Bootstrapping Caplet Volatilities from Cap Prices

Summary

The document outlines how to infer caplet implied volatilities from a sequence of cap prices and swap rates, as a step toward calibrating a Hull–White interest rate model for swaption pricing. It assumes caps are at the money and explains that the inputs should ideally include caplet prices at the relevant strikes.

The proposed procedure starts with the shortest maturity cap, treating it as a single caplet and using Black’s formula to infer its implied volatility. For each longer cap, previously calibrated caplets are repriced and subtracted from the total cap price; the residual is used to infer the next caplet’s volatility. The sequence is repeated across maturities. The explanation is a sketch rather than a complete calibration recipe: it does not detail conventions, market data requirements, or how the resulting volatilities map into a Hull–White parameter fit. Between observed terms, it suggests piecewise constant volatility or interpolation.

Key ideas

• Cap prices can be decomposed into caplet prices across maturities.
• Black’s formula can translate a caplet price into an implied volatility.
• Bootstrapping uses earlier caplet estimates to isolate later caplet prices.
• Volatility between quoted maturities requires an assumption such as constancy or interpolation.
• The outlined process is an implied volatility extraction step, not a full Hull–White calibration specification.

Tags

# How to calibrate the Hull-White model using cap prices?


# How to calibrate the Hull-White model using cap prices?












I'm given cap prices and swap rates, and i'm trying to calibrate the Hull-White model to them. I then want to use the model in order to price a swaption.

I know that the model can be calibrated from implied volatilities but, how do you do so with prices?

Is there a way to find the volatilities from the cap prices?

## Answer by compilation-error (score 2)

https://quant.stackexchange.com/a/15523

Given that you have swap rates and Cap prices (ATM, I assume), you can back out the IVs for the time periods using by bootstrapping. Strictly speaking, you would need Caplet prices for the given strikes.

In such a case,

- You would look at the shortest dated cap and (assume) it is made up of only one caplet.

- You can then use black's formula and back out the IV for this price.

- Once this is fixed, we move on to the next cap. This cap is again broken down in to two caplets (say), where the IV for the first caplet is as calculated in step 2 and the caplet is re-priced for the new strike.

- Now that you have the Cap price, the first caplet price, you can back out the price of the remaining caplet and use that to back out the IV for the second period.

- This method is repeated till the longest dated cap.

for in between terms, you can assume the IV to be constant (piece-wise constant volatility) or some form of interpolation.

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.