  We study the leading term in the small-time asymptotics of at-the-money call
option prices when the stock price process $S$ follows a general martingale.
This is equivalent to studying the first centered absolute moment of $S$. We
show that if $S$ has a continuous part, the leading term is of order $\sqrt{T}$
in time $T$ and depends only on the initial value of the volatility.
Furthermore, the term is linear in $T$ if and only if $S$ is of finite
variation. The leading terms for pure-jump processes with infinite variation
are between these two cases; we obtain their exact form for stable-like small
jumps. To derive these results, we use a natural approximation of $S$ so that
calculations are necessary only for the class of L\'evy processes.
