  We prove that the variance swap rate (fair strike) equals the price of a
co-terminal European-style contract when the underlying is an exponential
Markov process, time-changed by an arbitrary continuous stochastic clock, which
has arbitrary correlation with the driving Markov process, provided that the
payoff function $G$ of the European contract satisfies an ordinary
integro-differential equation, which depends only on the dynamics of the Markov
process, not on the clock. We present examples of Markov processes where the
function $G$ that prices the variance swap can be computed explicitly. In
general, the solutions $G$ are not contained in the logarithmic family
previously obtained in the special case where the Markov process is a L\'evy
process.
