I noticed that I had the button to award a bounty even on my own answer, so I couldn't resist and awarded the bounty to myself, just to see what happens. The result is that the bounty is awarded, but I did not get any reputation for myself.
It makes sense that you don't get reputation for your own bounty, but I think the option to award it should not be there in the first place.
The option also appeared for answers posted before the bounty was started, I thought a bounty could only be awarded for new answers?