There isn't a way to change the structure of a theme on WordPress.com directly using HTML or PHP, so you can't add a video into the header. But what you can try to do is re-position one that is already on the page. For this to work, the home page would always have to have the same post (or you could setup a static front page). Also, this example targets the post on your home page right now titled "Caravan on the Silk Road."
#post-57 .embed-youtube {
position: absolute;
width: 760px;
height: 480px;
top: -800px;
left: 180px;
z-index: 100000;
}
Note that this is just a starting point and you should adjust the CSS width, height, top, and left values until it looks good to you.
In that post, the YouTube video you embedded is too large for the post container area, and that's why the ratio aspect looks off. I would recommend changing the video shortcode to smaller dimension such as 640 x 480 (or something else you pick) to fix that. See this help page for more info: http://en.support.wordpress.com/videos/youtube/