I'm having a difficulty with calibrating the drift of the continuous version of the BDT to the yield-curve.

I have first calibrated a discrete constant-volatility BDT (tree) to a set of interest-rate derivatives, this gave me a reasonable value of the log volatility [$]\sigma(t)=\sigma = 0.59[$]. 

I, however, need a continuous version of the BDT because I want to use it together with a continuous default intensity process to price defaultable claims using PDE methods. In my case, the log of the short rate in BDT follows
[$]d\ln r(t) = \theta (t)dt + \sigma dW(t)[$], 
where [$]\theta(t)[$] is a deterministic function of time ensuring a perfect fit to the yield curve. What I did to find [$]\theta (t),t \in [0,{T^*}][$] is that I constructed a PDE for a bond price [$]B(t,{T^*}) = B(t,{T^*},x)[$], where [$]x = \ln r[$], which is
[$]\frac{{\partial B}}{{\partial t}} + \theta (t)\frac{{\partial B}}{{\partial x}} + \frac{1}{2}{\sigma ^2}\frac{{{\partial ^2}B}}{{\partial {x^2}}} - {e^x}B = 0[$] and given [$]\sigma[$] is known I iteratively looked for [$]\theta ({t_i})[$] (where [$]i[$] is an index of a time horizon) such that the PDE result matches the observed bond price in every time horizon [$]t_i[$]. (This is done in the same way as bootstrapping, i.e. I'm using previous values of [$]\theta(t),t<t_i[$] as fixed inputs and calculate only the value of [$]\theta(t_i)[$] that relates to [$]t_i[$] by matching the observed zero bond price, maturing at [$]t_i[$].)

I, however, get very wild values of [$]\theta(t)[$], reaching even the value 100 which I suspect is a too high number. I would be happy if someone could give me a hint about how to calibrate the [$]\theta(t)[$] appropriately so that its values don't explode.

I did some analysis now and it looks that the problem is surprisingly complicated. I did the analysis on one-factor Hull-White model where [$]\theta(t)[$] has an explicit form and thus can be compared with the numerically obtained [$]\theta(t)[$] through the PDE.

The main issue here is that [$]\theta(t)[$] ensuring a perfect fit to a bond maturing at time [$]t[$] might actually complicate the fit to bonds maturing later than [$]t[$]. Then, [$]\theta[$] tends to explode because it has to change drastically to fit the bond prices given some prior behaviour of [$]\theta[$] is fixed.

To make a good fit with reasonable behaving [$]\theta[$] the best compromise is to find such [$]\theta[$] that does not wiggle too much overall, reducing the requirement of the perfect fit to a 'reasonably good' fit to bond prices. This is, however, still tricky to be formulated technically.

Finally got it. It's tricky.

Here is the BDT-fitted vs observed discount curve:

The function [$]\theta[$] (notice that I had to limit the range of [$]\theta(t)[$] to prevent 'explosions'):

And the BDT 5Y-bond pricing surface (notice the stepwise pattern due to oscillating [$]\theta(t)[$]. Not perfect yet suffices):

The bottomline: don't allow [$]\theta[$] to oscillate arbitrarily just because you want a perfect fit to the yield curve. Rather, find a reasonable compromise between the fit quality and the oscillations in [$]\theta[$].

That looks like a nice learning experience! As an aside, I would struggle to think of a less suitable model for any real-world interest modeling purpose than a continuous time BDT model. But that may have been a choice imposed on you by others...

I'm having a difficulty with calibrating the drift of the continuous version of the BDT to the yield-curve.

[$]\frac{{\partial B}}{{\partial t}} + \theta (t)\frac{{\partial B}}{{\partial x}} + \frac{1}{2}{\sigma ^2}\frac{{{\partial ^2}B}}{{\partial {x^2}}} - {e^x}B = 0[$]

 We had a discussion in some other thread about the instantaneous explosion of the money market account with the log-normal continuous-time interest rate process. So, while the bond prices may exist, the explosion in the MM account renders the model perhaps untenable. 

p.s. here was the other thread:
viewtopic.php?f=4&t=99922

Bearish - 
You are right, the model is inappropriate for continuous time modeling. I think some explosion discussion when using the (continuous?) BDT are provided by Clewlow and Strickland [1998] but I no longer have the book so I can not check it. In my case I just needed to verify some special tree implementation. If I knew that I waste so much time with the continuous-time model, I wouldn't have done this exercise. But at least it is some new practical experience.

Alan -
Thanks for the reference! It looks that the article Weintraub and Hogan [1998] discusses "nearly" my case so it is very likely that no "nice" solutions exists for the continuous lognormal IR models...surprising. Fortunately, I don't need to use the log-normal IR models in practice as the rates are strongly negative here in the Europe  

I did a bit of fixing of the BDT model once for an MBS application. The approach was similar to C&S applied to the yield curve data (not volatility data which was constant) for the short rate. The cash flows were positive; however, a small finite range of volatility crashed Newton Raphson (after looking for needle in haystack). The seed 5% as in C&S did not work. I then used Bisection to roughly bracket the solution and then use NR. The vol was tested in range [0, 100].
For negative cash flows, the method could lead to multiple solutions (or none)..

What was thought to be simple software error ("it crashes") turned out to be a numerical analysis problem.

The model also has the annoying property that the mapping from the short rate to the full yield curve is hard to calculate. This is a first order problem in a lot of practical applications. Superficially, if you want to price a 1-year option on a 30-year bond you need a 30-year lattice. 

I entered the fray from the other end! a) a software fault(event) in Excel/dll when it crashed for a tiny percentage of volatilities > 60. One could always assume that it was some software library revision issue but back-pedaling to Newton Rapshon we have a defect in that choosing the C&S seed value for yield (5%) does not always ensure converge. Is the error due to the fact that the interest rate curve humped now? Has something changed in the fixed income last few years?

//
For project estimation (fixed price) how can we determine if we have a fault, defect or error on our hands?
With BDT it is so that using constant volatility cannot model mean reversion?

I did some analysis now and it looks that the problem is surprisingly complicated. I did the analysis on one-factor Hull-White model where [$]\theta(t)[$] has an explicit form and thus can be compared with the numerically obtained [$]\theta(t)[$] through the PDE.

The main issue here is that [$]\theta(t)[$] ensuring a perfect fit to a bond maturing at time [$]t[$] might actually complicate the fit to bonds maturing later than [$]t[$]. Then, [$]\theta[$] tends to explode because it has to change drastically to fit the bond prices given some prior behaviour of [$]\theta[$] is fixed.

To make a good fit with reasonable behaving [$]\theta[$] the best compromise is to find such [$]\theta[$] that does not wiggle too much overall, reducing the requirement of the perfect fit to a 'reasonably good' fit to bond prices. This is, however, still tricky to be formulated technically.

You diagram does not indicate that [$]\theta[$] explodes. Is it not the problem that the short rate explodes depending on drift and vol etc.?
For PDE a wiggly theta is not the end of the world IMO.

Cuchulainn:
The figures that I posted show [$]\theta[$] after I imposed some restrictions on it. If I didn't, it would explode. Without the restrictions the sequence of [$]\theta[$] in time would be e.g. like [$]\theta ({t_0}) = 0.1,\theta ({t_1}) = 0.2,\theta ({t_2}) =  - 0.3,\theta ({t_3}) = 0.8,\theta ({t_4}) =  - 2.0,....[$]. That is the sequence is oscillatory and the amplitude increases with every time step. For some time [$]t_n[$] I was getting crazy numbers.

If one instead accepts an approximative fit instead of the perfect fit, the evolution of [$]\theta[$] is much more reasonable. The problem is actually quite complex as one is looking for the answer to: how to optimally choose [$]\theta(t)[$] so that I prevent the oscillations of [$]\theta(s),s>t[$] while the fit to the discount curve is good. Honestly I didn't expect this to be this painful.