Explanation:

- Despair means a complete loss of hope.

- Option 1: Definite - means clearly stated or decided; not vague or doubtful. This is unrelated as it's not an opposite of despair.

- Option 2: Vile - means extremely unpleasant or morally bad. This isn't related to the concept of hope or despair.

- Option 3: Hope - signifies the feeling of expectation and desire for a certain thing to happen. It’s directly opposite of despair.

- Option 4: Annoy - means to irritate someone. This doesn't relate to despair directly.

- Therefore, the most appropriate antonym of "Despair" is Hope.

- Correct Answer: Option 3 - Hope