If we have a yield curve and also a swaption matrix, how would one go about calibrating the one factor Hull White model? I see that Bloomberg keeps the mean reversion parameter constant but let volatility be time-dependent. And that they calibrate "Diagonal to moneyness", which I assume means "to swaptions of the form XyXy", ie to 1y1y, 2y2y, 3y3y swaptions...My question is, if the volatility of the short rate is taken to be time-dependent, how would you break time up? By distinct forward rates, ie every quarter or six months? Or by maturity of the calibrating instruments?Thanks

by expiry times of the calibrating swaptions.

expiry dates of the calibrating instruments

Thanks for your reply. I am wondering if I may follow up with a few more questions...How would be go about approxiamting f(0,t)? ie the instantaneous forwards. Do we simply take the central difference of the discoutn factors? ie take (D(0,t+eps)-D(0,t+eps))/(2*eps) as a proxy? Or do we take the daily continuous forwards? ie take f(0,t)=-ln(D(0,t+ 1 day)/D(0,t))/eps where eps is the time length of one day, if it is taken this way, then f(0,t) would be piece-wise flat where each piece correspond to a day.As for the volatility parameter, do people normally take it as piecewise flat or piecewise linear?Thanks again for your help on this...

my 2 cents1. Brigo and Mercurio book?2. Here is some comments on this issuehttp://math.ut.ee/~spartak/papers/hwtree.pdf3. From a numerical analysis and accuracy viewpoint, interpolation is an issue (e.g. cubic spline bad for sparse data). See the article by Hagan and the late Graeme West. http://finmod.co.za/Hagan_West_curves_AMF.pdfI found Akima, Hyman/Dougherty/Hagan-West good.