Skip to content
All library documents

Building Cap Volatility from SABR Caplet Volatilities

Article Quant Q&A · Author: Hasek
Read the original

Summary

The document addresses how to derive an implied volatility for an interest rate cap after calibrating SABR to caplet quotes. Caplets are options on forward rates, and separate strike and maturity quotes can be used to fit SABR parameters and produce caplet implied volatilities. A cap, however, consists of multiple caplets, so its implied volatility cannot be obtained by simply treating the cap as one option on a forward rate or by taking a straightforward average of caplet volatilities.

The recommended procedure is to price each constituent caplet using its implied volatility, sum those prices to obtain the cap price, and then invert the cap pricing model to find the cap’s implied volatility. The answer acknowledges that a cap volatility behaves like a weighted combination of caplet volatilities, but the weights depend on strike and maturity in a complex way. It offers no direct closed-form averaging shortcut and does not specify calibration details, pricing conventions, or model assumptions for the inversion.

Key ideas

  • SABR can be calibrated to caplet quotes across strikes and maturities to generate caplet implied volatilities.
  • A cap is a series of caplets, so its implied volatility is not directly the volatility of a single forward-rate option.
  • Price the individual caplets and add their prices to obtain the cap price.
  • Invert the cap pricing model to recover the cap implied volatility from that total price.
  • The effective weighting of caplet volatilities depends on both strike and maturity.

Tags

Fixed incomeOptionsVolatilityDerivatives pricing
Full text
# How to build a volatility surface for caps from the SABR model?


# 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.

## Answer by dm63 (score 2, accepted)

https://quant.stackexchange.com/a/66643

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.