Extreme Spillover Effects of Volatility Indices

Transcription

1 Yue Peng and Wing Lon Ng / Journal of Economic Research 17 (2012) Extreme Spillover Effects of Volatility Indices Yue Peng 1 ING Bank, United Kingdom Wing Lon Ng 2 University of Essex, United Kingdom Abstract In this study, we analyse contagion effects and extreme comovements of equity and volatility indices in major international markets. The tail dependence coefficients (TDCs) increase during a financial crisis, especially for the lower TDCs of stock index returns and upper TDCs of volatility index returns. This indicates that crashes and fluctuations are easier to transmit between markets during turmoils, implying the existence of contagion. In particular, we find that the crash events transmit from the Japanese market to other markets, whereas booms are more likely to transmit from the US and Europe to Japan. Keywords : Spillover effects; Financial Crisis; Time-varying Copulas; Volatility Indices JEL classification : C32; C58; G01; G15 1 Quant Risk Manager, ING Bank, London, UK. 2 Corresponding author, Centre for Computational Finance and Economic Agents, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK, [email protected]. We are very grateful to the Centre for the Study of Finance and Insurance of the Osaka University for providing the data of the Volatility Index of Nikkei 225 (VXJ). We also would like to thank the Editor and two anonymous referees for their helpful comments and suggestions that led to an improvement of the paper. First received, October 27, 2011; Revision received, March 10, 2012; Accepted, March 13, 2012.

2 2 Extreme Spillover Effects of Volatility Indices 1 Introduction The existing literature on financial contagion mainly concentrates on the time-varying correlation between different stock market indices. In this study, we apply the dynamic copula framework proposed by Patton (2006) to investigate the relationships, particularly the tail dependence, between daily returns of both equity and volatility indices. The results show that both the degree of dependence and the tail dependence increase during the periods of financial turmoil, implying the existence of financial contagion particularly for volatility indices. Furthermore, the extreme movements between volatility indices significantly rise after the middle of 2006, which cannot be found between stock indices. A volatility index is constructed with index option prices and indicates the expected future market volatility; it can be seen as a gauge to measure investors fears over the future market (Whaley, 2000). The volatility index level indicates the willingness of investors to pay for hedging the market downside risk. A higher volatility index level implies a higher expected future realised volatility and more fear about the future market. The advantage of considering the volatility index is that (a) the implied volatility captures the investors expectation on future market volatility more accurately than the second moment of returns, and (b) it provides more market information than the realised volatility (e.g. Blair et al., 2001; Giot, 2005). The model free volatility index methodology introduced by Chicago Board Option Exchange in September 2003 following Carr and Madan (1998) and Demeterfi et al. (1999) is robust and can perfectly replicate the volatility derivative the variance swap. It contains market information for all strike prices and does not depend on any pricing model, providing an excellent tool to measure market volatility. With the increasing popularity of volatility indices, researchers have started analysing cross-market relationships with volatility indices in different index markets (e.g. Nikkinen and Sahlstrom, 2004; Äijö, 2008). Recent research shows that dependence structures in international financial markets are not symmetric but asymmetric (e.g. Xu and Li, 2009; Ammann and Süss, 2009). In most cases, news impacts show an asymmetric effect on the cross-market correlation. The correlation increases much more with bad news than with good news of the same magnitude. Moreover, the literature suggests that market returns have an asymmetric correlation with volatility, that is, negative returns have larger effects on volatility than positive returns (Black, 1976; Christie, 1982; Pindyck, 1984; Schwert, 1989). In general, financial analysts are interested in models which can capture the asymmetric dependence in financial markets, such as asymmetric GARCH models (Kroner and Ng, 1998), Markov Switching models (Ang and Chen, 2002), asymmetric Stochastic Volatility models (Centeno and Salido, 2009) or extreme value the-

3 Yue Peng and Wing Lon Ng / Journal of Economic Research 17 (2012) ory models (Longin and Solnik, 2001), as ignoring the asymmetry in market dependence might lead to suboptimal international diversification. This study further contributes to this research area, using copulas to measure the dependence between different volatility indices across major international stock exchanges over different time periods. Financial contagion has been controversially discussed in the literature. Most common procedures test whether the cross-market correlation of returns increase during a period of crisis. However, Forbes and Rigobon (2002) criticised that market co-movements alone merely indicate interdependence but do not prove the existence of contagion as the correlation coefficient can be upward biased due to the volatility of returns over the time period under consideration. They argued that high market co-movements during periods of financial turmoil can only be seen as a continuation of existing strong cross-market linkages, but not as contagion which would imply an (unbiased) increase in the linkage. The advantage of copula models is that they are very flexible and can also model dependence other than the linear correlation. Moreover, it can capture the extreme co-movements (tail dependence) that a simple linear correlation fails to model (Forbes and Rigobon, 2002), allowing for a more accurate investigation of the extreme spill-over effects between financial markets. Although volatility indices are becoming more and more important in derivative pricing, volatility hedging and risk management, there are rarely investigations on the asymmetry and the time-varying of dependence between different volatility indices during extreme financial events. With respect to asymmetric cross-market dependence and financial contagion, we contribute to the existing literature by analysing extreme nonlinear co-movements of volatility indices with copula models detect contagion and capture asymmetric dependence in higher moments. In particular, we estimate asymmetric dependence structures of stock market movements across the US, Europe and Japan. We take the effect of shifting time zones into account and estimate the model with contemporaneous as well as with a one day lead in the Japanese market returns. In contrast to the studies reported in the literature, we estimate stock market dependence not only with index returns but also with volatility index returns, building on the work of Rodriguez (2007), Xu and Li (2009), and Peng and Ng (2012). Our investigations provide results for both the static and dynamic case as only the latter can catch the time-varying tail dependence behaviour. In the following, we introduce the copula method used in this study. The remainder is organised as follows. Section 2 introduces the copula method and the dynamic extension proposed by Patton (2006). Section 3 discusses the data and the empirical results. Section 4 summarises the main findings of this work.

4 4 Extreme Spillover Effects of Volatility Indices 2 The Copula Concept The copula approach is a very flexible tool for capturing multivariate distributions and can, for example, be used to measure dependence structures between equity markets (Rodriguez, 2007) and foreign exchange markets (Patton, 2006). Since the volatility index returns are asymmetrically distributed and non-gaussian (Low, 2004), the flexibility of the copula approach is useful to model their correlations without relaying on the commonly assumed normal distribution. Also, the copula approach has the advantage that it can measure extreme co-movements (tail dependence) which simple linear correlation coefficient can not. As Forbes and Rigobon (2002) stated, correlation alone does not suggest contagion. The unique ability of the copula method to measure tail dependence is very useful in our analysis. Consider two random variables X 1 and X 2 with continuous univariate distribution functions F X1 (x 1 ) = P (X 1 x 1 ) and F X2 (x 2 ) = P (X 2 x 2 ), and their joint distribution function F X1,X 2 (x 1, x 2 ) = P (X 1 x 1, X 2 x 2 ). The theorem by Sklar (1959) suggests that there exists a function called copula C that merges the univariate distributions F X1 and F X2 to a bivariate distribution function F X1,X 2 (x 1, x 2 ) = C (F X1 (x 1 ), F X2 (x 2 )). (1) If the marginal distributions F X1 and F X2 are continuous, then C is unique, and the random variables X 1 and X 2 have a copula C given by eq. (1). Likewise, for any u 1, u 2 in [0, 1] 2 ( C (u 1, u 2 ) = F X1,X 2 F 1 X 1 (u 1 ), F 1 X 2 (u 2 ) ), (2) where F 1 X 1 (u 1 ) is the quantile function given by F 1 X 1 (u 1 ) = inf {x 1 : F X1 (x 1 ) u 1 }, respectively for F 1 X 2 (u 2 ). The copula C is a multivariate distribution whose marginal distributions are uniformly distributed on the unit interval. For a more detailed introduction to the copula concept, we refer the reader to Joe (1997) and Nelsen (2006). One of the advantages of the copula model over straightforward correlation analysis is that it can measure the probability of extreme co-movements with the so-called tail dependence coefficients (TDCs). For the bivariate joint distribution F X1,X 2 (x 1, x 2 ), the upper and lower TDCs, denoted as λ U and λ L, respectively, can be described as the limit of the conditional probabilities ) λ U = lim P (X 2 > F 1 X 2 (x 2 ) X 1 > F 1 X 1 (x 1 ) (3) 1 C (τ, τ) = 2 lim τ 1 1 τ (4)

5 Yue Peng and Wing Lon Ng / Journal of Economic Research 17 (2012) and λ L = lim P C (τ, τ) = lim τ 0+ τ ) (X 2 F 1 X 2 (x 2 ) X 1 F 1 X 1 (x 1 ) (e.g. Frahm et al., 2005). Consider the filtration F of a stochastic process. In order to extend the copula framework to the time series context to model the dynamic cross-market dependence of the volatility index returns, we employ the concept of conditional copulas introduced by Patton (2006). Let F be an 2-dimensional distribution function with continuous marginal distributions F 1 and F 2, then a modified version of the theorem by Sklar (1959) is ) F t (x 1,t, x 2,t F t 1 ) = C t (F 1,t (x 1,t F t 1 ), F 2,t (x 2,t F t 1 ) F t 1. (7) Because the observed (conditional) marginal distributions of the equity index returns and the volatility index returns are not constant over time but show the common stylised facts of volatility clusters, we fit an ARMA-GARCH model for each series (e.g. Nikoloulopoulos et al., 2010; Hu, 2010) and then compute the standardised returns X 1,t and X 2,t for the subsequent copula estimation. This interim step is essential to obtain standardised returns because otherwise the the copula could be misspecified if joint market movements result from serial autocorrelation and not cross-sectional dependence (see also Forbes and Rigobon, 2002). The dynamic SJC copula model, which is a modified version of the BB7 copula in Joe (1997), is then introduced at the end of this section. If the marginal distributions F 1,t and F 2,t, and the copula C are (twice) differentiable, then the density of the conditional copula can be obtained by ) c t (F 1,t (x 1,t F t 1 ), F 2,t (x 2,t F t 1 ) F t 1 (8) ) 2 C t (F 1,t (x 1,t F t 1 ), F 2,t (x 2,t F t 1 ) F t 1 =. F 1,t (x 1,t F t 1 ) F 2,t (x 2,t F t 1 ) The conditional joint density f t (x 1,t, x 2,t F t 1 ) is then given by f t (x 1,t, x 2,t F t 1 ) = f 1,t (x 1,t F t 1 ) f 2,t (x 2,t F t 1 ) (9) ) c t (F 1,t (x 1,t F t 1 ), F 2,t (x 2,t F t 1 ) F t 1, where f.,t (x.,t F t 1 ) is the univariate conditional density of the standardised returns from the ARMA-GARCH model. We estimate the copula model with the two-step Inference for Margins (IFM) approach introduced by Joe (1997). We select this method because (5) (6)

6 6 Extreme Spillover Effects of Volatility Indices of its good efficiency properties (see also Joe, 2005). First, for the bivariate random vector (X 1,t, X 2,t F t 1 ) n t=1 we estimate the parameters of the marginal distributions θ mar using a Maximum Likelihood (ML) approach, ˆθ mar = argmax n t=1 i=1 2 ln f i,t (x i,t F t 1 ). (10) In the second step, given the estimated parameters ˆθ mar, the parameter estimates of the copula θ cop can be obtained (again via ML), ˆθ cop = argmax n ( ln c t Fi,t (x i,t ) F t 1, ˆθ ) mar t=1 (11) (see also Xu and Li, 2009). As the common elliptic copulas such as the t- or the Gaussian copula do not account for asymmetric tail dependence, we employ the modified symmetrised Joe-Clayton (SJC) copula by Patton (2006) to get a better fit for the asymmetric dependence patterns within the data. We choose this particular model, because in contrast to mixed copulas models (Rodriguez, 2007; Peng and Ng, 2012) the TDC here can be uniquely related to the corresponding copula parameter. We later apply an improved goodness-of-fit test recently proposed by Genest et al. (2009) to show that our chosen copula is not misspecified. We specify the copula C in eq. (12) as the SJC copula which is defined as C SJC ( u, v λ U, λ L) (12) = 0.5 ( C JC ( u, v λ U, λ L) + C JC ( 1 u, 1 v λ U, λ L) + u + v 1 ), where ( C JC u, v λ U, λ L) ( = 1 1 { [1 (1 u) κ ] γ + [1 (1 v) κ ] γ 1} 1/γ ) 1/κ, ( (13) is the original Joe-Clayton copula with κ = 1/ log ) 2 2 λ U (, γ = 1/ log ) 2 λ L, λ U (0, 1), λ L (0, 1) (see also Joe (1997)). The SJC copula has two parameters, λ U and λ L, which measures the tail dependence. As Patton (2006) mentioned, the SJC copula can account for symmetric dependence when λ U = λ L, whereas the original BB7 copula employed in Xu and Li (2009) would be biased towards asymmetry by construction. The dynamic structure for the SJC copula as proposed by Patton (2006) has two ARMA(1,10) processes for the time-varying TDCs λ U t = Λ U ω U + β U λ U 1 10 t 1 + α U u t j v t j (14) 10 j=1

7 Yue Peng and Wing Lon Ng / Journal of Economic Research 17 (2012) and λ L t = Λ L ω L + β L λ L t 1 + α L j=1 u t j v t j, (15) whereby the logistic transformation Λ(a) = ( 1 + e a) 1 (16) restricts the TDCs between (0, 1). The advantage of this specification is that the forcing variable in the evolution equation can be modelled by the MA term, whose expectation in this case is directly inversely related to the concordance measure of the copula (Patton, 2006). 3 Empirical Results In January 1993, Chicago Board Option Exchange (CBOE) first launched and disseminated the Market Volatility Index (VXO) based on at-the-money implied volatility from S&P 100 index options using the method described in Whaley (1993). In September 2003, CBOE constructed a new volatility index (VIX) from S&P 500 index options based on the methodology suggested by Carr and Madan (1998) and Demeterfi et al. (1999) which provides a more accurate estimation of future volatility. Nowadays, various volatility indices for different stock markets are calculated and released by CBOE, and other exchanges mainly adopt these two approaches. New VDAX, VSTOXX and VSMI were released with the same methodology of VIX. More recently, VCAC and VFTSE are constructed with the same methodology of VIX. Likewise, the Center for the Study of Finance and Insurance in Japan released VXJ as a benchmark of future 30 days volatility for Nikkei 225 with the same methodology of VIX and the new VDAX. This methodology is different to at-the-money implied volatility from Black and Scholes (1973) model used in the conventional literature. It captures the whole volatility skewness in stock markets and gives a more precise expectation of short term future market volatility. Table 1 gives an overview of 10 popular volatility indices. However, analysing all possible pair combinations would be beyond the scope of this paper. As we are more interested in spillover effects of volatility indices across different geographical regions, we arbitrarily select two indices from the United States (VIX and VXN), two from Europe (VDAX and VFTSE), and the VXJ for further analysis, also comparing the results with the corresponding equity indices. The chosen volatility indices have the same time to maturity of 30 days, allowing for a better comparison. The sample period ranges from 2nd February 2001 to 29th January It is to be noted that the international

8 8 Extreme Spillover Effects of Volatility Indices Table 1: Summary of Volatility Indices in US, Europe and Japan Index Market Underlying Type Exchange Methodology VXO US S&P 100 American CBOE VXO method Whaley (1993) VIX US S&P 500 European CBOE VIX method Carr and Madan (1998), Demeterfi et al. (1999) VXN(new) US NASDAQ 100 European CBOE VIX method VXD US DJIA European CBOE VIX method VDAX(new) Germany DAX European Deutsche Demeterfi et al. (1999), Börse Deutsche Börse, similar to VIX VSTOXX Euro DJ EURO European Eurex Demeterfi et al. (1999), STOXX 50 Deutsche Börse, similar to VIX VSMI Switzerland SMI European SWX Demeterfi et al. (1999), Swiss Exchange Deutsche Börse, similar to VIX VCAC France CAC 40 European NYSE VIX method VFTSE UK FTSE 100 European NYSE VIX method VXJ Japan Nikkei 225 European CSFI VIX method markets considered in this study have time zone differences, requiring an additional sampling scheme if their operating hours do not overlap. For example, most European markets open after the Asian markets have closed; similarly, most Asian markets usually open after the US markets have closed. Consider the prices of an asset that can be traded in both Tokyo and New York on the same weekday (e.g. Monday). Dependence of contemporaneous prices corresponding to the same calendar day will only reflect spill-over effects from east-to-west, but not vice versa. Therefore, especially when comparing the Japanese market with the US/European markets, we additionally consider the Japanese market returns led by one day to account for possible effects from US and Europe (e.g. Monday) to Japan (e.g. Tuesday). We will only focus on the results for the TDCs in this section. Further detailed estimation results of the copulas are provided in the Appendix. For the static SJC copula (eq.(12)), we estimate its upper and lower TDCs with the IFM method and report them in Table 2. As it can be seen, the SJC copula also show asymmetric upper and lower TDCs. For the equity index return pairs, 9 out of 14 have higher values for the lower TDCs and the

9 Yue Peng and Wing Lon Ng / Journal of Economic Research 17 (2012) Table 2: Estimated TDCs for the static SJC copula Panel A: Stock index returns Panel B: Volatility index returns SP 500 VIX λ U λ U (se) (0.007) NASDAQ 100 (se) (0.008) VXN λ L λ L (se) (0.000) (se) (0.012) λ U λ U (se) (0.014) (0.019) DAX 30 (se) (0.007) (0.025) VDAX λ L λ L (se) (0.013) (0.025) (se) (0.027) (0.023) λ U λ U (se) (0.028) (0.174) (0.012) FTSE 100 (se) (0.024) (0.027) (0.010) VFTSE λ L λ L (se) (0.026) (0.010) (0.018) (se) (0.019) (0.021) (0.011) λ U λ U (se) (0.019) (0.016) (0.021) (0.018) Nikkei (se) (0.015) (0.021) (0.029) (0.026) VXJ λ L λ L (se) (0.015) (0.017) (0.023) (0.013) (se) (0.016) (0.000) (0.017) (0.018) λ U λ U (se) (0.017) (0.022) (0.031) (0.029) Nikkei (se) (0.029) (0.018) (0.023) (0.022) VXJ λ L λ L (lead) (se) (0.023) (0.007) (0.014) (0.026) (lead) (se) (0.015) (0.021) (0.009) (0.007) This table lists the estimated upper (λ U ) and lower (λ L ) tail dependence coefficients (TDC) of the SJC copula (see eq.(12)) in order to describe the extreme spill-over effects for all possible pairwise combinations of the 5 markets, considering both their equity index returns (Panel A) or volatility index returns returns (Panel B). Values in parentheses are the corresponding standard errors (se). rest have values for the upper TDCs which are close to the lower TDCs. The upper TDCs for the volatility index returns are almost always higher than the lower TDCs for volatility index. This finding indicates evidence of a negative correlation between market return and its volatility and is consistent with most of the literature stating that market downside dependent risk is generally higher than the upside one. The results of the goodness-of-fit tests are shown in Table 4 in the Appendix, implying that the copula models are not misspecified. Figures 1 and 2 plot selected time series of the estimated TDCs (equations (14) and (15) in the dynamic extension of the SJC copula. Table 3 provides a summary of the maximum, minimum and mean values of the TDCs. For

10 10 Extreme Spillover Effects of Volatility Indices Table 3: Summary of TDCs for the dynamic SJC copula Panel A: Stock index return Panel B: Volatility index return S&P 500 VIX λ L λ U λ L λ U Min Min Mean NASDAQ 100 Mean VXN Max λ L λ U Max λ L λ U Min Min Mean DAX 30 Mean VDAX Max λ L λ U Max λ L λ U Min Min Mean FTSE 100 Mean VFTSE Max λ L λ U Max λ L λ U Min Min Mean Nikkei 225 Mean VXJ Max Max Min Min Mean Nikkei 225 (lead) Mean VXJ (lead) Max Max This table lists the minimum, mean, and maximum of the dynamic upper (λ U t, eq.(14)) and lower (λ L t, eq.(15)) tail dependence coefficients (TDC) of the SJC copula in order to describe the range of extreme spill-over effects for all possible pairwise combinations of the 5 markets, considering both their equity index returns (Panel A) and volatility index returns (Panel B).

11 Yue Peng and Wing Lon Ng / Journal of Economic Research 17 (2012) Figure 1: TDCs of dynamic SJC copula between German and UK markets 9/11 Invasion of Iraq 0.8 Markets DAX 30 FTSE 100 Tumbled Lehman Brother Bankcruptcy TDC / Markets Invasion of Lehman Brother 9/11 Tumbled Iraq Bankcrupt 0.8 VDAX VFTSE TDC / Upper Tail Lower Tail Figure 2: TDCs of dynamic SJC copula between UK and Japanese markets Invasion of Markets 9/11 Iraq FTSE 100 Nikkei 225 Tumbled 0.8 Lehman Brother Bankcruptcy TDC / /11 Invasion of Iraq VFTSE VXJ Markets Tumbled Lehman Brother Bankcruptcy TDC / Correlation ρ of dynamic t copula for volatility index returns Upper Tail Lower Tail example, the cross-market relationships between the US and Japan as reflected in contemporaneous return pairs for both equity and volatility index is always weaker than compared to the one day lead of Japanese return. This finding suggests that the market movements are generally propagated from the US to Japan, not vice versa. We take the joint markets of Germany and the UK with dynamic SJC copula as another example (see Figure 1). The upper TDC (black) of DAX 30 - FTSE 100 is obviously lower than the lower TDC (red), and the upper TDC (black) of VDAX - VFTSE is obviously higher than the lower TDC (red) (also see Table 3). During financial crises their TDCs increase, indicating the existence of contagion. Between the German and UK markets, the market downside dependence risk is almost always more significant than the upside one. When one market becomes extremely volatile, the other one has a greater chance of becoming volatile; when one market changes from a turbulent to a calm state the other market is less likely to be stable. For most joint mar-

12 12 Extreme Spillover Effects of Volatility Indices kets between Japan and the US/Europe, we can also find similar asymmetric properties with their TDCs. We take another example of cross-market relationships between the UK and Japan (see Figure 2). For the FTSE Nikkei 225 pair, the mean of its lower TDC is higher than the mean of the upper TDC, and in most situations when the joint market dependence increases, the change in their lower TDC lasts longer and increases more obviously than the upper TDC. For the VFTSE - VXJ pair, the upper TDC is higher than lower TDC for the most time periods, and after the market crises, only the upper TDC significantly increases. Between the UK and Japan, the lower TDC of the equity index return pair and the upper TDC of the volatility index return pair dominate the dynamic joint tail dependence. Our findings indicate that the dependence between volatility indices is time variant. In most cases, their correlations are higher in the bear market and lower in the bull market, implying the existence of financial contagion. Interestingly, the extreme value movements between volatility indices significantly rise after the middle of 2006, which cannot be found between stock indices. The dependence structure between Japanese returns and 1 day lagged US/European returns rise significantly around the end of 2008 for both stock indices and volatility indices, which indicates that the crisis during the end of 2008 is transmitted from the US to Japan, and not the other way. In most cases, the estimated tail dependence coefficients (TDCs) are negatively related with market trends, which emphases that the extreme co-movements of markets are higher during financial turmoil than the booming periods. The estimated TDCs also show that the big changes of volatility index returns transmit less frequently between US and Europe, and between Japan and US/Europe, than compared to stock index returns. 4 Summary Recently, volatility indices and volatility products have become very popular. Investors pay more and more attention to the volatility trading and hedging; especially during bear markets these financial instruments are even more attractive. This study investigates time-varying dependence between daily returns of equity indices and their corresponding volatility indices between five major international markets. In particular, we apply a copula framework in order to account for tail dependence and, hence, to measure the extreme spillover effects that the standard correlation measure can not detect. Generally the results of volatility index returns are quite similar to their corresponding stock index returns in most situations. However, there are also different findings specific for volatility index returns. For example, the tail

13 Yue Peng and Wing Lon Ng / Journal of Economic Research 17 (2012) dependence decreases during the financial crisis at the end of 2008 for stock index returns. However, this phenomena can not be observed for volatility index returns across US and European markets, and between Japanese and US/European markets. This also reflects on their tail dependence coefficient patterns. Considering for most cases of volatility indices across US and European markets, and between Japanese and US/European markets, the correlations increase at the end of Volatility indices are much more volatile than stock indices and their changes are quite time sensitive. Our main findings are that the correlations of stock indices and volatility indices increase during the periods of financial crisis, which supports the existence of financial contagion. The results from the tail dependence coefficients confirm our hypothesis. In particular, there are more extreme co-movements between volatility indices after the middle of 2006, but this phenomenon cannot be found for stock indices. References Äijö, J., Implied volatility term structure linkages between VDAX, VSMI and VSTOXX volatility indices, Global Finance Journal 18(3), 2008, Ammann, M. and Süss, S., Asymmetric dependence patterns in financial time series, European Journal of Finance 15(7-8), 2009, Ang, A. and Chen, J., Asymmetric correlations of equity portfolios, Journal of Financial Economics 63(3), 2002, Black, F., The pricing of commodity contracts, Journal of Financial Economics 3(1-2), 1976, Black, F. and Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy 81(3), 1973, Blair, B. J., Poon, S. H., and Taylor, S., Forecasting S&P 100 volatility: the incremental information content of implied volatilities and highfrequency index returns, Journal of Econometrics 105(1), 2001, Carr, P. and Madan, D., Towards a theory of volatility trading, In Volatility: New Estimation Techniques for Pricing Derivatives, RISK Publications, London

14 14 Extreme Spillover Effects of Volatility Indices Centeno, M. G. and Salido, R. M., Estimation of asymmetric stochastic volatility models for stock-exchange index returns, International Advances in Economic Research 15(1), 2009, Christie, A. A., The stochastic behavior of common stock variances : Value, leverage and interest rate effects, Journal of Financial Economics 10(4), 1982, Demeterfi, K., Derman, E., Kamal, M., and Zou, J., A guide to volatility and variance swaps, Journal of Derivatives 6(4), 1999, Forbes, K. and Rigobon, R., No contagion, only interdependence: measuring stock markets comovements, Journal of Finance 57(5), 2002, Frahm, G., Junker, M., and Schmidt, R., Estimating the taildependence coefficient: Properties and pitfalls, Insurance: Mathematics and Economics 37(1), 2005, Genest, C., Remillard, B., and Beaudoin, D., Goodness-of-fit tests for copulas: A review and a power study, Insurance: Mathematics and Economics 44(2), 2009, Giot, P., Relationships between implied volatility indexes and stock index returns, Journal of Portfolio Management 31(3), 2005, Hu, J., Dependence structures in chinese and us financial markets: a time-varying conditional copula approach, Applied Financial Economics 20(7), 2010, Joe, H., Multivariate Models and Dependence Concepts. Chapman & Hall, London Joe, H., Asymptotic efficiency of the two-stage estimation method for copula-based models, Journal of Multivariate Analysis 94(2), 2005, Kroner, K. F. and Ng, V. K., Modeling asymmetric comovements of asset returns, The Review of Financial Studies 11(4), 1998, Longin, F. and Solnik, B., Extreme correlation of international equity markets, Journal of Finance 56(2), 2001, Low, C., The Fear and Exuberance from Implied Volatility of S&P 100 Index Options, Journal of Business 77(3), 2004,

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16 16 Extreme Spillover Effects of Volatility Indices Appendix Table 4: The p-values of the goodness-of-fit test for the copulas Stock index returns Volatility index returns S&P 500 VIX NASDAQ VXN DAX VDAX FTSE VFTSE Nikkei VXJ Nikkei 225 (lead) VXJ (lead) Following Genest et al. (2009) the null hypothesis is that the copula models are not misspecified. Since all p-values for all return pairs are higher than the conventional 10% significance level, we conclude that the null hypothesis can not be rejected.

17 Yue Peng and Wing Lon Ng / Journal of Economic Research 17 (2012) Table 5: Estimated parameters of the dynamic SJC copula Panel A: Stock index returns Panel B: Volatility index returns S&P 500 VIX ω U ω U (se) (0.048) (se) (0.000) α U α U (se) (0.222) (se) (0.000) β U β U (se) (0.130) (se) (0.001) ω L NASDAQ100 ω L VXN (se) (0.002) (se) (0.074) α L α L (se) (0.043) (se) (0.295) β L β L (se) (0.007) (se) (0.059) llk llk ω U ω U (se) (0.616) (0.092) (se) (0.478) (0.046) α U α U (se) (0.443) (1.075) (se) (1.487) (0.238) β U β U (se) (0.295) (0.300) (se) (0.199) (0.082) ω L DAX 30 ω L VDAX (se) (0.366) (0.325) (se) (0.500) (0.042) α L α L (se) (2.110) (0.944) (se) (1.493) (0.313) β L β L (se) (0.078) (0.640) (se) (1.948) (0.237) llk llk ω U ω U (se) (0.068) (0.063) (0.002) (se) (0.296) (0.154) (0.068) α U α U (se) (0.204) (0.956) (0.008) (se) (1.079) (0.979) (0.045) β U β U (se) (0.085) (0.338) (0.003) (se) (0.281) (0.954) (0.014) ω L FTSE100 ω L VFTSE (se) (0.506) (0.121) (0.000) (se) (0.060) (0.307) (0.015) α L α L (se) (2.693) (0.413) (0.006) (se) (0.447) (1.217) (0.032) β L β L (se) (0.327) (0.475) (0.001) (se) (0.440) (0.241) (0.041) llk llk ω U ω U (se) (3.045) (0.888) (0.443) (0.080) (se) (0.695) (1.295) (0.886) (0.601) α U α U (se) (6.889) (1.736) (1.409) (0.306) (se) (1.461) (5.721) (2.362) (2.515) β U β U (se) (8.346) (23.393) (2.432) (0.298) (se) (79.336) (4.525) (5.235) (1.421) ω L Nikkei 225 ω L VXJ (se) (0.822) (1.012) (0.611) (0.115) (se) (2.931) (3.932) (3.714) (3.123) α L α L (se) (0.230) (3.028) (3.061) (0.560) (se) (7.493) (32.008) (10.828) (6.114) β L β L (se) (50.342) (70.254) (0.575) (0.061) (se) (36.803) (41.721) (17.612) (3.689) llk llk ω U ω U (se) (0.068) (0.411) (0.087) (0.243) (se) (0.455) (0.072) (0.708) (0.320) α U α U (se) (0.152) (1.075) (0.244) (1.741) (se) (2.695) (1.243) (1.422) (0.843) β U β U (se) (0.013) (0.172) (0.162) (0.751) Nikkei 225 (se) (0.770) (0.531) (0.323) (1.500) VXJ ω L (lead) ω L (lead) (se) (0.045) (0.197) (0.036) (0.119) (se) (3.441) (0.828) (1.320) (1.591) α L α L (se) (0.460) (0.363) (0.122) (0.314) (se) (6.763) (2.208) (0.772) (4.083) β L β L (se) (0.263) (0.450) (0.441) (0.461) (se) (15.500) (2.944) (10.597) (6.286) llk llk The dynamic SJC copulas for all pairwise combinations are estimated with equations (14) and (15). Values in parentheses are the corresponding standard errors (se), llk is the log likelihood function value.