# Pricing Variance Swaps on Time-Changed Markov Processes

## TL;DR

This paper establishes a theoretical link between variance swap rates and European option prices for exponential Markov processes time-changed by stochastic clocks, extending previous Levy process results.

## Contribution

It generalizes the pricing relationship to arbitrary time-changed Markov processes with explicit solutions in certain cases, broadening the scope of variance swap valuation.

## Key findings

- Variance swap rate equals European contract price under specified conditions.
- Explicit solutions for the pricing function G in certain Markov process examples.
- Generalization beyond Levy processes to more complex Markov models.

## Abstract

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.