This paper studies stock price comovements in two key regions [the United States and Europe, which is represented by three major European developed countries (France, Germany, and the United Kingdom)]. Our paper uses recent high-frequency data (HFD) and investigates price comovements in the context of “normal times” and crisis periods. To this end, we applied a non-Gaussian Asymmetrical Dynamic Conditional Correlation (ADCC)-GARCH (Generalized Autoregressive Conditional Heteroscedasticity) model and the Marginal Expected Shortfall (MES) approach. This choice has three advantages: (i) With the development of high-frequency trading (HFT), it is more appropriate to use HFD to test price linkages for overlapping and nonoverlapping data. (ii) The ADCC-GARCH model captures further asymmetry in price comovements. (iii) The use of the MES enables to measure systemic risk contributions around the distribution tails. Accordingly, we offer two interesting findings. First, while the hypothesis of asymmetrical and time-varying stock return linkages is not rejected, the MES approach indicates that both European and US indices make a considerable contribution to each other's systemic risk, with significant input from Frankfurt to the French and US markets, especially following the collapse of Lehman Brothers. Second, we show that the propagation of systemic risk is higher during the crisis period and overlapping trading hours than during nonoverlapping hours. Thus, the MES test is recommended as an indicator to help monitor market exposure to systemic risk and to gauge expected losses for other markets.

In this paper, we determine some sufficient conditions for an $A\,+\,XB\left[ X \right]$ domain to be an $\text{HFD}$. As a consequencewe give new examples of $\text{HFDs}$ of the type $A\,+\,XB\left[ X \right]$.