“The effective convexity of a callable bond can be negative in some circumstances, but the effective convexity of a putable bond is always positive”
why not always for callable bond?
Draw the price vs. yield curve for a callable bond.
When the call option is way out of the money, a callable bond acts very much like a straight bond; in particular, it has positive convexity.
IO strips from MBSs are even more interesting: at low interest rates they have negative effective duration.