Calibrating Hull–White to Volatility Data: Standard Deviation or Variance
Summary
The question concerns calibrating a Hull–White interest-rate model to observed spot-rate changes using an objective function that appears to subtract observed sample variance from theoretical conditional volatility. The accepted response argues that the paper’s expression is likely a typographical error: it should compare like quantities, such as theoretical and observed standard deviations or theoretical and observed variances.
The reasoning is based on the paper’s notation, which distinguishes variance, written as squared volatility, from standard deviation. Comparing a standard deviation with a variance mixes units and can make the objective poorly responsive to observed volatility; it also fails to behave correctly when model and observed standard deviations match. The response notes that the same expression appears in the paper’s two-factor model, suggesting the error may have been copied. This is a brief interpretation of the equations rather than a reproduced calibration or empirical test, so checking the original definitions and derivation remains useful.
Key ideas
- A calibration objective should compare theoretical and observed quantities expressed on the same scale.
- The response interprets the standard-deviation-minus-variance expression as a likely typo.
- The paper’s notation reportedly uses squared sigma for variance and sigma for standard deviation.
- Mixing variance and standard deviation can distort the objective’s sensitivity to observed volatility.
- The repeated expression in the two-factor model may reflect a copied error.
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Full text
# Calibrating Hull-White using volatility data
# Calibrating Hull-White using volatility data
I would like to calibrate Hull-White model using volatility data.I am using [Park (2004)] paper as a reference.
He suggests to minimize the following objective function:
where the first term is theoretical (H-W) conditional volatility [st. dev.] of changes of the spot rates and the second term is defined as:
which is sample variance of observed market data.
My question is:
- why do we subtract variance from volatility[standard deviation] in the objective function? (i.e. not variance - variance).
NOTE: Initially, I thought this was a mistake, but the same expression is used for the two factor model as well (formula (158) in the paper). In addition, I tried to calibrate the model using both (standard deviation - variance) and (standard deviation - standard-deviation) approaches. It seems like the results from (standard deviation-variance) case, as in Park(2004), make more sense.
Thank You
## Answer by Probilitator (score 3, accepted)
https://quant.stackexchange.com/a/10296
I took a look at the paper and would contend that it is a typo. I would assume he just copy-pasted the equation - for it is exactly the same for the two factor model cf. eq (157) and eq (41)
If you follow his reasoning and his notation it would make no sense to use the observed sample variance. He always denotes the variace by $\sigma^2$ and the standard-deviation by $\sigma$ or $\sigma(t)$
Also, it would make no sense to compare standard deviation to variance - your objective function would be not very sensitive to changes in the observed variances. For variances being the squared number of a $\sigma <1$ will always be much smaller. Also you objective function would not evaluate the case $\sigma = \sigma^{obs}$ properly, with $\sigma > (\sigma^{obs})^2$.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.