No, the system won't allow you to award a bounty to your own answer.
If you don't manually choose an answer to award your bounty to within 24 hours of it ending, it will be automatically awarded. It goes to the accepted answer, or, if there is none, or it is your own, then half the bounty will be awarded to the highest scoring answer meeting the following conditions:
The answer must have been given after the bounty was started
The answer must have a score of at least +2
The answer must not have been written by the bounty starter
The definitive source: http://meta.stackexchange.com/a/16067/233816