  We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$
started from step initial data and obtain the exact Lyapunov exponents for
$H_0(t)$, the integrated current of ASEP. As a corollary, we derive an explicit
formula for the upper-tail large deviation rate function for $-H_0(t)$. Our
result matches with the rate function for the integrated current of the totally
asymmetric simple exclusion process (TASEP) obtained in [Johansson
00](arXiv:math/9903134).
