ogoluwa

Your understanding is not quite correct... generally speaking a rise in volatility increase the value of an option, both for a call and a put.

--> for a callable bond the buyer of the bond is short an option, so with an increase in vol the value of the bond decreases
--> for a puttable bond the buyer of the bond is long the option, so with an increase in vol the value of the bond will increase

A rise in interest rates will increase the value of a call and decrease the value of a put... but the question of how it applies to the value of a callable/puttable is tricky because not only the option component of the bond changes, but also the bond component.