Right answer: está ecribiendo
In this case, if you are referring to a friend, you must use the 3r person in singular él/ella (he/she) and make the correct conjugation of the verb in its verbal tense and according to the subject in gender and number.
According to this, the correct option is:
Está ecribiendo
or
(Él/Ella) está ecribiendo
(He/She) is writing
Note in the first sentence the subject is not explicitly written. This is because in Spanish grammar is not always necessary that the subject is explicitly in the sentence, it is called Tacit Subject, Implicit Subject or Null Subject.
This is because the subject is encoded in the verb conjugation.
The other options are incorrect because do not match with the 3r person in singular:
-están ecribiendo (they are writing)
-estoy ecribiendo (I am writing)
-estás ecribiendo (You are writing)
