Last week it was finally time to upgrade my mobile plan to LTE after I've been using an iPhone 5 for about a month now.
LTE is pretty fast and I was thinking about cancelling my home internet line, since I might be fine with the tethering option of the iPhone to work at home. I wanted to give it a try and noticed that the Personal Hotspot option had disappeared though. I remembered that when Three Hong Kong switched me over to LTE, I didn't have internet access at all, since some of the cellular data settings were not correct. Rebooting the iPhone fixed this problem. I revisited these settings again and it turns out that you only need to change one setting for Internet Tethering.
Under Settings > General > Cellular > Cellular Data Network change the Internet Tethering APN from share.three.com.hk to share.lte.three.com.hk and the Personal Hotspot option appears again. Magic.