If you've looked at a pricing formula by Alex Lipton, he almost always uses square-root-of-spot as the numeraire. That symmetrizes between the original numeraire and taking the asset as numeraire. Of course he doesn't write it that way : he moves to PDE space and does changes of coordinates there. But if you translate back, that is the equivalent. Oh, and the payoff S^2 conditional on S>K is not far-fetched. In FX payoffs essentially like this trade as a self-quantoed vanilla, also called an "inverse" payoff.
Last edited by travisf
on March 1st, 2016, 11:00 pm, edited 1 time in total.