Probability random walk remains bounded

Question

Consider a simple unbiased random walk on the discrete line starting at $0$. Fix a number $n$. As a function of $k$, what is the probability that the walk remains bounded in $\{-n,\ldots,n\}$ for the first $k$ steps? Alternatively, what is a good upper bound for this probability? I would like a stronger bound than $(1-2^{-{2n}})^k$.