This study derives a reactivity balance equation and criteria for preventing core melting during an Unprotected Transient Overpower (UTOP) accident in fast reactors using the assumption of the asymptotic state. In fast reactor design, it is necessary to perform transient analyses that account for potential accidents to ensure core safety. On the other hand, a simplified approach can also be used to evaluate core safety by applying static nuclear parameters, such as reactivity coefficients, based on the reactivity balance equation in the asymptotic state. While this simplified method is valuable for preliminary core design evaluations, the assumptions used in the derivation have not been fully discussed. Therefore, this study clarifies the derivation of the reactivity balance equation in the asymptotic state, considering the fuel temperature, coolant temperature, and axial and radial expansions. Using this equation, criteria are established for avoiding both coolant boiling and core melting during UTOP conditions. A key approximation made in this derivation is that the shape of the power distribution remains unchanged through the transient event.