  Recently Carr and Wu (2004, 2005) and also Huang and Wu (2004) show that most
stochastic processes used in traditional option pricing models can be cast as
special cases of time-changed L\'evy processes. In particular these are models
which can be tailored to exhibit correlated jumps in both the log price of
assets and the instantaneous volatility. Naturally similar to a recent work of
Barndorff-Nielsen and Shephard (2001a, b), such models may be used in a
likelihood based framework. These likelihoods are based on the unobserved
integrated volatility, rather than the instantaneous volatility. James (2005)
establishes general results for the likelihood and estimation of a large class
of such models which include possible leverage effects. In this note we show
that exact expressions for likelihood models based on generalizations of Carr
and Wu (2005) and Huang and Wu (2005), follow essentially from the arguments in
Theorem 5.1 in James (2005) with some slight modification. This serves to
formally verify a claim made by James (2005).
