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" ? Feb 18 '14 at 10:19

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