  Carr and Wu (2004), henceforth CW, developed a framework that encompasses
almost all of the continuous-time models proposed in the option pricing
literature. Their main result hinges on the stopping time property of the time
changes, but all of the models CW proposed for the time changes do not satisfy
this assumption. In this paper, when the time changes are adapted, but not
necessarily stopping times, we provide analogous results to CW. We show that
our approach can be applied to all models in CW.
