  The hour-glass-like dispersion of spin excitations is a common feature of
underdoped cuprates. It was qualitatively explained by the random phase
approximation based on various ordered states with some phenomenological
parameters; however, its origin remains elusive. Here, we present a numerical
study of spin dynamics in the $t$-$J$ model using the variational Monte Carlo
method. This parameter-free method satisfies the no double-occupancy constraint
of the model and thus provides a better evaluation on the spin dynamics with
respect to various mean-field trial states. We conclude that the lower branch
of the hour-glass dispersion is a collective mode and the upper branch is more
likely the consequence of the stripe state than the other candidates.
