We develop a novel class of time-changed Levy models, which are tractable and readily applicable, capture the leverage effect, and exhibit pure jump processes with finite or infinite activity. Our models feature four nested processes reflecting market, volatility and jump risks, and observation error of time changes. To operationalize the models, we use volume-based proxies of the unobservable time changes. To estimate risk premia, we derive the change of measure analytically. An extensive time series and option pricing analysis of sixteen time-changed Levy models shows that infinite activity processes carry significant jump risk premia, and largely outperform many finite activity processes.