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?
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2$\begingroup$ A bit of more careful reading on p.1 and the question is answered: Bergomi distinguishes explicitly between Barrier Options that allow for fairly stable hedges and newer exotics he is focusing on where that is not the case (Napoleons, reverse cliquets - whatever they are). Peter Carr has published some work about static hedging which probably explains why this is so. Can easily be googled. $\endgroup$– Kurt G.Apr 24 at 10:34
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$\begingroup$ @KurtG.It was clear to me why Napoleons need to have their vega hedge rebalanced. What was less os was why the same does not seem to hold for barrier options, or at least not to the same extent. $\endgroup$– fwd_TApr 24 at 11:34
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2$\begingroup$ This can surely not be seen from a purely theoretical perspective. Bergomi writes that vega hedge stability "is usually the case for barrier options." If I remember correctly Carr's work makes this clearer and is highly recommended. $\endgroup$– Kurt G.Apr 24 at 11:56
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1$\begingroup$ @KurtG. If I could put in a single vote to reopen this one I would but I can't but I wonder: Isn't "Why is the Vega hedge of barrier options almost immune to volatility surface dynamics?" an interesting question on itself without reference to a paper or book? $\endgroup$– Bob Jansen ♦Apr 26 at 19:09
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1$\begingroup$ @BobJansen It is and that's why I referred OP to Carr. Astonishing how fast those papers were read in the mean time. For those who want to discuss that single sentence from Bergomi I voted to reopen the question. $\endgroup$– Kurt G.Apr 27 at 4:41
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1 Answer
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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.
