Hull White Tree Calibration 2

Question

This is actually to extend the question I asked previously and to follow up Bernd's answers. This is the original link: Instruments for calibrating Hull White Model

1.

As Bernd mentioned, it's generally a good idea to price a products using Curves/Models that are calibrated from quotes of the same (or similar) kind of products. My real scenario is to price hundreds and thousands of bermuda callable bonds. Does that mean ideally I have to calibrate trees for all of them based on their market prices?

2.

The main goal is to solve OASs, given it is an unknown variable, how can I calibrate the tree to derive mean reversion and volatilities? Mathematically seems to be unfeasible. As I have to optimize three variables - OAS, alpha and sigma.

3.

Another further question I have is assuming I am calibrating a quarterly tree for 30 years use swaptions. The calibration process is essentially solving forward volatilities for every path iteratively, right? What if there is no swaption from the market that exactly matches one of the paths, how can I derive that forward volatility? For example, I want to solve the volatility for the path of 29th year to 29th year and a quarter, but there is no such swaption in the market, how can I calibrate that part of the tree?

4.

Last question is if I want to calibrate a tree for 30 years, what's the ideal path number should be used? I am right now using quarterly path that is totally 120 paths.

Thanks!

Answer 1

1.

Concerning you first question of pricing callable bonds, I would suggest to use swaption volas instead of cap/floorlet volar. In my viewthe product type, i.e. swaption, is closer to that of a callable bond. Imagin a swaption with the right of termination after half of the tenor period. Such a product could be "engineered" by putting together a swap and a swaption.

Your problem of bonds wich included options seams to be related. However, I don't know if your products could be "engineered" from swaptions and linear products. If yes, you should definitly use swaption volas. If not, you have to derive a method to construct callable bonds from some other products that have quoted volatilities. Mybe this link helps: Lyuu (2002), section 27.3.1

But it has at least nothing to do with capping or flooring anything. I guess.

2.

I still don't understand what is OAS? What it unfeasable and why?

3.

The calibration process is essentially solving forward volatilities for every path iteratively, right? The calibration process is not solving anything. It works in two steps: First you generating a tree for a fixed parameterization (alpha and sigma). Then you are looking for the parameterization that best fits current market quotes.

What if there is no swaption from the market that exactly matches one of the paths, how can I derive that forward volatility? For example, I want to solve the volatility for the path of 29th year to 29th year and a quarter, but there is no such swaption in the market, how can I calibrate that part of the tree? It doesn't matter. You don't need it for spanning the tree (step one). You also don't need it for the optimization process. The objective function simply depends on the swaptions you receive from the market.

4.

I think this is basically a matter of time. Ideally you would like as many as possible. But since you need to implement numerical routines to find the best parameters, you should care about computation time.

You should have a look at this post. The author is calibrating Hull/White and other models using the QuantLib-Python libraries: http://gouthamanbalaraman.com/blog/short-interest-rate-model-calibration-quantlib.html