A note on exact likelihoods of the Carr-Wu models for leverage effects and volatility in financial economics

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).