Black Ops III isn't that bad stock, but it can be an under-performer and the mouse can certainly lag behind with the wrong settings. To fix mouse smoothing and acceleration, we suggest these basics:
The “Controls” options page has what you need, probably. Disable “Frame Smoothing,” set Mouse Acceleration to 0, and set mouse filtering to 0. This gets us raw mouse input without game modification.
If that hasn't resolved the issue, the next likelihood is a tie-in to FPS. You'll want to keep your FPS at or above 60 at all times. In Black Ops III, we've noticed mouse movement speed ties more heavily to framerate than in some other titles. Even running at 50FPS, we see “laggy” mouse input that impacts predictable game movements. The next step, then, is to lower settings and try to hit that sometimes-elusive 60+ FPS.
If you've still got problems, we'd then suggest checking the V-Sync (“Synchronize Frames”) and Max FPS settings in the graphics options. Disabling V-Sync is recommended. Capping FPS can sometimes be helpful to mouse input on some configurations – you'll just have to play with that to see if it works for yours.
- Steve “Lelldorianx” Burke.