Volatility Spillover across Global Equity Markets. International market linkages are important for a variety of investment and risk

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1 Volatility Spillover across Global Equity Markets 1. Introduction International market linkages are important for a variety of investment and risk management decisions. For example, a shock in the U.S. market should make investors adjust their exposure to other international markets because of the volatility spillover from the U.S. market to the other markets. The interest of studying the international market linkage can be traced back to the market crash of Until then the question of why the market around the world fell simultaneously and with surprisingly uniform was largely ignored (King and Wadhwani, 1990). A series of financial turbulence in the 1990s, Mexican Tequila crisis in 1994, East Asian Asian flu in 1997, Russian virus in 1998, and Brazil s currency crisis in 1999 arouse great attention among researchers and investors. Our research is motivated by the financial turmoil during Especially, since the collapse of Lehman Brothers in September 2008, we have seen the plummet of stock markets in almost every market around the world. Although the cross market linkage is a topic of ongoing interest to researchers and practitioners, it seems that we are still in the preliminary stage to fully understand the cross market linkage, and far from being able to prevent the crisis transmitting across markets. The current crisis provides us with a unique opportunity to examine the market correlations and the information spillover across global stock markets. One aspect of the cross-market linkage is closely related to the contagion literature when a sudden shock occurs in one market. Although a number of papers, such as Longin and Solnik (1995) and Bae, Karolyi, and Stulz (2003), find evidence of contagion across markets, Forbes and Rigobon (2002) argue that there is no contagion and only interdependence exists across markets after adjusting the correlation coefficients for the volatility heteroskedasticity. We do 1

2 not provide direct tests on contagion in the paper. However, based on our observation of the stable cross-market linkage, the results provide supportive evidences on the no contagion but high interdependence argument. The international financial market correlations are affected by factors such as political, economic, and financial integration, as well as the degree of removal of impediments to global investment. Goetzmann, Li, and Rowenhorst (2005) show the dramatic changes on international equity correlation based on tests over a 150-year horizon. Longin and Solnik (1995) also find that unstable correlation structure during However, our tests demonstrate the stability of correlation structure between the U.S. and other foreign markets from 1999 to In the last decade, the connection among markets has been significantly enforced due to the development of trading technologies. Trading overseas stocks from any country was used to be not particularly easy. Trading technology now makes it just as easy for a European investor to trade stocks, say Japan as in Europe. The financial markets show high degree of correlation, which should be caused by the various linkage channels that connect them and through which the information shall diffuse across market. One argument is that shocks are transmitted by fundamental channels, such as trade link (Dornbusch, Park, and Claessens (2000)), balance sheet link (Kiyotaki and Moore (2002)), and price information discovery process (King and Wadhwani (1990)). The other argument is that information is transferred through channels such as volatility (Hamao, Masulis, and Ng (1990), Nikkinen and Sahlstrom (2004), and Aijo (2008)) and liquidity (Allen and Gale (2000), Brunnermeier and Pedersen (2009)). Volatility linkages between markets represent an important avenue through which the fluctuation on markets can affect security prices and alter investors behavior. When an 2

3 investor s operation spans over more than one market, his net volatility exposure depends on the cross market correlations of volatility changes. The investors should incorporate such information into their risk measurement. Nikkinen and Sahlstrom (2004) and Aijo (2008) have found strong volatility linkage across several volatility indexes. We confirm the strong linkage through volatility indexes; meanwhile, we have shown the structural stability of such linkage in the past decade. Our study focuses on the cross market volatility linkage in foreign markets with the U.S. market. The foreign markets include German, Japanese, Swiss, and the European markets. These markets represent the most important stock markets for global investors. Estimated from the market prices of stock index options and providing a method to measure investors expectation of uncertainty regarding future, implied volatility should be reflected in expectation on other markets. Therefore, the degree of market integration can be investigated by examining interaction in implied volatility across various stock markets. In this paper, we address two specific questions about the volatility linkage of international stock markets: Does the volatility correlation across markets change over time, especially whether correlation increases in the periods of high market volatility in the past decade? What is the volatility linkage structure among U.S. and other stock markets? Our interest also includes investigating whether the structure of such linkage is consistently stable. Our sample periods cover from 1999 to 2009, using the implied volatility indexes across five stock markets. In addition, we include the most recent crisis in our study, which is not available in the current literature. The paper adds to the existing literature in several respects. Firstly, we consider the cross market linkage in the most up-to-date crisis. Secondly, we investigate the implied volatility and 3

4 the realized volatility linkage across the U.S. and other foreign stock markets. Thirdly, we indentify the consistency and stability on market linkage in the last decade. The remainder of the paper is organized as follows. Section 2 describes the data and descriptive statistics. Section 3 tests the dynamics of implied volatility correlation. Section 4 presents the tests of information spillover across implied volatility indexes. The robustness tests with realized volatility are provided in section 5, and section 6 concludes. 2. The Data We use the implied volatility of stock exchange indexes, VIX, VSTOXX, VDAXNEW, VXJ, and VSMI, to proxy the expected volatility of U.S., European, German, Japanese, and Swiss markets. The data are collected from Reuters. The calculation of implied volatility is based on the observable option prices and the model of option pricing, which is widely accepted since the introduction of Black-Scholes model. The CBOE volatility index (VIX), which was introduced in 1993, measures the implied volatility of S&P500 index options, the components of which are near and next-term put and call options of S&P 500 index, and VIX represents one measure of the market s expectation of volatility for S&P 500 index over the next 30 days. The calculation of VSTOX, VDAXNEW, VXJ, and VSMI are the same as that used for VIX *, and they represent the forward-looking 30 day volatilities for Dow Jones Euro Stoxx 50 (Stoxx 50), DAX, Nikkei225, and SMI. VSTOXX is based on Dow Jones Euro Stoxx 50 real time option prices. Dow Jones Euro Stoxx 50 covers 50 stocks from 12 Euro zone countries, and provides a blue-chip representation of super-sector leaders in the Eurozone. VDAXNEW index is based on DAX (Germany Stock Index) options. DAX consists of the 30 major German * The calculation methods of these indexes are based on 2003 revised methodology for the CBOE VIX. 4

5 companies trading on the Frankfurt Stock Exchange. VSMI is based on SMI (Swiss Exchange Index). VSTOXX, VDAXNEW, and VSMI were introduced in April 20, 2005 by Eurex. Daily data of VSTOXX and VSMI are available back to January 1999, and VDAXNEW back to January VXJ, presented by Center for the Study of Finance and Insurance (CSFI), is based on the Nikkei 225 options traded on the Osaka Securities Exchanges, and the daily data are available back to These indexes are selected for the following reasons. The U.S. market is selected since it is the world s largest and represents the global shock. Stoxx 50 represents the overall performance of European market. German and Japanese markets are the largest in Europe and Asia. The Swiss market is used as a representative of a small country s stock market. In addition, we also require the markets have long enough implied index constructed by an agent to guarantee the quality of data. Taiwan and South Korea also have the similar indexes, but those are not included in our sample, since both of them are not longer than two years. In addition, the similar calculation methodology offers a comfortable way to compare the volatilities between the U.S. and these foreign markets. Our sample covers the period from January 1999 to December 2009, with total 2,831 daily observations. Figure 1 illustrates the implied volatility indexes during the sample period. The rationale for the sample period is based on the data availability and that it coincides with the launching of the Euro in January 1999, and also covers the several financial crises. Figure 1 clearly identifies several global financial events, that is, the 911 attack in 2001, dot-com burst in 2002, and the subprime mortgage crisis in Table 1 presents the summary statistics for the daily volatility indexes for the five markets, which shows that the magnitude of these indexes is comparable. 5

6 3. The Dynamics of Implied Volatility Correlation 3.1. Unadjusted Volatility Correlation The conventional way to check the international market integration is to look at the correlation matrix of stock exchange index on returns (Longin and Solnik (1995), Solnik, Bourcelle, and Fur (1996), and Goetzmann, Li, and Rouwenhorst (2005)). We provide a different analysis by examining evolution in the implied volatility across various equity markets. Implied volatility index is widely accepted as the forward looking of uncertainty regarding future price movement. As on the expectation of uncertainty of the international markets, the implied volatility of different markets should interact with each other. Implied volatility index correlation fluctuates over time. Figure 2 plots the cross-country rolling correlation (U.S. vs. other markets) of the implied volatility index for the sample period. Rolling correlations are calculated over a backward-looking window of 260 trading days. This figure illustrates a similar pattern to each correlation series, namely, correlations have changed dramatically over the past eleven years. All the correlation series show a certain common peaks and troughs that would be explained by the global events, such as the 911 attack and 2008 financial turmoil. However, when we compare the correlations of all the four foreign markets with the U.S. market, the movement of pair-wise correlations is far from synchronization. For example, we see several unique troughs for U.S.-Swiss correlation. The correlation non-synchronization could be related to the national business cycles and economic growth (Erb, Harvey, Viskanta (1994)). Are the variations in the correlation structure consistent in the last decade? We are interested in checking whether the cross market linkage, in terms of correlation, is stable over time. In order to answer the question, we test the consistency of correlation using Jennrich tests. 6

7 Jennrich (1970) develops a test based on the difference between two correlation matrices. Jennrich test is a Chi-square test based on overall matrix equality and two independent correlation matrices. Let R 1 and R 2 be sample correlation matrices for two independent subsamples of size n 1 and n 2 from two p-variant normal populations with correlation P 1 and P 2. If P 1 and P 2 have a common non-singular value P, then as n, and n, random matrix. L L n 1 ( R 1 P) Y and n 2 ( R 2 P) Y, where Y is a symmetric zero diagonal The null hypothesis of Jennrich test is P 1 =P 2. The following summarizes the computation of 2 χ. 1 2 R = ( nr + nr)/( n + n ), for R = r ), the inverse R = ( r ), Let ( ij 1 ij 2 c = n n /( n + 2 ), Z = c 1/ R R R ), n 1 ( 1 2 ij 1 if i = j S = ( σ ij + rij r ), where σ ij =, 0 otherwise Then the test statistics 2 χ is defined as: χ = tr( Z ) diag ( Z ) S 1 diag( Z ) 2 As to two correlation matrices, according to the Jennrich test, we need to compute the value, degrees of freedom and the p-value. If the p-value is greater than the test significance, we will accept the null hypothesis that the two matrices are equal. In our test, we first calculate the unconditional correlation matrix each calendar year between VIX and other implied volatility indexes, and then compute p-value based on the 2 χ For detail, see Jennrich (1970). 7

8 Jennrich test. We show the p-value of the test statistics in Table 2. As we have expected from Figure 2, the annual correlation is highly statistically different for all markets. Especially, most of the correlation matrices for any two consecutive years are all significantly different; the exceptions are for VIX VDAXNEW and for VIX-VSTOXX and VIX- VXJ. In addition, we find the correlation matrices between VIX and other markets are very similar between 2004 and In sum, our tests indicate the unadjusted implied volatility correlation structure highly fluctuates from 1999 to The result is consistent with investors common expectation that the correlations of global stock market tend to vary over time. (Solnik, Bourcelle, and Fur (1996), and Goetzmann, Li and Rouwenhorst (2005)). We confirm such expectation through the implied volatility correlation matrices Adjusted Volatility Correlation Based on a statistical perspective, one would expect the biased correlation coefficients because of the heteroskedasticity. Forbes and Rigonbon (2002), among many others, provide a formal proof for that. Forbes and Rigobon (2002) present that once adjusting the conditioning bias, they find small changes in correlation coefficients during the 1997 East Asia crises, the 1994 Mexican peso collapse, and the 1987 U.S. stock market crash. We adopt a multivariate GARCH model to capture some of the evolution in the conditional correlation structure. The general VAR -VECH (1,1) GARCH model is used as follows: V = C1 + φ 11V 1( 1) + φ12v2 ( 1 1) 8

9 V = C2 + φ 21V 1( 1) + φ22v2 ( 2 1) H t = Ω + A ε t 1ε t 1 + B H t 1 where v 1, v 2 are the difference of volatility index. H t is the time varying conditional covariance matrix. Assuming conditional normality, the log likelihood function is given by: L t = ' 1 mlog(2π ) Log( H t ) ε th t 2 2 ε t where m is the dimension of the model and ε t is the m vector of equation residuals. We report the conditional covariance in Figure 3 and conditional correlation in Figure 4. The conditional covariance diagrams clearly capture several common peaks, for example, in 2002 and 2008 across different countries. As to 2008 crisis, the adjusted covariance diagrams show the market linkage tend to turn normal in later 2009, which indicates the recovery of global market after the 2008 crisis. After adjustment, the conditional correlations of VIX with other volatility index demonstrate no time varying patterns. Therefore, different with Longin and Solnik (1995) who find the covariance and correlation metrics of the international equity returns are unstable over time from 1960 to 1990, our tests show that from 1999 to 2009 the conditional correlations of volatility indexes between U.S. and other four indexes are quite stable. 4. The Transmission of Implied Volatility Index We use a vector autoregressive analysis (VAR) to estimate a dynamic simultaneous equation system that helps us analyze the transmission of uncertainty between implied volatility indexes. The VAR model is expressed as: We conduct the analysis for the difference variables to avoid stationarity based on unit root tests. 9

10 P Y = C + A Y + ε t k t k t k = 1 where Y t being a vector of two daily volatility indexes with first difference; C and A k are matrices of coefficients to be estimated while p is the lag length, and ε t is a vector of innovation. The model is estimated using OLS. We choose the VAR lag length based on the AIC criterion, Schwarz s criterion, and likelihood ratio tests, and we adopt 5-lag length. Table 3 presents the VAR estimation for the VIX and other implied volatility indexes. The adjusted R 2 ranges from 4.13% to 22.13%. VIX and the corresponding implied volatility index demonstrate two-way impact based on the coefficient significance. In general, VIX has relatively stronger impact on other indexes. This could be proven by the variance decomposition analysis. Variance decomposition analysis reveals what fraction of the innovation of variables could be explained by the shock from its different step ahead and other variables. The results show that the forecast variance of VIX is only caused by its own innovation. Meanwhile, the innovation of VIX explains about 20% to 40% of the ten days ahead forecast error variance of VSTOXX, VDAXNEW, VXJ, and VSMI. The VAR test and variance decomposition analysis indicate the strong influence linkage across markets. Are the linkage structures stable over time? We use the Quandt-Andrews breakpoint test (Andrews (1993)) for structure change with unknown timing to test for the stability of the estimated parameters. We report the valid p-values in Table 4. With checking all possible break points, the results suggest no structure break for any equations. In sum, based on the tests, we find the strong information transmission structure across markets using the implied volatility index over the last decade. And such linkage structure is stable during the sample period even when we have experienced several financial crises. 10

11 5. Robustness Tests Our previous tests are based on the implied volatility index, which represents the forward-looking on stock market uncertainty. Theodossiou and Lee (1993) and Stoll and Whaley (1990), among many others, have revealed the realized volatility is related to the rate of flow of information to the market. We shall expect that under the same market integration scenario, the realized volatility, which represents the backward-looking on the stock market uncertainty, should demonstrate a similar linkage with that of the implied volatility index. We conduct the robustness tests using the realized volatility of stock markets. We calculate the market return using the daily stock market indices at closing time, in terms of local currency units, for the U.S., German, Japanese, Swiss, and European markets. The sample covers the period from January 4, 1999 to December 31, The indices used are the S&P 500 for the United State, Frankfurt DAX 30 for Germany, Nikkei 225 for Japan, and SMI for Switzerland. And we use Dow Jones Euro Stoxx 50 (Stoxx 50) to proxy the European market. S&P 500 index and Stoxx 50 are denominated with US dollar. We adjust DAX, Nikkei 225, and SMI index with daily Dollar/Euro, Yen/Dollar, and SF/Dollar exchange rate. Rates of change of the data series are calculated as following: R it = t 100 ln( Pi, t / Pi, 1) where P i,t is the price level (in dollar) of market i at time t. We calculate the realized volatility using data for the previous 20 trading days. The standard deviations are annualized. Figure 5 plots the cross-country rolling correlation of the realized volatility. Rolling correlations are calculated over a backward-looking window of 260 trading days. The figures show similar fluctuation as that of the implied volatility index. The Jennrich tests for annual unconditional correlation matrix for the realized volatility also show the 11

12 statistical significant difference (Table 5) and are the two periods in which the annual unconditional correlations show similarity across VIX and other implied volatility indexes. We use VAR -VECH (1, 1) GARCH model to evaluate the adjusted correlation for the realized volatility. The conditional covariance and the conditional correlation between S&P 500 realized volatility and other markets realized volatilities are reported in Figure 6 and 7. Similar to that of the implied volatility index, the conditional covariance (Figure 6) captures several common peaks, such as the dramatic increase in 2007 and 2008, which was obviously caused by the sharp increase of stock market turmoil. And we also find the covariance goes back to normal in later In the figure of conditional correlation (Figure 7), we do not find the unusual change in Although in the four correlation series we do find the increase of correlation in the current crisis, such high correlations are not unusual and not even the peak in the sample period. The stability tests also show the adjusted correlation series are stationary at 5% confidence level. We also examine the stability of information transmission structure by using Quandt- Andrews unknown breakpoint test and report the results in Table 6. Similar to the tests of the implied volatility index, the 5-lag has been used in the estimation. We also document the similar two way information flow between U.S. and other stock markets. In addition, U.S. has relatively stronger impact to other markets. The Quandt-Andrews breakpoint test for structure change with unknown timing clearly shows no breakpoint. The results are similar to those in previous tests with implied volatility indexes. 12

13 6. Conclusion We test the global market integration through the volatility channel. By investigating the stock market implied volatility (using the implied volatility indexes), we demonstrate the stability of the global market linkage over the past eleven years. Although the unadjusted market correlations show high fluctuations, which is widely reported and accepted by researchers and investors, the adjusted correlation reveals the stable linkage across markets. The adjusted correlation provides a real picture of market linkages since Forbes and Rigobon (2002) have shown the defects using unadjusted correlation. We test the information transmission between U.S. and other foreign markets, which shows the U.S. market plays a stronger role in a two-way information flow structure. We find no breakpoint of such linkage structure in the sample period using Quandt-Andrews tests. Our study provides the evidence to show that in the last decade, there could be no structural changes in the global market linkages. Especially, several financial crises could not significantly change the highly integrated global market linkage. In addition, the tests on the implied volatility index have been confirmed by the robustness tests using the realized volatility, since under the same integration scenario, the implied and the realized volatility linkage should demonstrate similar pattern. 13

14 References Äijö, J. (2008). Implied volatility term structure linkages between VDAX, VSMI and VSTOXX volatility indices. Global Finance Journal, vol. 18, Allen, Franklin, and Douglas Gale, 2000, Financial contagion, Journal of Political Economy 108, Andrews, D., (1993). Tests for parameter instability and structural change with unknown change point. Econometrica, 61, Bae, Kee Hong, Andrew Karolyi, and Rene Stulz, 2003, A new approach to measuring financial market contagion, Review of Financial Studies 16, Brunnermeier, Markus K. and Lasse H. Pedersen, 2009, Market liquidity and funding liquidity, Review of Financial Studies, 22 (6), Dornbusch, Rudiger, Yung Chul Park, and Stijn Claessens, 2000, Contagion: understanding how it spreads, The World Bank Research Observer 15, Erb, C., Harvey, C., & Viskanta,T. (1994). Forecasting International Correlation. Financial Analysts Journal, vol. 50, Forbes, Kristin, and Rigobon, Roberto, 2002, No contagion, only interdependence: measuring stock market comovements, Journal of Finance, 57 (5), Goetzmann, W., Li, L., & Rouwenhorst, G. (2005). Long-Term Global Market Correlations. The Journal of Business, 2005, vol. 78, no. 1, Hamao, Yasushi, Ronald W. Masulis, and Victor K. Ng, 1990, Correlations in price changes and volatility across international stock markets, Review of Financial Studies, 3, Jennrich, R., (1970). An Asymptotic Chi-square Test for the Equity of Two Correlation Matrices. Journal of the American Statistical Association, 65, King, Mervyn A., and Sushil Wadhwani, 1989, Transmission of volatility between stock markets, Review of Financial Studies 3, Kiyotaki, Nobuhiro, and John Moore, 2002, American Economic Review: Papers and Proceedings 85, Longin, Francois M., and Bruno Solnik, 1995, Is the correlation in international equity returns constant: ? Journal of International Money and Finance 14,

15 Nikkinen, J., and Sahlström, P. (2004). International transmission of uncertainty implicit in stock index option prices. Global Finance Journal, 15, Solnik, B., Bourcelle C., & Fur Y. (1996). International Market Correlation and Volatility. Financial Analyst Journal, vol. 52, no. 5, Stoll, H., and Whaley, R. (1990). Stock market structure and volatility. Review of Financial Studies, 3, Theodossiou, P., and Lee, U. (1993). Mean and Volatility spillovers across major national stock markets: future empirical evidence. Journal of Financial Research, XVI,

16 Figure 1: Daily Implied Volatility Indexes ( ) The figure depicts the time series of daily implied volatility indexes from The indexes include: the implied volatility index for S&P 500 index options (VIX); the implied volatility index for Dow Jones Euro Stoxx 50 index options (VSTOXX); the implied volatility index for DAX index options (VDAXNEW); the implied volatility index for Nikkei 225 index options (VXJ); and the implied volatility index for SMI index options (VSMI). 100 VIX VSTOXX

17 100 VDAXNEW VXJ VSMI

18 Figure 2: The cross-country rolling correlation (implied volatility index) The figure depicts the correlations between VIX and VSTOXX, VIX and VDAXNEW, VIX and VXJ, VIV and VSMI. Correlations are computed using data for the previous 260 days. The sample period covers from 1999 to corr(vix,vstoxx) corr(vix,vdaxnew) corr(vix,vxj) corr(vix,vsmi)

19 Figure 3: Conditional Covariance (implied volatility index) The figure depicts the conditional covariance between (DVIX, DVSTOXX), (DVIX, DVADXNEW), (DVIX, DVXJ), (DVIX, DVSMI) after adjustment for VAR-GARCH model. Cov(DVIX,DVSTOXX) Cov(DVIX,DVDAXNEW) Cov(DVIX,DVXJ) Cov(DVIX,DVSMI)

20 Figure 4: Conditional Correlation (implied volatility index) The figure depicts the conditional correlation between (DVIX, DVSTOXX), (DVIX, DVADXNEW), (DVIX, DVXJ), (DVIX, DVSMI) after adjustment for VAR-GARCH model. Cor(DVIX,DVSTOXX) Cor(DVIX,DVDAXNEW) Cor(DVIX,DVXJ) Cor(DVIX,DVSMI)

21 Figure 5: The Cross- country Rolling Correlations (the realized volatility) The figure depicts the correlations for the realized volatility between S&P 500 index and STOXX, S&P 500 index and DAX, S&P 500 index and Nikkei 225, and S&P 500 index and SMI. The realized volatility of each market is computed using data for the previous 20 days. Correlations are computed using data for the previous 260 days. The sample period covers from 1999 to corr(vix, vstoxx) corr(vix, vdaxnew) corr(vix, vxj) corr(vix, vsmi)

22 Figure 6: Conditional Covariance (realized volatility) The figure depicts the conditional covariance of realized volatilities between S&P 500 index and STOXX, S&P 500 index and DAX, S&P 500 index and Nikkei 225, and S&P 500 index and SMI after adjustment for VAR-GARCH model. The realized volatility of each market is computed using data for the previous 20 days. Cov(DVIX,DVSTOXX) Cov(DVIX,DVDAXNEW) Cov(DVIX,DVXJ) Cov(DVIX,DVSMI)

23 Figure 7: Conditional Correlation (realized volatility) The figure depicts the conditional covariance of realized volatilities between S&P 500 index and STOXX, S&P 500 index and DAX, S&P 500 index and Nikkei 225, and S&P 500 index and SMI after adjustment for VAR-GARCH model. The realized volatility of each market is computed using data for the previous 20 days. Cor(DVIX,DVSTOXX) Cor(DVIX,DVDAXNEW) Cor(DVIX,DVXJ) Cor(DVIX,DVSMI)

24 Table 1 Summary Statistics for the daily implied volatility index The table presents the descriptive statistics of VIX, VSTOXX, VDAXNEW, VXJ, and VSMI, covering from January 1999 to December The implied volatility indexes are reported in percentage. VIX VSTOXX VDAXNEW VXJ VSMI Mean Median Maximum Minimum Std. Dev Skewness Kurtosis Observations 2,831 2,831 2,831 2,831 2,831 24

25 Table 2 Jennrich tests for the unconditional correlation matrix (implied volatility index) The table reports the p-value of Jennrich test between each calendar year correlation matrix. We compute the correlation between two markets implied volatility index of each calendar year. The entry is the p- value of Jennrich tests for the correlation of the left hand side year and the correlation of the year at the top. vix-vstoxx vix-vdaxnew

26 Table 2 (Continued) vix-vxj vix-vsmi

27 Table 3: VAR estimation (implied volatility index) The table presents the VAR estimations for (DVIX, DVSTOXX), (DVIX, DVADXNEW), (DVIX, DVXJ), (DVIX, DVSMI) respectively. The 5- lag length has been chose based on the various lag structure criteria. The sample period covers Jan 1999 to Dec T-statistics are reported in bracket. DVIX DVSTOXX DVIX DVDAXNEW DVIX DVXJ DVIX DVSMI DVIX(-1) *** 0.490*** DVIX(-1) *** 0.390*** DVIX(-1) *** 0.443*** DVIX(-1) *** 0.315*** [-5.97] [ 21.12] [-6.76] [ 18.28] [-6.74] [ 23.63] [-5.00] [ 21.04] DVIX(-2) *** 0.248*** DVIX(-2) ** 0.236*** DVIX(-2) *** 0.165*** DVIX(-2) *** 0.200*** [-2.85] [ 9.40] [-2.27] [ 9.97] [-5.05] [ 8.00] [-3.02] [ 12.02] DVIX(-3) *** DVIX(-3) *** DVIX(-3) *** DVIX(-3) *** [ 0.73] [ 5.51] [ 0.87] [ 5.50] [ 0.04] [ 7.39] [ 0.38] [ 4.28] DVIX(-4) *** 0.085*** DVIX(-4) *** 0.108*** DVIX(-4) *** 0.128*** DVIX(-4) *** 0.049*** [-4.44] [ 3.24] [-3.50] [ 4.59] [-3.87] [ 6.20] [-2.92] [ 2.89] DVIX(-5) *** DVIX(-5) 0.046* 0.152*** DVIX(-5) *** DVIX(-5) 0.043* 0.119*** [ 0.45] [ 4.82] [ 1.97] [ 6.91] [ 1.02] [ 7.16] [ 1.91] [ 7.57] DVSTOXX(-1) *** DVDAXNEW(-1) 0.065** *** DVXJ(-1) ** *** DVSMI(-1) ** *** [ 0.85] [-14.31] [ 2.72] [-10.13] [-2.53] [-17.03] [-2.77] [-6.18] DVSTOXX(-2) *** *** DVDAXNEW(-2) *** *** DVXJ(-2) *** *** DVSMI(-2) *** *** [-4.62] [-10.02] [-7.32] [-9.33] [-3.94] [-8.81] [-4.11] [-6.05] DVSTOXX(-3) *** DVDAXNEW(-3) * *** DVXJ(-3) * *** DVSMI(-3) ** [-1.73] [-6.03] [-1.88] [-5.34] [-1.87] [-5.57] [-0.01] [-2.60] DVSTOXX(-4) 0.063** DVDAXNEW(-4) 0.047* ** DVXJ(-4) DVSMI(-4) *** [ 2.72] [-0.19] [ 1.93] [-2.16] [-0.33] [-1.55] [-1.57] [-4.39] DVSTOXX(-5) ** *** DVDAXNEW(-5) *** *** DVXJ(-5) DVSMI(-5) ** *** [-2.20] [-5.45] [-4.82] [-5.97] [ 0.06] [ 1.29] [-2.20] [-4.21] Intercept Intercept Intercept Intercept [-0.08] [-0.15] [-0.10] [-0.27] [-0.08] [-0.01] [-0.12] [-0.32] Adj. R % 16.07% Adj. R % 13.06% Adj. R % 22.13% Adj. R % 17.67% *** Significant at the 0.01 level. ** Significant at the 0.05 level. * Significant at the 0.10 level. 27

28 28

29 Table 4: Quandt-Andrews Unknown Breakpoint Test (implied volatility index) The table presents the Quandt-Andrews unknown breakpoint test for 2 variables VAR model with 5 lags. We report the tests for four VAR model respectively, including Panel A (VAR for DVIX and DVTOXX), Panel B (VAR for DVIX and DVDAXNEW), Panel C (VAR for DVIX and DVXJ), and Panel D (VAR for DVIX and DVSMI). P-values are reported in the parentheses. Panel A VAR(DVIX, DVSTOXX) Panel B VAR(DVIX, DVDAXNEW) Statistic DVIX DVSTOXX DVIX DVDAXNEW Maximum LR F-statistic (0.9944) (0.8696) Maximum Wald F-statistic (0.9944) (0.8696) Exp LR F- statistic Exp Wald F- statistic (0.9995) (0.9995) (0.9861) (0.9861) (0.8047) (0.8047) Ave LR F- statistic (0.9995) (0.9682) (0.7981) (0.9999) Ave Wald F- statistic (0.9995) (0.9682) (0.7981) (0.9999) Panel C Panel D VAR(DVIX, DVXJ) VAR(DVIX, DVSMI) Statistic DVIX DVXJ DVIX DVSMI Maximum LR F-statistic (0.9888) (0.9999) (0.9781) Maximum Wald F-statistic (0.9888) (0.9999) (0.9781) Exp LR F- statistic Exp Wald F- statistic (0.9999) (0.9999) (0.8841) (0.8841) (0.9969) (0.9969) (0.9149) (0.9149) Ave LR F- statistic Ave Wald F- statistic (0.9978) (0.9978) (0.8011) (0.8011) (0.9876) (0.9876) (0.8201) (0.8201) 29

30 Table 5: Jennrich tests for the unconditional correlation matrix (realized volatility) The table reports the p-value of Jennrich test between each calendar year correlation matrix. We compute the correlation between two markets realized volatility index of each calendar year. the correlations for the realized volatilities include the correlations between S&P 500 index and STOXX, S&P 500 index and DAX, S&P 500 index and Nikkei 225, and S&P 500 index and SMI. The realized volatility of each market is computed using data for the previous 20 days. The entry is the p-value of Jennrich tests for the correlation of the left hand side year and the correlation of the year at the top. vix-vstoxx vix-vdaxnew

31 Table 5 (Continued) vix-vxj vix-vsmi

32 Table 6: Quandt-Andrews unknown breakpoint test (realized volatility) The table presents the Quandt-Andrews unknown breakpoint test for 2 variables VAR model with 5 lags. We report the tests for four VAR model respectively, including Panel A (VARs for the realized volatilities between S&P 500 index and STOXX), Panel B (S&P 500 index and DAX), Panel C (S&P 500 index and Nikkei 225), and Panel D (S&P 500 index and SMI). P-values are reported in the parentheses. Panel A VAR(DVIX, DVSTOXX) Panel B VAR(DVIX, DVDAXNEW) Statistic DVIX DVSTOXX DVIX DVDAXNEW Maximum LR F- statistic (0.9999) Maximum Wald F- statistic (0.9999) Exp LR F-statistic Exp Wald F-statistic Ave LR F-statistic Ave Wald F-statistic Panel C VAR(DVIX, DVXJ) Panel D VAR(DVIX, DVSMI) Statistic DVIX DVXJ DVIX DVSMI Maximum LR F- statistic Maximum Wald F- statistic Exp LR F-statistic Exp Wald F-statistic Ave LR F-statistic Ave Wald F-statistic