well, i have ribbon that will fray as well, and i don't even finish it.
but i you really want to, just but some sort of wet glue on the end or i have heard that some people use a clear nail polish, or there is a sewing notion called fray check.
also you can burn the edges and that'll work...but i am very hesitant of doing so. 1. i don't want to end up burning down the house or 2. don't want young eyes seeing me do this so that they think that it is ok to burn stuff so they don't burn down the house.
i'd use the glue or fray check.