The correct option is A will have written
The sentence indicates that the action will have been completed (finished or "perfected") at one point in the future as mentioned in the phrase "By the end of this month", hence this sentence is in the future perfect tense and is of the form will+have+past participle. The past participle here is "written", (a past participle is a form of a verb that ends in -ed and is used in perfect tenses). Option A is of the future perfect tense will+have+written, hence it's correct. Options B, C and D are incorrect as 'wrote' is in the simple past tense and indicates an action that took place in the past. 'Will write' is in the simple future tense that indicates an action that will begin in the future but does not mention its completion. 'Have written' is in the present perfect tense that links the past to the present, the action began in the past and ended in the present. None of these three options are of the same tense as the sentence.