At-the-money call when prices follow Bachelier dynamics

Suppose a non-dividend stock follows arithmetic Brownian motion, $dS_t = \sigma\, dW_t$ with $\sigma$ measured in dollars, and interest rates are zero. Derive the value of a call struck at the current price $S_0$ with maturity $T$.

$C = \sigma\sqrt{T/(2\pi)} \approx 0.4\,\sigma\sqrt{T}$. With zero rates the stock is already a martingale, so $S_T - S_0 = \sigma\sqrt{T}Z$ with $Z$ standard normal. Then $C = \sigma\sqrt{T}\,\mathbb{E}[\max(Z,0)] = \sigma\sqrt{T}\,\phi(0) = \sigma\sqrt{T/(2\pi)}$.