Motivated by the lack of research on price efficiency dynamics of green bonds and the impact of the COVID-19 on the pricing of fixed-income securities, this study investigates the comparative efficiency of green and conventional bond markets pre- and during the COVID-19 pandemic applying asymmetric multifractal analysis. Specifically, the multifractal scaling behaviour is examined separately during upward and downward trends in bond markets using the asymmetric multifractal detrended fluctuation analysis (A-MF-DFA) approach. The empirical findings confirm the presence of asymmetric multifractality in the green and traditional bond markets. Not surprisingly, inefficiency in both bond markets significantly escalated during the COVID-19 outbreak. Furthermore, our results indicate a higher level of efficiency of the conventional bond market over the full sample period. However, the green bond market is more efficient during a black swan event, such as the COVID-19 global pandemic, showing the potential of green bonds to become an effective diversifier for investors in traditional assets in times of extreme market turmoil. The results of the study can have important implications for investors and policymakers.