Are spot starting exotics like callables sensitive to forward skew ( skew dynamic) the same way a forward starting option like a cliquet is sensitive to the forward skew ?
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Yes but more so to forward vol than forward skew. With cliquet you are averaging the BS price at different possibilities of the strike. So the convexity matters (i.e. what is the BS price at a higher strike w.r.t a lower strike).
With callables the forward vol determines exercise. If you move around skew a bit, the exercise boundary is still comparing the immediate gain with the option cost, rather than one option cost with the other. So you can create skew but ultimately you're not comparing the high strike call with low strike call, so skew is less important.
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$\begingroup$ I am’interested in the whole surface not particularly the smile. Can you pls develop on how we are averaging all vols for a cliquet ? And from a replication pov how does unwinding a called payoff fit with this pls ? $\endgroup$– bigInnerCommented Jun 26 at 9:23
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$\begingroup$ If spot hits K when strike is determined, then you have BS price with strike K in the future. The current price is average of all these prices at different K. So think of the payoff being a BS Call instead. Pr(strike=k)*BS(k) summed up. $\endgroup$– ArshdeepCommented Jun 26 at 14:10