 ,..., Gk (7)

When the constraint condition of the original hypothesis is established, the ST-VCopula model degenerates into a general static Copula model. Suppose the maximum likelihood function is f and the maximum likelihood function is x in the unconstrained case (ST-VCopula model). According to the likelihood ratio test, the following test statistics are constructed. · La−x αP ch α d 4R f = − · f 2 x (8) πd La πd sh α · d The PDE of the VIX index call option is derived from the following: f 0 = f − f − cσ (9)

Because M is not a trading asset price, we allow the market risk price S to be related to the VIX index. If the stochastic ﬂuctuation process of VIX obeys the stochastic differential equation k, then the risk-neutral process obeys the following stochastic differential equation:  k/(M1 ∗ M2)(x, y) ∈ S  ρ(x, y) = , (10)   0 (x, y) ∈/ S

Thus, a new risk neutral process is obtained:

h i 1 M ∗ M E d2 = 1 2 (11) toCH 2π k Since M itself is not the price of a portfolio, whether E is a nonlinear function of the price of other securities or not, it will not affect the nature of the result. The coefﬁcient T test between VIX index and S&P500 and the coefﬁcient U test between VIX index and S&P500 are shown in Tables3 and4.

Table 3. T test of the coefﬁcient between VIX index and S&P500.

Correlation Coefﬁcient n t Value 0.148 159 15.64 0.294 186 16.64 0.291 168 17.56

Table 4. U test of the coefﬁcient between VIX index and S&P500.

Correlation Coefﬁcient n t Value 0.641 162 13.26 0.418 168 15.19 0.694 191 17.19

The monthly data of VIX index and S&P500 are selected as the analysis data of the time series. At the same time, in order to eliminate the heteroscedasticity in the data and reduce the volatility of the data, the time series data is logarithmic, making it easier to become a wide stationary time series. Sustainability 2018, 10, 4113 9 of 12

In the model setting, we use Y to represent the logarithm of the VIX exponent and N to indicate the logarithmSustainability of 2018 S&P500., 10, x FOR Moreover, PEER REVIEW since the determination of the lag order is two orders above, the9 data of 12 is substituted into the model and the estimated models are as follows: Table 5. Test for residuals of the regression equation. Y(s) G (s)G(s) = D (12) Test Critical ValuesN(s) 1 + C(s)H(s) t-Statistic 10% level 0.26 Then, the residual in the regression15% level equation of the model tested 0.29 to determine whether there is an equilibrium relationship between20% VIX level index and S&P500. In this paper,0.28 we use the ADF test, and the results are shown in Table5. As shown in Table 5, the residual sequence rejects the original hypothesis at the 99% confidence Table 5. Test for residuals of the regression equation. level, indicating that the residual sequence is stable. It can be explained that the model is reliable and reasonable and that there is a balancedTest Critical relationship Values betweent-Statistic VIX index and S&P500. In the model, the G and H values10% can level only be used to show 0.26 whether the change of one variable has a significant statistical effect on the15% change level of the dependent 0.29 variable under the premise that other variables remain unchanged. However,20% level it cannot be used to 0.28 explain the positive and negative relationship between variables and how long the impact will last. In this paper, the problem is characterizedAs shown by in Tablethe impulse5, the residual response sequence function rejects graph the and original the variance hypothesis decomposition at the 99% graph confidence of the level,model, indicating as shown that in Figure the residual 7. The sequence X-axis shows is stable. the Ittime can required be explained for the that pulses the model to cause is reliable the S&P500 and reasonablefluctuations and due that to therethe impact is a balanced of the VIX relationship exponent, between and the VIX Y-axis index represents and S&P500. the impact of the VIX exponentIn the to model, cause thethe GS&P500and H fluctuations.values can only be used to show whether the change of one variable has aIn significant Figure 8, statistical the variance effect decomposition on the change method of the dependentis used to measure variable underthe contribution the premise of thatVIX otherexponential variables shocks remain to S&P500 unchanged. changes However, and to itdeterm cannotine be the used importance to explain of the VIX positive exponential and negative changes. relationshipThrough the between above analysis, variables we andcan howsee that long there the is impact an internal will last.relationship In this between paper, the the problem changes isof characterizedVIX index and by S&P500, the impulse which response is the trend function of the graph stock and market. the variance That is decomposition to say that the graph volatility of the of model,investor as sentiment shown in Figurein the market7. The Xwill-axis indeed shows have the timea very required far-reaching for the impact pulses on to the cause future the S&P500trend of fluctuationsthe market. dueThe toX-axis the impact showsof the the time VIX when exponent, the VIX and exponent the Y-axis shocks represents the S&P500 the impact and ofthe the Y-axis VIX exponentrepresents to the cause contribution the S&P500 of fluctuations.the VIX exponent shocks to the S&P500.

FigureFigure 7. 7.Impulse Impulse response response function function of of S&P500 S&P500 ﬂuctuation fluctuation caused caused by by the the impact impact of of the the VIXVIX exponent.exponent.

In Figure8, the variance decomposition method is used to measure the contribution of VIX exponential shocks to S&P500 changes and to determine the importance of VIX exponential changes. Through the above analysis, we can see that there is an internal relationship between the changes of VIX index and S&P500, which is the trend of the stock market. That is to say that the volatility of investor sentiment in the market will indeed have a very far-reaching impact on the future trend of Sustainability 2018, 10, 4113 10 of 12 the market. The X-axis shows the time when the VIX exponent shocks the S&P500 and the Y-axis representsSustainability the2018 contribution, 10, x FOR PEER of REVIEW the VIX exponent shocks to the S&P500. 10 of 12

Figure 8. Impact ofof thethe VIXVIX index’sindex’s impactimpact onon S&P500S&P500 changes.changes. 4. Conclusions 4. Conclusions Based on the idea of the STCC model, this paper establishes the ST-Vcopula model, considering the Based on the idea of the STCC model, this paper establishes the ST-Vcopula model, considering influence of exogenous variables on Copula parameters. Based on the maximum likelihood estimation the influence of exogenous variables on Copula parameters. Based on the maximum likelihood method, the model parameters are estimated and the model is selected by the likelihood ratio test estimation method, the model parameters are estimated and the model is selected by the likelihood method. The development of China’s financial system is still imperfect, therefore, the importance of ratio test method. The development of China’s financial system is still imperfect, therefore, the building volatility index in China cannot be ignored. China’s linkage with the international economy is importance of building volatility index in China cannot be ignored. China’s linkage with the weaker due to policy reasons. However, with the continuous advancement of economic globalization, international economy is weaker due to policy reasons. However, with the continuous advancement inter-state linkage has increased in times of crisis. It is very difficult for a country to be completely of economic globalization, inter-state linkage has increased in times of crisis. It is very difficult for a out of the way. Intermarket linkage analysis is of great significance to investors in risk prediction and country to be completely out of the way. Intermarket linkage analysis is of great significance to hedging. The construction of the VIX index also indicates that China has made a great breakthrough investors in risk prediction and hedging. The construction of the VIX index also indicates that China in risk prediction and hedging, which also means that China has taken a solid step on the road to has made a great breakthrough in risk prediction and hedging, which also means that China has international risk hedging. The VIX index can be regarded as one of the best measures of the future taken a solid step on the road to international risk hedging. The VIX index can be regarded as one of market trend. Further, its great ability in information transmission enables us to use this as a starting the best measures of the future market trend. Further, its great ability in information transmission point to analyze the impact of changes in the VIX index on the linkage between stock markets, and then enables us to use this as a starting point to analyze the impact of changes in the VIX index on the to predict the possibility of contagion and spread of the crisis. linkage between stock markets, and then to predict the possibility of contagion and spread of the Funding:crisis. This research was funded by the Fundamental Research Funds for the Central Universities, China (Grant No. 2412018QD028), Jilin Scientific and Technological Development Program, China (Grant No. 20180418095FG) andFunding: Social This Science research research was project funded of Jilinby the Province Fundamental Education Research Department, Funds Chinafor the (Grant Central No. Universities, JJKH20180057SK). China Acknowledgments:(Grant No. 2412018QD028),Great thanks Jilin to Scientific the anonymous and Techno reviewerslogical and Development editors for their Program, constructive China comments (Grant No. to this20180418095FG) manuscript. and Social Science research project of Jilin Province Education Department, China (Grant No. ConflictsJJKH20180057SK). of Interest: The author declares no conflict of interest. Acknowledgments: Great thanks to the anonymous reviewers and editors for their constructive comments to Referencesthis manuscript.

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