Indeed to be granted a certificate for an officially certified chronometer, a watch has to keep time to within a minute a month. According to the booklet that came with my watch, the watch will either gain or loose a few seconds a day, depending on its position (that is whether it is laying on its back, or is vertical), so with normal arm movement the two effects will largely cancel over the course of a day.

I don't know about current Rolex watches, but mine is an old model and is not an automatic.