  We perform a fully non-linear calculation of the production of supermassive
Grand Unified Theory (GUT) $X$ bosons during preheating, taking into account
the fact that they are unstable with a decay width $\Gamma_X$. We show that
parametric resonance does not develop if $\Gamma_X$ is larger than about
$10^{-2} m_X$. We compute the nonthermal number density of superheavy bosons
produced in the preheating phase and demonstrate that the observed baryon
asymmetry may be explained by GUT baryogenesis after preheating if $\Gamma_X$
is smaller than about $10^{-3} m_X$.
