Step 1: Read the description carefully.
The clue is: one who collects and studies coins. The key word here is coins.
Step 2: Recall the field name.
The study and collection of coins and currency is called numismatics. So the person doing it has a closely related name.
Step 3: Check option Philatelist.
A philatelist collects postage stamps, not coins. So this is wrong.
Step 4: Check option Archivist.
An archivist looks after historical records and documents. This is about papers and files, not coins, so it is wrong.
Step 5: Check option Antiquarian.
An antiquarian collects or studies antiques in general. This is too broad and not specific to coins, so it is wrong.
Step 6: Pick the matching word.
Numismatist is the exact word for a person who collects and studies coins. \[ \boxed{\text{Numismatist}} \]