  Let $X_1,X_2,...$ be independent random variables with zero means and finite
variances, and let $S_n=\sum_{i=1}^nX_i$ and $V^2_n=\sum_{i=1}^nX^2_i$. A
Cram\'{e}r type moderate deviation for the maximum of the self-normalized sums
$\max_{1\leq k\leq n}S_k/V_n$ is obtained. In particular, for identically
distributed $X_1,X_2,...,$ it is proved that $P(\max_{1\leq k\leq n}S_k\geq
xV_n)/(1-\Phi (x))\rightarrow2$ uniformly for $0<x\leq\mathrm{o}(n^{1/6})$
under the optimal finite third moment of $X_1$.
