Impact of the interest rate volatility in the valuation of a bond

Question

I am currently valuating a bond whose cupons have the following structure:

$\left\{ \begin{array}{rcl} H_j-2\% & \mbox{if} & R_j<H_j-2\% \\ R_j & \mbox{if} & H_j-2\%\leq R_j\leq H_j+2\% \\ H_j+2\% & \mbox{if} & R_j>H_j+2\% \end{array}\right\}$

where $R_j$ is the rate for a given period and $H_j$ is the forward rate today for the same period.

I have valuated it destructuring each cupon into a fix payment of $H_j-2\%$, a long position on a caplet with strike $H_j-2\%$ and a short position in a caplet with strike $H_j+2\%$. Now I am shifting the volatility curve upwards and downwards to analyze its impact on the value of the bond. When I move the volatility upwards the value of the bond decreases and when I shift it downwards the value increases. I have spent some time thinking about the cause of this result but I cannot realize why is it. I have tried reasoning based on the vega of each caplet but since the forward rate is always equally distant from the strike of both caplets I see it as a dead end path.Could somebody give me a hint?

Answer 1

It depends what type of interest rate model you are using. If rates are normally distributed, the situation should be as you describe, so there should be minimal exposure to implied volatility. If rates are lognormally distributed, the higher strike option has greater time value, and has a greater volatility exposure, than the lower strike option, hence the behavior you see.