In times of macroeconomic or financial turmoil volatility typically jumps suddenly to extreme values. We develop a distribution for volatility that allows for fat tails and that generalizes the Gamma Autoregressive Process. We show that this specification allows for a simple and efficient MCMC algorithm. By conditioning on some auxiliary variables, all volatilities can be sampled jointly from the joint conditional posterior. Because of its simplicity and efficiency, the MCMC algorithm can be used to tackle long time series. In applications to real data we show that the extension to fat tails is empirically relevant.