# Callable bond price sensitivity to Hull-White volatility changes

I'm using classic Hull-White model for short term interest rate dynamic: $$dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$$ (Notation is quite intuitive, anyway I am using the same as Wikipedia whether any doubt should occur).

I would like to have some feedback from users about the price sensitivity to constant volatility of a callable fixed rate bond priced under such model.

Let I have a fantasy bond with following features:

• termination date $=$ 1,825 days starting from today
• coupon payment tenor $=$ annual
• face amount $= 100$
• redemption $= 100\%$
• coupon $= 3\%$ of face amount

That above is a callable bond whose call schedule says it can be called at $100$ every year at the same date of maturity, hence we have $4$ scheduled call dates (e.g. if maturity were 28-Sep-2020, we would have following call dates: 28-Sep-2016, 28-Sep-2017, 28-Sep-2018, 28-Sep-2019).

Hull-White model parameters:

• $\alpha=0.03$
• $\sigma=132\%$

For the sake of this example, term structure employed in Hull-White model is a snapshot of (linearly interpolated) current euro swap curve - which I attach here below:

LAST_PRICE  Tenor
-0,135%     0D
-0,145%     1W
-0,113%     1M
-0,040%     3M
0,029%      6M
0,012%      11M
0,025%      1Y
0,029%      18M
0,051%      2Y
0,122%      3Y
0,230%      4Y
0,354%      5Y
0,484%      6Y
0,619%      7Y
0,748%      8Y
0,860%      9Y
0,963%      10Y
1,058%      11Y
1,142%      12Y
1,331%      15Y
1,481%      20Y
1,516%      25Y
1,520%      30Y
1,523%      35Y
1,522%      40Y
1,498%      45Y
1,475%      50Y


No credit risk!

Regardless of days conventions, day counters and trinomial tree engine grid size - which should account just for small variations in final results - my questions are:

1. what is the clean price Hull-White model is supposed to return with such parameters (mean reversion, volatility and term structure)?
2. Is it possible that halving volatility has a huge impact on clean price - about 15 ~ 20 points?
3. If volatility drops from $132\%$ to, say, $7\%$ does clean price go from very small default-like values (below $10\%$) to "normal" values (around $100\%$)?
4. If (2) and (3) were correct, how would you qualitatively justify such huge volatility sensitivity of the clean price?