How Rate Distribution Assumptions Shape a Capped Coupon’s Volatility Exposure

Summary

The document analyzes a bond coupon that follows a reference rate within a band and is capped at either end. The questioner decomposes the payoff into a fixed lower amount, a long caplet at the lower boundary, and a short caplet at the upper boundary. Although the forward rate is equally distant from both strikes, increasing the volatility curve lowers the bond’s value in the valuation described, while decreasing volatility raises it.

The answer says the result depends on the assumed interest-rate model. Under a normal rate distribution, the two option exposures should roughly offset, leaving little implied-volatility sensitivity. Under a lognormal model, the higher-strike caplet has greater time value and volatility exposure than the lower-strike caplet, so the short position in the higher-strike option can dominate and explain the observed direction. The explanation is qualitative; the document provides no numerical valuation, model calibration, or assessment of other rate models and market conventions.

Key ideas

• A coupon bounded above and below can be decomposed into a fixed payment, a long lower-strike caplet, and a short upper-strike caplet.
• Equal distances between the forward rate and the two strikes do not guarantee offsetting volatility exposures.
• Under a normal rate model, the two option exposures may approximately offset.
• Under a lognormal model, the higher-strike caplet can have greater time value and volatility sensitivity.
• The explanation depends on the assumed rate distribution and is not numerically demonstrated.

Tags

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


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












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 by dm63 (score 0, accepted)

https://quant.stackexchange.com/a/27699

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.