We propose a time-changed Lévy LIBOR market model to jointly price caps and swaptions. We design an additive separable model such that each component has its own distinct role in matching features from the observed implied volatility smiles in both the cap and swaption market. Our model is analytically tractable, parsimonious, and allows for fast calibration. Using the unscented Kalman filter, we perform a joint estimation and pricing analysis of caps and swaptions. Using a comprehensive data set spanning the recent financial crisis, we are able to accurately match different empirical features of cap and swaption implied volatilities across moneyness, maturity, and time. Finally, we study the link of actual and implied volatilities to the market for mortgage backed securities and to the looming sovereign debt crisis.