Ds1 (now 3) was a hitter (and still can be at times). It is so hard - he slept well and ate well but I'd have rather traded either of those things for a little boy who let me enter each child-orientated social situation without a coat of mental armour. He reached his hitting peak around 20-24months after which his speech developed and it became a little easier to reason with him.
At 15 months, there is very little you can do, except grow a thick skin. You can start being consistent (removing him from the situation for time out, or going home, if appropriate) but he won't really understand it at that age - it's more about social conditioning until they start to understand through verbal reasoning. Ds1 didn't really 'get' the consequences until he was at least two. Rather than leave planned social occasions where I was meeting friends, I took him to a playgroup on my own on two separate occasions, from which I knew we could leave easily without letting anyone down - on the first occasion, I suspected he would hit - and he did - so I gave him a warning and then when he did it again, I brought him home. We went back the following week, he hit again, I took him home again. We went back the week after, and several times since, and he hasn't hit. He has had the occasional altercation at other social occasions where his peers are present, but nothing on the grand scale it used to be. I suspect he will always be a little alpha male and act with his fists, but I have faith in his own intelligence that he will not use it in a neanderthal way for the rest of his life.
The one thing to remember, for your own sanity, is that it isn't about personality at 15 months. The part of the brain which would moderate impulses in adults isn't very well developed in a small child, so young children don't have the inhibitions us adults do, particularly when they are growing, when this part of the brain is even more under siege. Sharing is not actually a natural part of human nature if you think about it, whether it be a toy or personal space, so it's not surprising some kids find it incredibly difficult to deal with, especially when they don't have the words to articulate their feelings.
Suck it up for the time being, it's blimmin hard, but take heart in the knowledge that you'll be a better parent later on, and your ds will have grown out of it. You'll more than likely find your friends' dcs will go through this too at a later stage. Be smug in the fact you're the test case
, they'll quietly be smug until it comes to bite them on the arse later- and then who will have the smug certificate...!