I have a few basic question about calibrating current term structure in the Hull white model to cap/swaption. What I have a an input to the model is bootstrapped blended curve, with discount factors for a set of grid dates. Please note that the curve has been bootstrapped using log linear df method. But for fitting the term structure, I need to get instantaneous forward curve; for which the discount factor will have to differentiated numerically. So the question is do I need to impose some sort of differentiability condition on the discount curve? Can anyone provide insights on fitting the current term structure part of the problem, please.

try a Google search with "grant vora hull white" shows how to fit a trinomial tree to the current term structure

If you use e.g. the curve interpolation with cubic splines (which in principle connects your points with cubic polynomials so the entire term structure is continuous and smooth at the knots which are fixed), then you can derive the derivative of the discount factor at any point.

Thanks for the responses! Darry, yes cubic interpolation will ensure smoothness at the knots. But now I am thinking, If one uses, log linear Df method along with forward differentiation, effect of non smoothness can be done away with, is it a valid conclusion or I need to take care of something else? Also with Brigo method of separating r(t) into deterministic alpha(t) and stochastic x(t), and alpha(t) reflecting term structure at at time 0, f(0,t) would not be needed to be differentiated as it seems from the expression of theta. I will try to get some numerical example of calibration working and then discuss the results. Any comments/ suggestions?

I haven't applied a df method - I work mostly with Monte Carlo.In the context of Monte Carlo I never calculate [$]\theta[$] - just directly [$]\alpha[$], and I need derivatives to compute f(0,t). Calculating first [$]\theta[$] and the deriving f(0,t) from it is according to me at least double work and is much less stable.

A simpler approach is to consider the zero coupons. In the HJM framework, the ZC are automatically calibrated to the current curve thanks to the ratio of ZC at the initial time that occur in the formula.Then you will diffuse the ZC in your numerical method (MC or finite differences). Indeed in practice, it is better to avoid working with short rates.