I've got the exact same problem here, and I solved it.
My rig : AMD FX-8320, 8G DDR3-1600 CL9, GeForce GTX 650.
You're overclocked, and that's what causes AC4's instability.
You need to undo all overclocking and test your memory with memtest86+.
if your CPU is at stock frequency and your memory passes memtest with no errors, and you still have 15 fps, you need to lower your graphic memory's clock. Overclocking graphic memory on low end cards always causes instability in games, and sometimes GPU overclocking can cause instable fps without other graphic glitches...
I've set my CPU and memory to stock freq and my GPU is now doing great at 1199 MHz @ 1.137v and memory at stock speed.
a tip : when you start the game at recommended settings w/o vertical sync, the introduction videos AND all the menus should be 60 fps or more.
if at some point, it drops to 30-40, no need to go ingame, you'll get 15.