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Calibrating a One-Factor Hull–White Model to Caps or Swaptions

Article Quant Q&A · Author: Hui
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Summary

This exchange discusses choosing market instruments to calibrate a one-factor Hull–White interest-rate model, including a trinomial-tree implementation. It recommends calibrating to either cap and floor volatilities or swaption volatilities, with the choice guided by which product family the model will price. The two sets should not be fitted simultaneously, and plain Hull–White calibration should use at-the-money quotes because the model does not capture volatility smile effects.

For a model intended to represent rates broadly, the answer suggests using available volatility quotes across maturities up to the tree’s horizon; a particular market segment can receive more weight if that is the main pricing need. A calibrated model can technically be applied to products such as callable bonds, but the resulting prices may not match market levels. The exchange does not specify an optimal number or grid of expiries and tenors, nor does it provide a detailed tree-building procedure, so those design choices remain product- and market-dependent.

Key ideas

  • Choose cap and floor quotes or swaption quotes according to the products the model is meant to price.
  • Do not try to fit both instrument families at once.
  • Use at-the-money volatility quotes for the plain one-factor Hull–White model, which does not represent smile effects.
  • Use a broad range of maturities when broad curve coverage is desired, while prioritizing the relevant market area for specialized use.
  • A model calibrated on one product set may price other products, but its prices may not align with their market values.

Tags

Fixed incomeDerivatives pricingVolatilityOptions
Full text
# Instruments for calibrating Hull White Model


# Instruments for calibrating Hull White Model












I have a few questions regarding hull white calibration, specificly for the trinomial tree model.

1.I am wondering what are the ideal instruments could be used for hull white model calibration? Cap, Floor or Swaptions? Also for swaptions, what strikes should be used, ATM?

2.Also, if I want to calibrate a 30 year hull white trinomial tree, ideally, how many and what expiries and tenors should be used to construct an optimal tree? As granular as possible?

3.Lastly, can I use the tree calibrated from the above instruments to price other instruments (or solve OAS), for example, bermuda callable bonds. Or, I have to calibrate a specific tree for every instrument I want to price?

## Answer by Bernd (score 4, accepted)

https://quant.stackexchange.com/a/40313

I assume you are asking for the popular Hull/White one-factor model.

You could eiter calibrate them to Cap/Floor Volas or to swaption volas. Don't try to fit a model to both at the same time. You should decide this by the products you want to price. If you want to price caps/floors with the model, calibrate it to cap/floor volas and vice versa.

Calibrate it to ATM. There are models that could handle the smile effect (volas away from ATM). But the plain vanilla Hull/White can not. If you are interested in the modeling of smile effects, you should read about the SABR model maybe.

The answer might depend on the products you want to price. If you want the model to fit one area of the market better than others, you could think about mainly calibrating it to those. However, in general situations, you should take all volas that you have up to 30 years maturity.

You could (theoretically) use such a calibrated model to price other products. But it is not a good idea. Your prices would not be in line with 'the market'

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