The most common and most likely answer to this crossword clue is the 4 letter word VAIN
I've seen this crossword clue in Financial Times
V
A
I
N
More clues leading to the result VAIN
- Something to drive about India without success
- Conceited individual originally carried in commercial vehicle
- One in vehicle looking in the mirror a lot?
- Arrogant article plugging wine from Bordeaux
- Futile humour broadcast
- One in a leading position is full of himself
- Futile state to get in
- Report of mood being thoughtless
- Conceited, one in delivery vehicle?
- One in front gets conceited
Buddy explains!

The straight part
We know that "Arrogant" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case arrogant = vain.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does VAIN mean?
VAIN adjective- Overly proud of oneself, especially concerning appearance; having a high opinion of one's own accomplishments with slight reason.
- Having no real substance, value, or importance; empty; void; worthless; unsatisfying.
- Effecting no purpose; pointless, futile.
- Showy; ostentatious.