We introduce generalisations of the COGARCH model of Klüppelberg et al. (J. Appl. Probab. 41:601–622 2004) and of the volatility and price model of Barndorff-Nielsen and Shephard (J. R. Stat. Soc., Ser. B Stat. Methodol. 63:167–241 2001) to a Markov-switching environment. These generalisations incorporate exogenous jumps of the volatility at the times of a regime switch. Both models are studied within the framework of Markov-modulated generalised Ornstein–Uhlenbeck processes which allows deriving conditions for stationarity, formulas for moments as well as the autocovariance structure of volatility and price process. It turns out that both models inherit various properties of the original models and therefore are able to capture basic stylised facts of financial time series such as uncorrelated log-returns, correlated squared log-returns and non-existence of higher moments in the COGARCH case.