# How to build a volatility surface for caps from the SABR model?

My end goal is to build a volatility surface for caps. It's well known that SABR model has Hagan approximation formulas for log-normal and normal implied volatilities of options, e.g. caplets, therefore given quoted market volatilities of caplets with same maturities but different strikes one can use them to calibrate model parameters and obtain SABR implied volatilities. Repeating this procedure for different maturities allows one to build a volatility surface for caplets. However I do not see how to obtain a volatility of a cap which is a series of caplets. It would be wrong to take a quoted market cap volatility as an input and expect to get a proper implied volatility as an output because a cap isn't an option on a forward rate but rather a series of such options.

Is there an easy way to find implied volatility of a series of options? Of course one can compute implied volatilities of all caplets, use them to price caplets, get a price of a cap as a sum of all caplet prices and then extract the implied volatility from the cap price, but I would like to have a way around it which wouldn't involve calculating prices.

• Before extracting volatility, how do you define the forward of a Cap ? Aug 27 at 13:00
• The volatility of the cap/floor (also called flat-volatility) is not a true volatility, it's just a quoting mechanism. Caps/floors are portfolios of caplets/floorlets and each can be exercised independently from each other, so there is no "implied volatility for a series of options". Aug 27 at 14:31

## 1 Answer

The procedure you have specified in your last paragraph is the only reasonable way to do it. Clearly the cap volatility is some sort of weighted average of the constituent caplet volatilities, but the weighting is complex , having strike dependence as well as maturity dependence.

• +1, @Hasek, you can look here to get an idea of how the cap vol corresponds to the caplet vol (in Bloomberg). The cap vol itself is the flat vol which, when used as optlet vol in cap pricing, gives the deal the same premium. Sep 16 at 20:06