I need some help for the parametrization of the volatility parameter in the Hull-White model.
I have the necessary Caplet vols and I calibrated the HW model to match the Caplet and hence the Cap prices exactly.
But I calibrated the volatility as one factor for every Caplet. I didn't parametrize the volatility as a function of time for each T.
That could be the reason why my volatility is always higher than the volatility of the previous Caplet. I do not get a proper volatility term structure that is consistent with the market volatility structure, though my prices fit exactly.
I posted already something here: Cap price as bond options
The topic was a different (more fundamental) one, but the answer I am refering now to is that:
Hull-White calibration on cap volatilities The first step is to strip caps vol to get caplet vols. See for example: http://www.smileofthales.com/financial/cap-floor-pricing-stripping-the-basics/
Let's suppose you want to calibration on caplets with expiries T1
You start with the option with the nearest expiry T1, then determine the volatility σ(T1) that enables you to match the T1 caplets price.
Then, you move on to T2, the caplet price is a function of σ(T1) that is already known and σ(T2), so you determine the value of σ(T2) enabling you to match the T2 and so on, until you get to Tn, and you are done.
Now I am kind of confused.
1) How do I set up the parametrized volatility function and what do I calibrate for? For the time parameter or the other two parameters?
3) What are the corner points in time? I want to calibrate to each Caplet. Are my T the points in time for the Caplets? Starting with 0.5 for the first Caplet (6month tenor) and then going on to 1 - 1.5 - 2.0 - 2.5 and so on until every Caplet is calibrated?
I am really confused and would appreciate If some could serve me a fundamental great answer about the parametrization.
I hope you can help me.
Thanks in advance