Hi Denise,
If you have not received any money from this guest, then it is definitely not your responsibility to refund.
Airbnb will do it directly.
Guest can do partial payments to Airbnb, therefore the concept of deposit does exist within their system.
You are also right mentioning this could be the "security deposit" you require with your bookings; in any case if you have not received any funds then you should not refund what you did not receive...
Your cancellation policy also will influence the above.
Happy hosting 🙂