Abstract
This article proposes a framework to model the mutual volatility transmission between multiple assets and multiple trading places in different time zones. The model is estimated using a dataset containing daily returns of three stock indices – the MSCI Japan, the EuroStoxx 50, and the S&P 500 – each traded at three major trading places: the stock exchanges in Tokyo, London, and New York. Strong volatility transmission effects can be observed between New York and Tokyo, whereas current volatility in New York mostly depends on past volatility in New York. For the assets in consideration, spillovers are strong across trading zones, but weak across assets, suggesting a close connection between market places but only a loose volatility link between international stock indices. Volatility impulse response functions indicate a longlasting and comparably large response of volatility in Tokyo, whereas they suggest a quicker volatility decay in London and New York.
1 Introduction
Stocks of large international companies are often traded at more than one stock exchange. For instance, Apple stocks can be traded not only on Nasdaq but also on Xetra and many other exchanges. Since the exchanges may be located in different time zones, trading hours may only partially overlap or even be completely disconnected. Extant models of volatility spillover either (a) focus on the transmission of volatility between stocks but neglect the fact that they might be traded across many time zones or (b) pretend that stocks are only tradable on one (“home”) market and model the volatility transmission between exchanges in different time zones. For a deeper understanding of spillovers, it is, however, essential to distinguish transmissions between assets from transmissions between stock exchanges. This article fills a gap in the literature by introducing a model of volatility transmission that allows multiple assets to be traded on multiple markets. We explicitly allow for partially overlapping trading zones. Moreover, the joint return distribution is described by a multivariate copulageneralizedautoRegressiveconditionalheteroskedasticity (copulaGARCH) model that incorporates nonnormal innovations.
Statistical models of international volatility spillovers are of interest for two reasons: in portfolio management, they help optimize the asset weights of an international portfolio. And, even more importantly, they allows to better understand stock markets, in particular how news are digested by the markets. Is an asset’s volatility primarily driven by its lagged volatility on the same market or on markets in other time zones? How large is the volatility transmission stemming from other assets on the same market and on markets in other time zones? Such questions cannot be answered without explicitly modeling multiple assets traded on multiple markets.
Interest in volatility spillovers between different market places is not new. It first arose in the late 1980s and the early 1990s. Beginning with the work of Engle et al. [5], channels of volatility spillovers for different markets and different assets have been detected. Engle et al. [5] used a multivariate GARCH approach modeling a single asset that is traded in (three) different nonoverlapping trading zones over a 24h period. They distinguished between volatility transmitted from yesterday’s own trading zone (“heat waves”) and volatility transmitted from preceding foreign trading zones (“meteor showers”). The meteorological terms “heat wave” and “meteor shower” suggest that the quality of transmissions is different. A heat wave is local, and it will not spill over to other time zones. It also tends to be persistent – if it is hot today, it will likely still be hot tomorrow. In contrast, a meteor shower is shortlived and will not last until tomorrow. However, being observable in one time zone, one can also see it in the next time zone. Building on the work of Engle et al. [5], Hogan and Melvin [10] investigated the heat wave effects in the foreign exchange market by dividing the trading day into four nonoverlapping trading zones with data on the yen/dollar exchange rate, whereas Fleming and Lopez [6] examined the heat wave and meteor shower effects for the US treasury market.
Since the US markets are dominant in international finance, some studies make the simplifying assumption that volatility can only be transmitted from the US to other markets. Using a dynamic panel model, Berg and Vu [1] investigated the volatility spillover effects of US bond and stock markets on output growth in the developed economies. Focusing on the subprime crisis, Hemche et al. [9] studied the dependence of developed and emerging markets on the US market. Linton and Wu [13] faded out nonUS markets and captured their impacts by a separate model for overnight volatility.
Hamao et al. [8] modeled the volatilities of three assets (Nikkei, FTSE, and S&P 500) across three different markets (Tokyo, London, and New York) using a set of independent MA(1)GARCH(1,1) models. They allowed for volatility transmission from preceding trading zones to the current one. In contrast to our approach, they treated assets and trading zones equivalently, i.e., they assumed that the Nikkei can only be traded in Tokyo, the FTSE only in London, and the S&P 500 only in New York.
Booth et al. [2] investigated volatility spillovers for Scandinavian stock markets using a different approach. They used a multivariate GARCH model to describe the transmission between Denmark, Finland, Sweden, and Norway, i.e., trading zones that are almost completely overlapping. Hence, they also did not distinguish between trading zones and assets.
In a more recent approach, Karunanayake et al. [11] used a multivariate GARCH framework with four trading zones to model spillovers. Using weekly data, they excluded any intraday effects between different trading zones. Clements et al. [4] reinvestigated the results of Engle et al. [5]. They additionally estimated a set of different model specifications using realized volatility, decompositions into good and bad news, and separating realized volatility into continuous and jump components. However, Clements et al. [4] did not call into question the artificial construction of nonoverlapping trading zones. Besides, they only considered normally distributed innovations although it is generally accepted that financial returns could exhibit fat tails.
The remainder of this article is structured as follows: Section 2 introduces the volatility transmission model and discusses its statistical properties. Each asset is assumed to be tradable not only on its home market but also on each foreign market. In Section 3, we develop a Bayesian estimation approach that is based on the differential evolution Markov chain algorithm. Section 4 is the empirical application. We model the volatility transmission of three assets (MSCI Japan, EuroStoxx 50, and S&P 500) traded in three partially overlapping trading zones (Tokyo, London, and New York). Section 5 concludes.
2 Modeling volatility transmissions
Modeling multiple assets that are traded on markets in different trading zones enables us to distinguish between characteristics of the assets and characteristics of the trading zones. To highlight this, one asset is chosen for each trading zone that is mainly traded in this zone (the “home asset”) and, hence, subsumes a large share of the total news belonging to this zone. This allows us to investigate how market participants react to news regarding their home asset depending on volatility in other trading zones, past trading days, or other assets (“foreign assets”).
To identify the mutual transmission effects between both assets and trading places, we need a model that can capture both kinds of effects jointly. The general idea of this model is based on Engle et al. [5] and Clements et al. [4], extending their work to multiple assets that are traded in multiple time zones. Moreover, the model allows trading hours of different trading places to overlap.
2.1 Overlapping trading zones
To examine the volatility transmissions in overlapping trading zones, we first fix notation for the trading zones. To that end, consider a world that consists of only two trading zones. Trading in the first zone starts at time
Market movements are driven by new information. We assume that news are publicly available instantaneously worldwide. Hence, if two markets are both open when news arrive, they are affected at the same time and in the same way. Let
Accordingly, the correlation coefficient between the news of overlapping trading zones is as follows:
Obviously, the correlation only depends on the share of overlapping trading time relative to total trading times. As correlation is caused by identical information from both zones, this is in line with the market efficiency hypothesis. Generalizing the approach to more than two trading zones, which may or may not overlap, is straightforward.
2.2 Volatility transmission
Consider
Note that each asset appears
We proceed to describe the model parameters in turn. The matrix
All submatrices of
The timeinvariant correlation matrix
In contrast to most of the literature on volatility spillovers, we do not impose Gaussianity on the innovation vector
where
As our main focus is on modeling volatility transmissions, the model does not take into account nonzero expected returns. Hence, we assume for simplicity that each return timeseries
To assure stationarity of the return distribution, the parameters have to satisfy some restrictions. The onestepahead forecast of
Starting at equation (3), taking expectations and applying the law of iterated expectations result in
since
Accordingly, the
Evaluating
Equation (10) allows us to check for any given parametrization if the unconditional variances
3 Estimation
We propose a Bayesian estimation framework for the model parameters. To derive the joint posterior distribution of the parameters, we need a prior and the loglikelihood. Let
where
The number of model parameters is relatively large. For example, a model with three assets traded in three markets comprises 130 parameters.^{[2]} Estimation of such a large number of parameters is challenging. We suggest to use a differential evolution Markov chain (DEMC) – a combination of the differential evolution optimizer and a Markov chain Monte Carlo approach with a MetropolisHastings sampler – augmented by randomly clustering parameters into blocks as suggested by Chib and Ramamurthy [3]. DEMC offers two major advantages over ordinary MetropolisHastings algorithms: first, DEMC does not rely on a precise specification of the proposal distribution and thus can handle large parameter spaces more easily. Second, DEMC profits from running a large number of Markov chains in parallel and, thus, is perfectly suited to be used on a largescale computer cluster.
The general idea of the DEMC method is straightforward [14]:
where
where
In addition to the DEMC proposal, the parameters are randomly clustered into blocks according to Chib and Ramamurthy [3] in order to speed up convergence. At each iteration, the parameter space is split up into a random number of blocks. The parameters are then randomly assigned to the blocks. For each block in turn, a new draw of the parameters is generated using the MetropolisHastings algorithm. There is no clear theoretical underpinning how to choose the blocks. The larger the number of blocks, the more the algorithm resembles a parameterbyparameter MetropolisHastings Gibbs sampler and the more timeconsuming each iteration becomes. On the other hand, the smaller the number of blocks, the less likely the proposed draws are accepted. Therefore, choosing the number of blocks depends on the application at hand.^{[3]}
The estimation method results in
4 Three markets and three assets
We proceed to estimate the model for three assets traded in three markets. The market places are Tokyo, London, and New York, and the respective assets are the MSCI Japan, the EuroStoxx 50, and the S&P 500 indices.^{[4]} Trading hours of the stock exchanges are as follows: Tokyo, 12 a.m. to 6 a.m. greenwich mean time (GMT) (we ignore the lunch break between 2:30 a.m. and 3:30 a.m.), London, 8 a.m. to 4:30 p.m. GMT, and New York, 1:30 p.m. to 8 p.m. GMT. There is no time overlap between Tokyo and London nor between New York and Tokyo, but New York and London share three trading hours. As the indices are not directly traded as assets in all three markets, we proxy them by exchange traded funds where necessary. Opening and closing prices for the three stock exchanges are provided by Bloomberg.^{[5]} All prices have been converted to US dollars by the data provider.
Figure 2 shows the time series of daily closing prices of the three indices in the three markets. The observation period starts 1 May 2015 and ends 22 May 2020. All prices have been normalized to 1 at the start of the observation period. While the time series looks almost identical prima facie, a closer inspection reveals that there are differences. For example, the sharp drop in the closing price of the EuroStoxx, induced by the onset of the Corona pandemic, is less pronounced in the Japanese market than in the UK and US markets.^{[6]} For the statistical model, we need to restrict attention to tradings days where all three markets were active. The number of daily returns is 1,173 for each indexmarket combination. Figure 3 shows the corresponding daily return time series. Here, the differences between the markets are obvious. Since the markets open and close at different points of time, the return of an asset during a trading day, i.e., the change between opening and closing logprices, is not the same in the three markets.
Table 1 reports some descriptive statistics of the return time series separately for the three indices and the three markets. Panel (a) gives the annualized standard deviation, panel (b) the firstorder autocorrelation of the returns, and panel (c) the firstorder autocorrelation of the squared returns. The standard deviations of the returns of the S&P 500 and the MSCI Japan are highest in their home markets, suggesting that the relevant information flow is denser during the opening times of the home market. For the EuroStoxx, the return volatility in the Japanese market is slightly higher than in the UK. The firstorder autocorrelation of the daily returns is generally small. The large outlier (
Index  

MSCI  Euro  S&P  
Market  Japan  Stoxx  500 
(a) Annualized std. dev.  
JP  0.1682  0.1681  0.1064 
UK  0.1149  0.1371  0.0992 
US  0.0883  0.1111  0.1406 
(b) Return autocorrelation  
JP 

0.0943 

UK  0.0025  0.0365  0.0818 
US 

0.0038 

(c) Squared return autocorr.  
JP  0.2108  0.0751  0.2525 
UK  0.1413  0.1784  0.0886 
US  0.2343  0.2826  0.5002 
Table 2 reports the correlation coefficients of daily returns across the three markets. Parts (a) to (c) show the correlations for the three assets separately. Part (d) reports the correlations averaged over the assets. Overall, the correlations between the Japanese market and the markets in the UK and US are small, reflecting the lack of overlap between those time zones. In contrast, the correlation between the overlapping UK and US markets is notably higher, ranging between 0.22 (for the MSCI Japan) and 0.51 (for the S&P 500). There is one outlier: For the S&P 500 index, the (nonoverlapping) markets in Japan and the US display a relatively high correlation (0.32). While this value is substantially lower (
Index  Correlations between markets  

MSCI Japan  JP  UK  US  
JP  1.000  0.040  0.047  
UK  0.040  1.000  0.222  
US  0.047  0.222  1.000  
EuroStoxx 50  JP  UK  US  
JP  1.000  0.097  0.065  
UK  0.097  1.000  0.290  
US  0.065  0.290  1.000  
S&P 500  JP  UK  US  
JP  1.000  0.006  0.319  
UK  0.006  1.000  0.506  
US  0.319  0.506  1.000  
Average  JP  UK  US  
JP  1.000  0.059  0.128  
UK  0.059  1.000  0.336  
US  0.128  0.336  1.000 
The DEMC algorithm described in Section 3 has been run on a cluster computer.^{[7]} The first 10,000 draws of each chain are discarded as burnin phase. The prior distributions are flat. For the degrees of freedom parameters
i.e., the impact of a return shock in
where
Figure 4 shows the VIRFs of the three assets in the three markets for a shock of size
Following the idea of Engle et al. [5] to distinguish between heat waves and meteor showers, we aggregate all transmission parameters that belong to either the impact of the previous trading day (heat waves) or the preceding trading zone (meteor showers). In addition, we distinguish between the impact of home assets and foreign assets. Table 3 reports the aggregated impacts on the home asset of both the home asset and the foreign assets in the previous trading day and in the preceding trading zone, respectively. The entries in the table are average parameter values. For example, the average of the two parameters determining the impact of
Market  Asset(s)  “Heat wave”  “Meteor shower” 

(previous trading day)  (preceding trading zone)  
JP  Home  0.3955  0.3810 
Foreign 



UK  Home  0.0632  0.1253 
Foreign  0.0196  0.0469  
US  Home  0.3834  0.0828 
Foreign 

0.0812  
All  Home  0.2807  0.1964 
Foreign 

0.0389  
All  0.1021  0.1176 
Table 3 shows that both spillovers from other assets and transmissions originating in foreign trading zones, are present in the data. The volatility of home assets does not only depend on their own past volatility and squared return, but also on volatilities and returns of other assets and also on the preceding trading zone. This finding for international stock markets is in contrast to Engle et al. [5, p. 540], who found “that the empirical evidence is generally against the heat wave hypothesis” on foreign exchange markets.
Aggregating all parameters (of all markets and home as well as foreign assets), we find that the “heat wave” impact of the previous trading day (0.1021) is of the same order as the “meteor shower” impact of the preceding trading zone (0.1176). Zooming in on the parameters of the home asset, the aggregate heat wave effect (0.2807) is larger than the meteor shower effect (0.1964), but not substantially so. In contrast, the aggregate effects of foreign assets are much smaller.
The heterogeneity between the trading zones is notable. In Japan, both the heat wave and the meteor shower impact of the home asset are large. In the US, the meteor shower effect is much smaller, and in the UK, both effects are small. These results are in line with Hamao et al. [8], who also detected strong spillover effects from US and UK stock markets on the Japanese market but much weaker effects in the reverse direction. The volatility impact of foreign assets is rather small in all markets. Hence, ignoring trade in foreign assets in the home markets could be regarded as an acceptable simplification of international volatility transmission models.
5 Conclusion
This article sheds light on how volatility is transmitted both geographically between trading zones and between different assets. To that end, a new copulaGARCH framework that builds on the work of Engle et al. [5] and Clements et al. [4] was proposed and estimated using a novel combination of the differential evolution Markov chain sampler augmented by randomized clustering of the parameters into blocks as suggested in Chib and Ramamurthy [3].
The application in a setting of three assets (MSCI Japan, EuroStoxx, and S&P 500) that are traded over a 5year period (2015–2020) at three major trading places (Tokyo, London, and New York) reveals new insights. In Japan, volatility strongly and persistently responds to return shocks in the home asset, but also to shocks in the S&P 500, while shocks in the EuroStoxx have hardly any lasting volatility impact. In the UK, volatility responses to shocks are more limited and fade away more quickly. The impact of a return shock in the S&P 500 is more pronounced than in the other assets. In the US, the volatility response to shocks in Japan or the UK is very small and not persistent. These results stress the predominance of the US market for stock markets around the world.
Looking at the aggregated effects, it is apparent that, in general, spillovers from the previous trading day (heat waves) on the home asset volatility are somewhat stronger than from the preceding trading zones (meteor showers). However, both effects are roughly of the same order of magnitude. The presence of both types of effects, those associated with trading zones and those associated with home and foreign assets, calls for a model that can distinguish between assets and trading places in a thorough analysis of global volatility transmissions. Nevertheless, our empirical findings indicate that it might be justified to simplify volatility transmission models by only looking at home assets.

Funding information: We acknowledge the support from the Open Access Publication Fund of the University of Muenster.

Conflict of interest: The authors state no conflict of interest.
Appendix A Estimation results
We report the posterior means and standard deviations for all model parameters. The volatility equation
is split up into the three markets (JP, UK, and US). The vectors
and likewise for the past return vectors
The estimates for the correlation matrix are
The estimates for the degrees of freedom parameters of the marginal distributions are
Asset/Market  JP  UK  US 

MSCI JP 



EuroStoxx 50 



S&P 500 



For the single degrees of freedom parameter
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