We study the $k$:th raw moment of a variable $R$ following the binomial distribution $\text{B}(n, p)$, where $n/k \rightarrow β> 0$. It is known that $\mathbb{E}(R^k)$ is bounded both from below and from above by functions of the form $k^k Ψ^k$. We solve the asymptotically optimal value of $Ψ$ as a function of $p$ and $β$.