Recently, new optimal Families ${\cal S}$ and ${\cal U}$ of quaternary sequences have been presented, and the optimal binary sequence Family ${\cal V}$ obtained from Family ${\cal S}$ under Gray map has been investigated as well. The two sequence Families ${\cal U}$ and ${\cal V}$ are optimal with respect to the well-known Sidelnikov bound and Welch bound, but their exact correlation distributions are not known until now. In this paper, their exact correlation distributions are completely determined in some cases by making use of exponential sums and the theory of ${\bf Z}_4$-valued quadratic forms.