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Question:

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

A One bond will increase in value and the other will decrease.
Explaination

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.