# 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

• could you elaborate on what you mean by "make more sense" ? – Probilitator Feb 18 '14 at 10:19

## 1 Answer

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 $\sigma > (\sigma^{obs})^2$.

• Thanks. Finally, I have also concluded that it was a typo. But I have one more question. Is it possible to measure if the calibration was significant ? – Mstudent1 Mar 13 '14 at 2:49
• what exactly do you mean by significant ? a criteria to decide whether the calibration is any good ? – Probilitator Mar 13 '14 at 15:39
• Yes. Actually, the most natural way to do this is to compare the observed volatility and Hull-White implied volatility in a volatility-maturity space. But is there any rule which would allow to conclude that the calibration is successful (or significant), i.e. the difference between volatilities is small enough? – Mstudent1 Mar 14 '14 at 9:01
• it depends on what you are pricing. In most cases a calibration is considered to work well if it fits the most relevant vols or prices best. If you were pricing a cap with short maturity you would want to hit the short term implied vols best. – Probilitator Mar 14 '14 at 9:13