# Weighted average implied optionlet/swaptions volatility

Let an implied volatility curve/surface is made up by optionlets or swaptions Black's implied volatility.

If you wanted to price, say, a FRN with cap and/or floor, a CMS et cetera you would input the array/matrix filled by that IV curve/surface in you pricing model: regardless of the model, it's very likely it takes as input that curve/surface.

My questions:

• is it actually possible to find a constant volatility value which returns a fair value close enough to the one obtained via curve/surface? (*)
• If said value existed, how would it be? Could it be a weighted average of all the curve/surface implied volatilities? Or just the ATM ones?
• If said value existed and it could be expressed by a weighted average, how the weights should be? Maybe something related to swaptions options & swap tenors? Something else?

(*) When I say «close enough» I mean that I would be okay with a rough and/or heuristic proxy, too. I do not actually need any accurate value.

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Everything (capped floater, CMS) will have one "implied volatility" which recovers the price. I am not sure you can find this number by weighting pieces of the input implied volatility surface, especially since usually one hand-picks which particular caplets/swaptions to calibrate the model to in the first place. There might be some rough relationship between the implied volatility and the local volatility surface, when pricing with local volatilities, but in general, I am not aware of any such relationship. –  experquisite Apr 3 '14 at 14:13
Even some traders use weighted averages of surface volatilities as an initial stab at pricing, so you can certainly get in the ballpark that way. The weighting to use would depend heavily on the terms of the contract that needs pricing. For example if it pays off only in low-strike regions you would not necessarily weight ATM vols the highest. –  Brian B Apr 4 '14 at 12:31
As of your hint, Brian B, I was wondering how would you roughly average a curve for pricing if you had at your disposal the ATM values only and swaps OR optionlets tenors spanning from 1Y to 30Y... something more like a curve than a surface... thinking about how convexity adjustements work, maybe weighting the average more on the long term ATM volatilites could be a possibility? –  Lisa Ann Apr 4 '14 at 15:06