Terminology Question (Option embedded bonds)

As we know the value of an option embedded in a call is = value of a stright bond - value of a callable bond.

But isn’t the value of the option accounted for in a callable bond value? I get thrown off when the question metions a callable bond and a call option in a bond. Another way of asking is what do you call a callable bond with a call option provision? Is there an easy way to understand the relationship?

Thanks!

Yes, the value of the option is included in the callable bond value. That’s the reason that the value of the call option (by itself) is the difference between the value of the callable bond and the value of the straight bond: the first one has the value of the call option, while the second one does not.

What do you call the tall animal with a tall neck? A giraffe. Now you don’t have to say tall twice, you can just ask what do you call the animal with a tall neck. Similarly, what do you call a bond with call option provisions? A callable bond.