Wow. They Cayman Islands - congrats.
Normally they won't let you out of your contract.
1. If you meet a nice rep, and can prove that you are moving out of the country, you may get them to let you out due to the lack of AT&T service (I presume) in the Caymans. This is not an easy thing to do - at the time they sold you the contract, they should have given you a service map showing where they have service. But if you are nice, can prove that you are really leaving, and get a nice person, they occassionally will let you out of it. It's best to go to a real AT&T store - and practice puppy-dog eyes ahead of time.

2. You can pay to get out of your contract. This is usually somewhere in the region of $175, but it may have changed since I last had to deal with AT&T.
3. You can transfer your service to someone else. Find someone - usually a relative or friend - who wants to join AT&T, go to an official AT&T store (not a reseller), ask for a transfer of service, both parties must be present. This does not extend the contract.
Good luck.