I was re-reading Lorenzo Bergomi's paper Smile Dynamics I. On the first page, he makes the point that it is necessary for a model to match the vanilla smile observed in markets in order to incorporate correctly the price of a Vega hedge (e.g. for exotics). In the second paragraph of his introduction, he seems to imply that the Vega hedge is usually stable for barrier options. My question is: why is that so? Why is the Vega hedge of barrier options almost immune to volatility surface dynamics? Doesn't Vega hedging of barrier options require dynamic rebalancing of the Vega hedge in time dimension?
Too long for a comment. I find Bergomi's sentence vague, so here follows an equally imprecise attempt at an answer.
A claim that can be statically replicated in a model-free manner is in fact immune to volatility dynamics, agree? You could even argue a vega hedge is superfluous as the static hedge already hedges all risks.
For symmetric smiles in the absence of jumps a simple barrier option (eg KO or KI) can be statically hedged, with perhaps just one rebalancing instant. So from this perspective the (vega) hedge is stable as there is no vega hedge.
Smiles are rarely symmetric in practice, but assuming spot/vol correlation does not vary too much then the skew vega hedge (I like to decompose vega into level vega + skew vega + convexity vega) can be roughly implemented as an additional risk reversal in addition to the (semi-) static position in order to make up for the deviation from symmetry when the barrier is hit. I still wouldn't call this `stable', maybe manageable is a better word. The more the skew varies the more frequent this (skew vega) hedge would need to be rebalanced.
I think everything I've written has been alluded to in the other comments.