  We introduce a class of randomly time-changed fast mean-reverting stochastic
volatility models and, using spectral theory and singular perturbation
techniques, we derive an approximation for the prices of European options in
this setting. Three examples of random time-changes are provided and the
implied volatility surfaces induced by these time-changes are examined as a
function of the model parameters. Three key features of our framework are that
we are able to incorporate jumps into the price process of the underlying
asset, allow for the leverage effect, and accommodate multiple factors of
volatility, which operate on different time-scales.
