This paper studies the stock market return's volatility in the Eurozone as an input for evaluating the market risk. Stock market returns are endogenously determined by long-term interest rate changes and so is the return's conditional variance. The conditional variance is the time-dependent variance of the underlying variable. In other words, it is the variance of the returns measured at each moment t, so it changes through time depending on the specific market structure at each time observation. Thus, a multivariate EGARCH model is proposed to capture the complex nature of this network. By network, in this context, we mean the chain of stock exchanges that co-move and interact in such a way that a shock in one of them propagates up to the other ones (contagion). Previous studies provide evidence that the Eurozone stock exchanges are deeply integrated. The results indicate that asymmetry and leverage effects exist along with fat tails and endogeneity. In-sample and out-of-sample forecasting tests provide clear evidence that the multivariate EGARCH model performs better than the univariate counterpart to predict the behavior of returns both before and after the 2008 crisis.