Offensive Elbow.
 

Post player receives pass at free throw line near his basket. Looks to pass but has nothing. Keeps looking for open teammate. Throws elbow behind him while turning with the ball and connects with defense in the kisser.

You sound the whistle with the intention to call a foul.

What are your foul options? Why would you make them?

Rule set?

Rule set
 

Sorry I left that part out.
NFHS rules.

Options: Personal, intentional, or flagrant; based on normal factors.
If you think it was inentional or overly physical, you go with intentional.
If you think it was flagrant (like he was trying to loosen some teeth), go flagrant.
If not, you can go with personal.

This assumes that you've already determined it was a foul; no-call is an option depending on the circumstance.

You must mean common foul. Personal fouls are just live ball contact fouls which can be intentional or flagrant.

Naw, Snaqs meant personal foul. A personal foul can also be a common foul by rule.

NFHS rule 4-19-2- "A common foul is a personal foul..."

To be completely accurate, the options if you do call a foul are player-control foul, intentional personal foul and flagrant personal foul.

The definition for intentional and flagrant starts out basically the same way. A personal foul is just a foul which involves illegal contact while the ball is live. That personal foul can be intentional or flagrant. If it isn't intentional nor flagrant, nor against a player who is shooting, nor a double, multiple, or simultaneous foul (I think that is all of them) then it is a common foul. A player control foul is a type of common foul.

Under what circumstance would a no-call be an option?
Not challenging, just wondering.

A1 has the ball and pivots. B1 gets bumped slightly with the elbow at the end of the pivot, which is moving with A1's hips in his pivot. Neither player disadvantaged by the contact.

Or B1 is leaning over top of A1 who is bent over after getting the rebound, A1 pivots and straightens up, knocking B1 in the teeth with his elbow. Assuming A1's pivot equaled the speed of his elbows, could be a no-call.

Did he throw the elbow? Was he pivoting when the elbow was 'thrown'? And if so, as Snaqs pointed out, was he holding the ball with elbows outstretched and those elbows were not going any faster than the pivot?

Yes offense threw the elbow. No he was turning some but not pivoting.
Ball was being held and the elbows moved faster than the rest of the body.