# General assumptions for var swap replication

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?

• 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. – ilovevolatility Apr 28 at 8:30