In my tests, the video exceeded the screen on a mobile phone display. Large, but not good, either.
You can try and calculate the size depending on the screen size via JavaScript (more about that in a new question, if you need details).
What mobile phone did you use for testing, getting a small video?
It may well be that embedding a video from YouTube may not result in a perfectly satisfying result. But at least we can try and make it usable on any screen.