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Putable/Callable bond volatility

Can anyone explain this to me, I cant find anything about it in the schweser books besides that an increase in the volatility of the price of the underlying asset increases  the call/put option value

What effects will an increase in yield volatility have on the values of a putable bond and a callable bond?

Both bonds will increase in value.

Both bonds will decrease in value.

C. One bond will increase in value and the other will decrease. - correct

A callable bond is made up of a straight bond and a written call option. An increase in volatility increases the value of the call option and decreases the value of the callable bond. On the other hand, a putable bond is made up of an option-free (or straight) bond and a long put option. An increase in volatility increases the value of the put option and therefore increases the value of the putable bond.

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The level of interest rates affects the value of an option-free bond, but the volatility of interest rates doesn’t.  So for a callable or putable bond, interest rate volatility will affect only the value of the embedded option: the value will increase with increased volatility and decrease with decreased volatility.

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The way I thought about this question was that a putable bond has a price floor, and a callable bond has a price ceiling, so you are at the mercy of one-sided volatility either way. If the interest rate drops massively, you get a capped upside on the callable bond because the price won’t go above the call price (so the volatility doesn’t benefit you as the bondholder). But if the interest rate increases you will experience all of the pain of the bond price going down. Likewise, if the interest rate increases massively, the price of a putable bond won’t go below the put price so you are protected against that, but would benefit if the interest rates decreased and the bond price shot up.

Hope that helps.