The head of Pediatrics
________ four children of her own.
"has" is correct because the subject is “head” (singular) not Pediatrics, and it is in the present tense.
"will have" is incorrect because it is in the future tense and there is nothing to suggest this prediction can accurately be made.
"have" is incorrect because it is a plural verb and the subject is singular.
"has had" is incorrect because it is past perfect tense, which indicates an action that began in the past and ended before another past action. This is an ongoing action and, as such, should be in the present tense.