No. Neither of my daughters are called by shortened forms of their name. I've found people tend to follow your lead. By the time they hit 3 or so, the kids tend to correct others ("No, my name is not Cora, it is Cordelia!")
I was worried with Cordelia because of Cory. I hated the idea of a boyish nickname, but I've just always called her Cordelia so everyone else does as well.
I can't see it. Jasper has a soft 's' sound in it, not a 'z' sound, so shortening it to Jaz does not seem intuitive to me at all.
Anything is possible when it comes to names and their nicknames. With that said, throughout life, we're ALL called something other than our birth name, whether it's related or not. I say if you love Jasper, have your Jasper, call him Jasper, and let that be enough.