Price and Volatility Objectives in Interest Rate Model Calibration

Summary

The document compares calibration objectives for the Hull–White model and the Libor Market Model (LMM). It reports that Hull–White calibration is described as minimizing squared differences between model and market swaption prices, while LMM calibration is often framed as matching model and market volatilities. The question is why the targets differ.

The answer says the choice depends partly on which quantity is easier to calculate in a given model. Price-based and volatility-based objectives also apply different scales, so they can yield slightly different parameter estimates. It adds that weighting price residuals by quantities related to option vega can make price calibration roughly equivalent to volatility calibration. This is a brief conceptual response rather than a full derivation: it does not specify a weighting formula, discuss numerical implementation, or establish that the two approaches are equivalent in all settings.

Key ideas

• Calibration may target prices or implied volatilities depending on the model and calculation convenience.
• Price and volatility residuals use different scales and may produce different fitted parameters.
• Vega-dependent weights on price errors can make price fitting roughly resemble volatility fitting.
• The document gives no universal weighting rule or detailed comparison of model outcomes.

Tags

# Why do calibration objectives differ between LMM and Hull-White Models?


# Why do calibration objectives differ between LMM and Hull-White Models?












Hull, J., & White, A. (2001) suggest model volatilities are calculated to minimize the differences between model prices and market prices for benchmark Swaptions ("Specifically we...find the set of volatility parameters that minimizes the sum of the squares of the differences between the model prices and market prices for these options.") On the other hand, for LMM, the objective of calibration is to minimize the differences between the model volatilities and market volatilities. Could anyone explain why these objectives differ? Thank you very much! Reference: Hull, J., & White, A. (2001). The general Hull–White model and supercalibration. Financial Analysts Journal, 57(6), 34-43

## Answer by Jesper Tidblom (score 1)

https://quant.stackexchange.com/a/80768

It depends on several factors. In some models prices are easier to calculate than volatilities and vice versa. Also the objective function is a bit different due to the different quantities, resulting in slightly different results.

Calibrating prices with certain vega dependent weights multiplied by the least squares terms will be roughly equivalent to calibrating volatilities.

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.