This paper extends the studies published to date by performing an analysis of the causal relationships between sovereign CDS spreads and the estimated conditional volatility of stock indices. This estimation is performed using a vector autoregressive model (VAR) and dynamically applying the Granger causality test. The conditional volatility of the stock market has been obtained through various univariate GARCH models. This methodology allows us to study the information transmissions, both unidirectional and bidirectional, that occur between CDS spreads and stock volatility between 2004 and 2020. We conclude that CDS spread returns cause (in the Granger sense) conditional stock volatility, mainly in Europe and during the sovereign debt crisis. This transmission dynamic breaks down during the COVID-19 period, where there are high bidirectional relationships between the two markets.