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We study the leading term in the small-time asymptotics of at-the-money calloption prices when the stock price process $S$ follows a general martingale.This is equivalent to studying the first centered absolute moment of $S$. Weshow 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 finitevariation. The leading terms for pure-jump processes with infinite variationare between these two cases; we obtain their exact form for stable-like smalljumps. To derive these results, we use a natural approximation of $S$ so thatcalculations are necessary only for the class of L\'evy processes.

Small-Time Asymptotics of Option Prices and First Absolute Moments