volatility risk on putable /callable bond

why for a putable bond, volatility risk is the risk that expected yield volatility will decrease. and …call, …increase ?? isn’t any volitility of yield will add value to the option no matter it’s call or put?

Yes, higher volatility adds value to the option. If you hold a putable bond, an increase in the volatility would allow you a greater opportunity to put the bond to the issuer, so the price of the bond would rise. However, if you are a holder of a callable bond, an increase in volatility would decrease the total price of the bond because there is a greater chance the bond will be called by the issuer.

what he said

Just remember this formula: P0-C P0+P P0 is the price of an option free bond. If volatility increases, the price of the option increases, increasing the price of a putable bond but decreasing the price of a callable bond.