I want to set a bounty on my question within one day of asking it. I did not see an option for that. Is it possible? If so, how?
You must wait 2 days before you can offer a bounty on your question. You can also edit your question to bump it to the top of the front page.
See the FAQ
If, after 2 days, you still don't have an answer you like, and you have at least 100 reputation, you can offer a bounty.
I covered this topic extensively here
I'll summarize it here.
[Mentioned in] Podcasts #26 and #34, [...] They didn't want a bounty to interfere with the 'normal' [usage]. If you're allowed to offer it right away, then it [might] create an economy where people only focus on questions with bounties.
Without bounties in the first 48 hours, we have a StackOverflow economy that is based on providing timely, quality, relevant answers to questions.
Downsides to immediate bounties
[It could lead to users who] only answer the questions if the reward is rich enough.
The goal of StackOverflow is to judge technical questions based on their technical merits, [therefore] all questions should be treated equally.
There is no urgent tag
StackOverflow is not for people to come with time-sensitive questions. Most questions get answered very quickly, but it is not a guarantee and it shouldn't be relied on as such.
The Purpose Of Bounty
Bounties were created as a method to attract attention to questions which were missed on the initial run-through.
In the words of the FAQ
What if I don't get a good answer?
If, despite your best efforts, you feel questions aren't getting good answers, you can help by offering a bounty.
[Even] the FAQ says that "despite your best efforts". hat means if you've waited a few hours with no response, you can't just buy your way out of it. You should still be making an honest attempt to help the site by improving your question.
Bounties do not improve questions
Bounties do not encourage equality
Bounties do not encourage quality
Please read the original answer for full explanation of these points.
If your idea is that people will answer it faster if it is only one day, I think it doesn't work this way.
There generally an advantage to answering a question early, you will get the votes and be at the top (You get momentum). So people probably don't think, oh, I will answer that tomorrow. By leaving it up longer, you will probably just get the best answer over time, and maybe good ones right away.
This would require the bounty system to be restructured. It isn't a simple change. People would have to make decisions and implement them.
The problem is that a bounty question has a special status. For example, I'm a 10K+ SO user, and I can't vote to close a question with a bounty. That seems fair, but there are questions out there that really shouldn't be, that we like to close. The current rule is that if the question survives for two days, it's probably OK. (Yes, this rule fails from time to time.)
So, what should happen to a question with a bounty when it's voted closed? Can the questioner still accept an existing answer? Is the rep just refunded? What happens if the question gets deleted? These are questions we'd have to answer to allow immediate bounties.
In the meantime, I'd suggest looking at your question to see why it hasn't been attracting answers.