We prove that the variance swap rate  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 . The payoff function G of the European contract that prices the variance swap 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 G 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.
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.
A
(fair strike)
R
. The
, provided that the
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that prices the variance swap
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G
the function G that prices the variance swap