The verb "have" is used as an auxiliary verb in the present perfect tense. The present perfect tense is used to describe actions that started in the past and continue to the present or have recently finished.

In this sentence, the action of finishing the letter is in the past, but the speaker is emphasizing that the action is not yet complete. Therefore, we need to use the present perfect tense with the negative form, which is "have not."

The other options are incorrect because:

* "am not" is the present tense of the verb "to be" and does not indicate that the action is not yet complete.
* "had not" is the past perfect tense, which is used to describe actions that happened before another past action.
* "did not have" is the simple past tense, which is used to describe actions that happened in the past but are not related to the present.