I've seen claims that the standard static-hedge for a plain vanilla variance swap holds so long as the underlying doesn't jump, but every derivation I have seen begins by assuming the asset follows a GBM with fixed drift and volatility rates $\mu$ and $\sigma$. How general is the replication argument? Can $\sigma$ be stochastically time-varying? Can the underlying asset have any dynamics so long as it is without jumps?

  • $\begingroup$ I think you're asking if $\sigma$ can be stochastic and would the standard replication argument hold. The answer is yes, and I think even if $\sigma$ jumps but as along as $S$ doesn't you can use the replication formula. Also if $\sigma$ is a local vol the replication argument works. I don't think $S$ can have any dynamics (even if doesn't jump): for instance if the asset follows a time-changed process I am not sure anymore if the usual semi-static hedge formula holds. $\endgroup$ – ilovevolatility Apr 28 '19 at 8:30

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