Peter Carr’s Hall of Mirrors

Put-call symmetry may have been the start but, for Peter Carr, the importance of invariance extends into surprising realms of possibility. Dan Tudball traces Carr’s kaleidoscopic journey …

It’s the mid-1990s, the Wall Street of the Masters of the Universe is partying like it’s 1999, LTCM and the dotcom bubble are yet to spoil the good times. Peter Carr and Dilip Madan have developed a robust hedge for variance swaps using vanilla options such that static positions in the options combine with dynamic trading in the underlying. The amazing thing is that the hedge works in a big class of models.

Carr, then at Morgan Stanley, goes to a guy in marketing and says: “Wow! We should be selling these variance swaps; they’re simple contracts, people should want them, and they can be hedged in a big class of models!” The marketing guy says: “No one’s interested in variance. If you can create a volatility swap, that’s what we can sell.” And probably turns his attention back to his mid-morning glass of Petrus.

To a marketer, the difference between a variance and a volatility is just a detail in a contract. Carr, however, could not see past the square root which had just punched him in the face. The square root you have to take to go from a variance to a volatility, to someone who is trying to hedge in a model-free way, is a huge problem because it’s a nonlinear function. “Trust me,” Carr says, “It’s not a minor detail.”

But Carr’s overwhelming sense of diligence won’t allow him to drop it and he asks himself: “Is there an analog to this variance swap, where you can take the square root or more generally any nonlinear function? If you do a linear function, it’s obvious that it would work, but with a nonlinear function
could you still do it?”

Carr “struggled and struggled and struggled,” but couldn’t make it work robustly: “You could do it all in
theory but not in the absence of strong assumptions, let’s say.” The struggle would result in the piece of work of which Carr is most proud – even though it has never been published.

With the help of Roger Lee, the realization came that you could do it, if you were willing to make one more assumption compared to the standard theory. That one additional assumption is called ‘uncorrelatedness,’ a technical assumption. “It’s an assumption that feels wrong in many contexts, and
that’s what held me back from publishing it,” Carr says, but he “… was happy to make this progress because at least it was only one more assumption and not a lot more.”

Five years after the comment that sent him off on his quest, Carr then bumps into a different marketing
type and says: “Hey! I can do a vol swap now instead of a variance swap, semi-robustly!” “No one’s interested in vol swaps now,” the marketer replies.”Everybody’s doing variance swaps.” An appreciation of symmetry, in all its forms, is essential to understanding Peter Carr.

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Peter Carr's Hall of Mirrors