Mutual volatility transmission between assets and trading places

Andreas Masuhr; Mark Trede

doi:10.1515/demo-2022-0155

Article Open Access

Mutual volatility transmission between assets and trading places

Andreas Masuhr and Mark Trede

Published/Copyright: October 17, 2023

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From the journal Dependence Modeling Volume 11 Issue 1

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 long-lasting and comparably large response of volatility in Tokyo, whereas they suggest a quicker volatility decay in London and New York.

Keywords: international volatility spillovers; copula-GARCH models; intra-day; volatility impulse responses

MSC 2010: G15; G11; F37; C32; C51

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 copula-generalized-autoRegressive-conditional-heteroskedasticity (copula-GARCH) model that incorporates non-normal 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 non-overlapping trading zones over a 24-h 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 short-lived 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 non-overlapping 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 non-US 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 intra-day effects between different trading zones. Clements et al. [4] re-investigated 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 non-overlapping 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 z 1 a and ends at z 1 b , and trading times in the second zone are from z 2 a to z 2 b . The extant literature [4–6] imposes the restriction that the zones do not overlap, i.e., z 1 b ≤ z 2 a . We drop this assumption and instead assume that the two zones overlap for a time period of length^[1] t 12 = z 1 b − z 2 a (Figure 1). Let t 1 = z 2 a − z 1 a ( t 2 = z 2 b − z 1 b ) denote the duration when only the first (second) trading zone is active. Total trading hours are denoted by T 1 = z 1 b − z 1 a ( T 2 = z 2 b − z 2 a ).

Figure 1

Two overlapping trading zones.

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 η i denote the “standardized news” occurring in zone i , i.e., E ( η i ) = 0 and Var ( η i ) = 1 . Furthermore, denote the non-standardized news (i.e., the variance is proportional to the trading duration in the respective zone) occurring exclusively in the first, second, and overlapping zone by η 1 * , η 2 * , and η 12 * . Then, the former can be expressed as sums of the latter:

η 1 = η 1 * + η 12 * T 1 , η 2 = η 12 * + η 2 * T 2 .

Accordingly, the correlation coefficient between the news of overlapping trading zones is as follows:

Corr ( η 1 , η 2 ) = t 12 T 1 T 2 .

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 K ≥ 2 assets traded in J ≥ 2 trading places. Denote the return of asset k in trading place j in period t by r t , j , k . Define the vectors of returns

r t = ( r t , 1 , 1 , … , r t , 1 , K , r t , 2 , 1 , … , r t , 2 , K , … , r t , J , 1 , … , r t , J , K ) ′ .

Note that each asset appears J times in r t , namely, once on each market. The other vectors are stacked in the same fashion (i.e., the asset index is running fastest). The conditional covariance matrix of the returns r t is described by a constant correlation copula-GARCH model [12]:

(1) r t = H t 1 2 ε t ,

(2) H t = diag ( h t ) 1 2 M diag ( h t ) 1 2 ,

(3) h t = κ + Ah t − 1 + A ˜ h t + Gr t − 1 2 + Br t 2 ,

(4) M = R Z ⊗ R A ,

(5) ε t , j , k = t ν j , k − 1 ( u t , j , k ) ⋅ ( ν j , k − 2 ) ⁄ ν j , k ,

(6) u t ∼ t -copula ( I J ⋅ K , δ ) .

We proceed to describe the model parameters in turn. The matrix H t , appearing in (1) and (2), is the conditional covariance matrix of dimension ( J K × J K ) . It has a time-invariant correlation matrix M , but time-varying variances h t . In a typical GARCH fashion, the vector of conditional variances, h t in (3) depends on a constant vector κ , vectors of past conditional variances and vectors of past squared returns (the notation r t 2 represents the vector of element-wise squared return). The ( J K × J K ) matrices A and G display the transmission effects of the previous trading day referred to as heat waves in Engle et al. [5]. For trading zones that open later during a trading day, variances and squared returns from preceding zones are, of course, also past values. Hence, the vectors h t and r t 2 also appear on the right-hand side of equation (3). The matrices A ˜ and B account for the transmission effects of preceding trading places (referred to as meteor showers). Only transmissions from within the same trading place ( A and G ) and the immediately preceding trading place ( A ˜ and B ) are taken into account. Consequently, the matrices have the following shapes if there are three trading zones:

(7) A = A 11 0 A ˜ 13 0 A 22 0 0 0 A 33 , B = 0 0 0 B 21 0 0 0 B 32 0 G = G 11 0 B 13 0 G 22 0 0 0 G 33 , A ˜ = 0 0 0 A ˜ 21 0 0 0 A ˜ 32 0 .

All submatrices of A , A ˜ , B , and G have dimension K × K . Note that A ˜ 13 and B 13 account for effects of the immediately preceding trading place and, hence, describe the meteor shower patterns even though they appear in A and G .

The time-invariant correlation matrix M in (4) depends on the ( K × K ) cross-sectional correlation matrix R A between the assets and the ( J × J ) correlation matrix between trading zones due to overlapping trading hours, R Z . While it is conceivable to model R A as time-varying, we refrain from doing so to avoid inflating the number of parameters. If trading places are subject to different daylight-saving times, it is possible to use a separate matrices (say, R Z , 1 and R Z , 2 ) to account for varying overlapping times. In any case, the matrix R Z need not be estimated but can be computed given the opening and closing times of the markets. If two zones do not overlap, their correlation is zero.

In contrast to most of the literature on volatility spillovers, we do not impose Gaussianity on the innovation vector ε t in (5). Instead, we assume that the ( j , k ) -th component of the innovation vector ε t follows a (rescaled) t -distribution with ν j , k > 2 degrees of freedom. We denote its cumulative distribution function by F ν j , k ( ⋅ ) and the corresponding density function by f ν j , k ( ⋅ ) . The J × K parameters ν j , k are collected in the vector ν . The innovations for the assets are linked by a Student- t copula in equation (6) with the following structure:

C ( u t , 1 , 1 , … , u t , J , K ; I , δ ) = F I , δ ( F δ − 1 ( u t , 1 , 1 ) , … , F δ − 1 ( u t , J , K ) ) ,

where ( u t , 1 , 1 , … , u t , J , K ) ∈ ( 0 , 1 ) J × K , and F δ − 1 is the quantile function of the univariate standard Student- t distribution with δ degrees of freedom, and F I , δ is the multivariate standard Student- t distribution function with ( J K × J K ) identity correlation matrix I and δ degrees of freedom. The smaller the degrees of freedom parameter δ , the larger the upper and lower tail dependence of the copula. The copula component u t , j , k is transformed into the innovation ε t , j , k according to (5), where the scaling factor ensures that all innovations have unit variance. Note that the innovations do not follow a multivariate t -distribution even though they have a t -copula and each margin has a t -distribution (with different degrees of freedom). Furthermore, the identity correlation matrix does not imply independence, but it does ensure uncorrelatedness.

As our main focus is on modeling volatility transmissions, the model does not take into account non-zero expected returns. Hence, we assume for simplicity that each return time-series r 1 , j , k , … , r T , j , k is demeaned.

To assure stationarity of the return distribution, the parameters have to satisfy some restrictions. The one-step-ahead forecast of h t is E ( h t ∣ ℱ t − 1 ) , where ℱ t − 1 denotes all information up to, and including, t − 1 , i.e.,

ℱ t − 1 = { r 0 , … , r t − 1 , h 0 , … , h t − 1 } .

Starting at equation (3), taking expectations and applying the law of iterated expectations result in

E ( h t ∣ ℱ t − 1 ) = E ( κ + Ah t − 1 + A ˜ h t + Br t 2 + Gr t − 1 2 ∣ ℱ t − 1 ) = κ + Ah t − 1 + A ˜ E ( h t ∣ ℱ t − 1 ) + B E ( h t ∣ ℱ t − 1 ) + Gr t − 1 2

since diag ( H t ) = h t , and hence,

(8) E ( h t ∣ ℱ t − 1 ) = ( I − A ˜ − B ) − 1 ( κ + Ah t − 1 + Gr t − 1 2 ) .

Accordingly, the n ≥ 1 step ahead forecast is given by:

(9) E ( h t ∣ ℱ t − n ) = ( I − A ˜ − B ) − 1 ( κ + ( A + G ) E ( h t − 1 ∣ ℱ t − n ) ) .

Evaluating E ( h t − 1 ∣ ℱ t − n ) in (9) recursively for n → ∞ reveals that stationarity requires all eigenvalues of ( I − A ˜ − B ) − 1 ( A + G ) to lie inside the unit circle. For stationary processes, the vector of unconditional variances is

(10) h = lim n → ∞ E ( h t ∣ ℱ t − n ) = ( I − A − A ˜ − B − G ) − 1 κ .

Equation (10) allows us to check for any given parametrization if the unconditional variances h are nonnegative.

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 log-likelihood. Let θ contain all model parameters (i.e., κ , A , A ˜ , B , G , ν , δ , and the free elements of the correlation matrix R A ). For simplicity, the prior distributions of all parameters are assumed to be uniform and uninformative (if informative prior knowledge is available, it can be included in a straightforward way). The log-likelihood is [12]

ln L ( θ ; r 1 , … , r T ) = ∑ t = 2 T [ ln ∣ det H t − 1 ⁄ 2 ∣ + ln c ( F ν 1 , 1 ( ε t , 1 , 1 ) , … , F ν J , K ( ε t , J , K ) ) + ln f ν 1 , 1 ( ε t , 1 , 1 ) + … + ln f ν J , K ( ε t , J , K ) ] ,

where

(11) ε t = ( ε t , 1 , 1 , … , ε t , J , K ) ′ = H t − 1 ⁄ 2 r t .

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 Metropolis-Hastings sampler – augmented by randomly clustering parameters into blocks as suggested by Chib and Ramamurthy [3]. DEMC offers two major advantages over ordinary Metropolis-Hastings 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 large-scale computer cluster.

The general idea of the DEMC method is straightforward [14]: N Markov chains are run in parallel with θ i , j denoting the parameter vector of chain j in iteration i . The proposal for chain j is generated as:

θ p , j = θ i , j + γ ( θ i , a − θ i , b ) + η ,

where a and b (with a ≠ b ) are two randomly drawn elements from { 1 , … , N } \ j and η is drawn from a symmetric distribution with unbounded support to ensure irreducibility of the chains. In other words, DEMC randomly chooses two different chains θ i , a and θ i , b at each step, computes their difference, scales this difference by a factor γ , adds some random noise, and finally adds the resulting quantity to the parameter vector of the current chain. The proposal is accepted with probability

min 1 , p ( θ p , j ∣ r ) p ( θ i , j ∣ r ) ,

where p ( ⋅ ∣ r ) is the posterior distribution given the data r . If the proposal is not accepted, then θ i + 1 , j = θ i , j . For a large number of chains N and a small variance of the noise η , the proposal asymptotically (in this context, “asymptotically” refers to the number of chains N ) looks like θ p , j = θ i j + γ ε with E ( ε ) = 0 and Cov ( ε ) → 2 Ω , the covariance matrix of the posterior distribution [14].

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 Metropolis-Hastings algorithm. There is no clear theoretical underpinning how to choose the blocks. The larger the number of blocks, the more the algorithm resembles a parameter-by-parameter Metropolis-Hastings Gibbs sampler and the more time-consuming 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 N Markov chains with a unique stationary distribution that has density p ( ⋅ ∣ r ) N [14]. Thus, once the chains have converged, the draws from all chains can be used as posterior samples leading to a large reduction in computation time. In a simulation study on a 100-dimensional normal distribution, TerBraak [14] shows that convergence speed strongly depends on the number of parallel Markov chains and can be substantially increased by choosing a high number of chains. This underlines that DEMC is perfectly tailored for cluster computers with the ability to evaluate many chains in parallel. This argument becomes even stronger the more complex and time-consuming the evaluation of the likelihood is as the fraction of overhead computing costs diminishes the longer each individual processor needs for calculating the likelihood.

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 index-market 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 log-prices, is not the same in the three markets.

Figure 2

Daily closing prices of the MSCI Japan, EuroStoxx 50, and S&P 500 in the markets in Japan, the UK, and the US. All prices have been normalized such that each time series starts at price 1.0.

Figure 3

Daily returns of the MSCI Japan, EuroStoxx 50, and S&P 500 in the markets in Japan, the UK, and the US. Returns are calculated as the logarithm of the ratio of closing price to opening price.

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 first-order autocorrelation of the returns, and panel (c) the first-order 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 first-order autocorrelation of the daily returns is generally small. The large outlier ( − 14.9 %) for the S&P 500 in the US market is mainly caused by a very small number of erratic price changes in the second and third week of March 2020 as a reaction to the onset of the corona crisis. The level of the first-order autocorrelation in squared returns demonstrates the existence of volatility clustering.

Table 1

Annualized standard deviations of the returns (a), first-order autocorrelations of the returns (b), and first-order autocorrelations of the squared returns (c)

	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	‒ 0.0184	0.0943	‒ 0.0641
UK	0.0025	0.0365	0.0818
US	‒ 0.0303	0.0038	‒ 0.1490
	(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 (non-overlapping) markets in Japan and the US display a relatively high correlation (0.32). While this value is substantially lower ( ≈ 0.2 ) when the period of the corona pandemic is left out, it is still considerably higher than the corresponding correlation for the MSCI Japan and the EuroStoxx.

Table 2

Correlation coefficients of daily returns between trading places for the three indices MSCI Japan, EuroStoxx and S&P 500, and the average correlation

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 burn-in phase. The prior distributions are flat. For the degrees of freedom parameters ν , we impose uniform priors on the interval [ 1 , 40 ] . At the upper boundary of the interval, the t -distribution is close to a normal distribution. The posterior means of the parameters A , A ˜ , B , G , R A , ν , and δ (and the corresponding standard deviations) are reported in Appendix A. Looking at the estimates of the copula parameters, we find that the degree of freedom parameter of the t -copula is rather large ( ≈ 25 ), indicating that there is virtually no tail dependence in the return shocks. Moreover, the estimates of the degrees of freedom parameters of the marginal distributions are close to the upper limit (of 40) of their prior distribution. Hence, the t -copula is in effect much the same as a Gaussian copula. The point estimates of the cross-sectional ( 3 × 3 ) correlation matrix of the assets ( R A ) are all close to 0.6. Turning to the parameters of the volatility equations, we find that the posterior confidence intervals of many estimates include zero. The large number of parameters implies that their estimation precision is only limited. The point estimate of ( I − A ˜ − B ) − 1 ( A + G ) satisfies the stationary condition: all eigenvalues are inside the unit circle. The point estimates of all unconditional variances (see equation (10)) are positive. To visualize how return shocks influence the conditional volatility, we compute volatility impulse response functions (VIRF). Similar to Hafner and Herwartz [7], we define the VIRF as:

V t ( r 0 ) = E ( vech ( H t ) ∣ r 0 , h 0 ) − E ( vech ( H t ) ∣ h 0 ) ,

i.e., the impact of a return shock in t = 0 on volatilities and covariances given some initial state of volatility h 0 . In contrast to Hafner and Herwartz [7], we do not consider orthogonal shocks since innovations drawn from a t -copula are not independent anyway. Instead, we trace the impact of a given return vector r 0 (with all elements save one equal to zero) by comparing the variances and covariances to the benchmark case r 0 = 0 . Using equation (8), the VIRF can be computed as:

(12) V 1 ( r 0 ) = vech diag ( ( I − A ˜ − B ) − 1 ( κ + Ah 0 + Gr 0 2 ) ) 1 2 M diag ( ( I − A ˜ − B ) − 1 ( κ + Ah 0 + Gr 0 2 ) ) 1 2 − diag ( ( I − A ˜ − B ) − 1 ( κ + Ah 0 ) ) 1 2 M diag ( ( I − A ˜ − B ) − 1 ( κ + Ah 0 ) ) 1 2 ,

where I is a ( J K × J K ) identity matrix and diag ( x ) creates a diagonal matrix with the elements of vector x on its main diagonal. For t ≥ 2 , V t ( r 0 ) can be computed using the recursion in equation (9). The terms involving h 0 cancel when computing the impact on the variances (but not for the covariances).

Figure 4 shows the VIRFs of the three assets in the three markets for a shock of size r 0 , j , k = 0.1 in the respective home trading zones ( j = k ) with all other returns in t = 0 fixed to zero. We set the initial h 0 to the vector of the (estimated) unconditional variances. The plots reveal large and persistent reactions in Tokyo, particularly in response to shocks in the MSCI in Japan and the S&P 500 in the US. Shocks in the EuroStoxx index have rather limited and only short-term effects in all markets. The largest and most persistent responses can be observed for a shock in the S&P 500. This indicates that news appearing during US trading hours tend to have a global impact on volatility, reflecting the worldwide importance of the US economy.

Figure 4

VIRFs of the MSCI Japan, EuroStoxx, and S&P 500 in Japan, UK, and US. Shocks happen in the respective home trading zones and have a magnitude of 0.1.

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 h JP,MSCI-JP , t − 1 ( A 1 , 1 = 0.648 ) and r JP,MSCI-JP , t − 1 2 ( G 1 , 1 = 0.143 ) on the return of the MSCI Japan in Tokyo is 0.3955.

Table 3

Aggregated parameter values of the heat wave impact (previous trading day) and the meteor shower impact (preceding trading zone) of home and foreign assets on the home assets’ volatilities

Market	Asset(s)	“Heat wave”	“Meteor shower”
		(previous trading day)	(preceding trading zone)
JP	Home	0.3955	0.3810
	Foreign	‒ 0.1017	‒ 0.0114
UK	Home	0.0632	0.1253
	Foreign	0.0196	0.0469
US	Home	0.3834	0.0828
	Foreign	‒ 0.1473	0.0812
All	Home	0.2807	0.1964
	Foreign	‒ 0.0765	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 copula-GARCH 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 5-year 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

h t = κ + Ah t − 1 + A ˜ h t + Gr t − 1 2 + Br t 2

is split up into the three markets (JP, UK, and US). The vectors h t , JP , h t , UK , and h t , US collect the nine volatilities

h t , JP = h t , JP,MSCI-JP h t , JP,Eurostoxx h t , JP,S&P , h t , UK = h t , UK,MSCI-JP h t , UK,Eurostoxx h t , UK,S&P , h t , US = h t , US,MSCI-JP h t , US,Eurostoxx h t , US,S&P ,

and likewise for the past return vectors r t − 1 , JP 2 , etc. The first 10,000 draws from each chain are discarded as burn-in period. The number of draws from the joint posterior distribution is 4 million (=10,000 draws from 400 chains). For each parameter, we calculate the share of draws that are positive, f pos . Entries are printed in bold face if min ( f pos , 1 − f pos ) < 0.05 , i.e., if more than 95% of the draws are either positive or negative.

h t , JP = 7.352 ( 6.287 ) 18.020 ( 8.576 ) 2.693 ( 3.512 ) × 1 0 − 6 + 0.648 ( 0.223 ) 0.112 ( 0.251 ) − 0.603 ( 0.468 ) 0.169 ( 0.118 ) 0.501 ( 0.307 ) − 0.215 ( 0.424 ) 0.072 ( 0.056 ) 0.112 ( 0.093 ) − 0.138 ( 0.154 ) h t − 1 , JP + 0.604 ( 0.739 ) − 0.274 ( 0.615 ) − 0.041 ( 0.259 ) 0.067 ( 0.694 ) 0.020 ( 0.574 ) − 0.300 ( 0.341 ) − 0.586 ( 0.377 ) 0.181 ( 0.318 ) 0.384 ( 0.197 ) h t − 1 , US + 0.143 ( 0.078 ) 0.038 ( 0.030 ) 0.046 ( 0.049 ) 0.024 ( 0.022 ) 0.094 ( 0.049 ) 0.016 ( 0.051 ) 0.006 ( 0.011 ) 0.015 ( 0.012 ) 0.057 ( 0.033 ) r t − 1 , JP 2 + 0.158 ( 0.145 ) 0.069 ( 0.082 ) 0.200 ( 0.069 ) − 0.018 ( 0.105 ) 0.151 ( 0.078 ) 0.180 ( 0.085 ) 0.150 ( 0.080 ) 0.001 ( 0.033 ) 0.025 ( 0.035 ) r t − 1 , US 2 ,

h t , UK = 11.827 ( 7.258 ) 17.921 ( 10.771 ) 6.239 ( 8.067 ) × 1 0 − 6 + 0.070 ( 0.314 ) 0.005 ( 0.205 ) 0.052 ( 0.207 ) 0.133 ( 0.297 ) 0.107 ( 0.300 ) − 0.055 ( 0.278 ) − 0.152 ( 0.254 ) 0.082 ( 0.263 ) 0.273 ( 0.188 ) h t − 1 , UK + 0.065 ( 0.142 ) 0.125 ( 0.149 ) 0.257 ( 0.355 ) 0.012 ( 0.121 ) 0.258 ( 0.227 ) 0.123 ( 0.472 ) 0.061 ( 0.121 ) − 0.111 ( 0.129 ) 0.489 ( 0.377 ) h t , JP + 0.016 ( 0.028 ) 0.022 ( 0.018 ) − 0.043 ( 0.041 ) − 0.040 ( 0.026 ) 0.020 ( 0.021 ) 0.041 ( 0.067 ) 0.009 ( 0.015 ) − 0.001 ( 0.012 ) 0.083 ( 0.046 ) r t − 1 , UK 2 + 0.040 ( 0.020 ) − 0.010 ( 0.006 ) 0.023 ( 0.037 ) 0.010 ( 0.021 ) − 0.007 ( 0.010 ) 0.043 ( 0.038 ) − 0.008 ( 0.005 ) 0.006 ( 0.005 ) 0.038 ( 0.021 ) r t , JP 2 ,

h t , US = − 3.405 ( 4.002 ) − 0.447 ( 5.057 ) − 2.482 ( 5.096 ) × 1 0 − 6 + 0.038 ( 0.185 ) − 0.049 ( 0.125 ) 0.069 ( 0.107 ) 0.124 ( 0.284 ) − 0.098 ( 0.206 ) 0.059 ( 0.178 ) -0.626 ( 0.334 ) − 0.053 ( 0.263 ) 0.666 ( 0.226 ) h t − 1 , US + 0.290 ( 0.136 ) 0.026 ( 0.147 ) − 0.009 ( 0.145 ) 0.158 ( 0.191 ) 0.288 ( 0.188 ) − 0.017 ( 0.279 ) 0.326 ( 0.181 ) 0.012 ( 0.174 ) − 0.040 ( 0.276 ) h t , UK + 0.062 ( 0.035 ) 0.022 ( 0.018 ) 0.008 ( 0.015 ) − 0.076 ( 0.056 ) 0.041 ( 0.029 ) 0.056 ( 0.030 ) 0.072 ( 0.055 ) 0.017 ( 0.025 ) 0.101 ( 0.034 ) r t − 1 , US 2 + 0.005 ( 0.007 ) -0.013 ( 0.003 ) 0.117 ( 0.027 ) − 0.012 ( 0.017 ) 0.002 ( 0.011 ) 0.066 ( 0.032 ) − 0.008 ( 0.011 ) − 0.005 ( 0.008 ) 0.205 ( 0.045 ) r t , UK 2 .

The estimates for the correlation matrix are

R ˆ A = 1.000 − 0.625 ( 0.012 ) 0.585 ( 0.013 ) 0.625 ( 0.012 ) 1.000 − 0.573 ( 0.013 ) 0.585 ( 0.013 ) 0.573 ( 0.013 ) 1.000 − .

The estimates for the degrees of freedom parameters of the marginal distributions are

Asset/Market	JP	UK	US
MSCI JP	39.024 ( 1.017 )	39.088 ( 0.918 )	39.155 ( 0.903 )
EuroStoxx 50	38.498 ( 1.527 )	38.993 ( 1.110 )	39.101 ( 0.933 )
S&P 500	38.452 ( 1.517 )	39.129 ( 0.907 )	39.135 ( 0.914 )

For the single degrees of freedom parameter δ of the t -copula, we obtain the point estimate δ ˆ = 25.3 , standard deviation 3.5 and all draws are positive.

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Received: 2023-01-04

Revised: 2023-05-11

Accepted: 2023-06-14

Published Online: 2023-10-17

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/demo-2022-0155

Keywords for this article

international volatility spillovers; copula-GARCH models; intra-day; volatility impulse responses

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