I am thinking about implementing Hull-White model. I heard that Nelson-Siegel can be used to construct a twice differentiable yield curve and it also has some nice properties. I am planning to use Monte Carlo to simulate a process that involves short rate. I notice that there are two exponential terms in the second derivative of short rate of a Nelson-Siegel function. If I understand correctly, this second derivative should be a time-dependent function that shows up in the drift term. Will this make my Monte Carlo too slow in practice? If so, is there anyway to speed it up?Or maybe using a polynomial curve is better with Monte Carlo?

as far as i know, NS describes a set of forward rates at a given time, it suggests a family of forward rates.. it is not a model for evolution of forward rates like HW.

I was thinking the theta(t) (the time-varying function in the drift term) in Hull-White is to be expressed in terms of the instantaneous forward rate function and its derivative with respect to t. I thought I need a smooth yield curve for these two functions and this smooth yield curve can be the one a Nelson-Siegel curve. Am I missing something here?

Yes, this approach is often used. Actually, what will make you MC slow is the fact that you use small time steps in the discretization of the stochastic process.However, since the distribution of the short rate is known, it is also possible to use timesteps of 1 year, which will make your monte carlo much faster. Also, you don't need the Nelson & Siegel curve in this case. The discounting can be handled by simulating the integral of the short rate, of which the distribution is also known. The formulas that are necessary for this are in Brigo & Mercurio.

Thanks for your reply. I finally get a hand on the Brigo & Mercurio's 2006 edition.In pp. 115, it says there are two ways to do MC:1. Sample at each time step the exact transition density from r(s_i) to r(s_i+1), i=0...q-1, under the T-forward-adjusted measure;2. Discretize the SDE for r, under the T-forward-adjusted measure, via the Euler or the Milstein scheme.I think my approach is 2 and your approach is 1?

In general, yes. But for the Hull-White model specifically you don't need to simulate in the T-forward measure, you can also use the risk neutral measure. This is possible because you can simulate the integral of the short rate, needed for the discount factors.I suggest that you look at chapter 4, the 2-factor model, where they describe more details than in the paragraph about the 1F Hull White.

Thanks for your reply. I will give that a try. BTW, the B&M book is really a good read for a practioner. Thanks for your recommendation.