One-Sided Duration

An option is a rather one-sided thing.

It moves in the money, things happen. It doesn't, nothing happens. This is where a natural price limit comes from on callable and putable bonds. If a bond is callable at 101, and the price is 100.58, then a shift up or down in the yield curve isn't really the right thing to discuss. One is going to have a lot more impact on the price of the bond than the other. Which direction in the yield curve will have this larger impact?

Of course.

No. An upward shift in the yield curve will cause a larger price change to this callable bond.

The callable bond price hits a ceiling at 101, so it's not going much further no matter how far yields plunge. But if the yield curve rises, the bond will fall in value, eventually catching up to that of a straight bond.

This leads to the need for __one-sided durations__. It's fine to calculate effective durations for callable and putable bonds, but they are too high in one direction and too low in another to really give a good price change approximation when price is close to the exercise price. So there's the one-sided up duration for yield curve increases and the one-sided down duration for yield curve decreases. In this example, which duration measure do you think is smaller?

No. The one-sided down duration will be smaller.

You got it!

Since the callable bond is near its ceiling, the "down" shift in the yield curve would lead to a small price change, meaning a small one-sided duration. The one-sided up duration would be larger, with effective duration somewhere in the middle.

To summarize: [[summary]]

Finally, apply this same idea to a putable bond. Suppose you had an effective duration measure of 5 for a putable bond. How can you order the relevant duration measures?

Exactly! The one-sided up duration would be smaller than 5 since the bond's price is bounded by the exercise price from below. This means that the one-sided down duration must be greater than 5.

No. The effective duration would be in the middle here, too.

No. The one-sided up duration would have to be the smallest of the three.

Up

Down

The one-sided up duration

The one-sided down duration

One-sided up < one-sided down < effective

One-sided up < effective < one-sided down

One-sided up > effective > one-sided down

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