Key advantages of modeling security returns with time-changed Lévy processes: 

Generality:

• Lévy processes can generate pretty much any return innovation distribution.
•  Applying stochastic time changes on Lévy processes randomizes the return innovation distribution over time ⇒ stochastic volatility, and higher moments.

Explicit economic mapping by modeling returns with several time-changed Lévy components (versus models with hidden state vectors):

• Each Lévy component captures shocks from each economic source.
• Time change captures the time-varying intensity of its impact.
• ⇒ makes model design more intuitive, parsimonious, and to the point.

Tractability: A model is tractable for option pricing if we have under the risk-neutral measure Q

• tractable characteristic exponent for the Lévy components.
• tractable Laplace transforms for the activity rates underlying the time change.
• ⇒ any combinations of the two generate tractable return dynamics.