The Quick At-the-Money Option Price Formula

Full Black-Scholes has five inputs and no closed form you can run in your head. But for an at-the-money option, strike equal to the current price, there is a genuinely useful shortcut: the call (or put) premium is approximately $0.4 \times \sigma \times S \times \sqrt{T}$, where $\sigma$ is annualized volatility, $S$ is the stock price, and $T$ is time to expiry in years. The constant $0.4$ comes from a simplification of the normal distribution's density, and the approximation is accurate to within a few percent right at the money, which is exactly the case interviewers like to ask about.

Worked example: a stock trades at $100, implied volatility is 20% annualized, and the option expires in three months ($T = 0.25$, so $\sqrt{T} = 0.5$). Plugging in gives $0.4 \times 0.20 \times 100 \times 0.5 = 4$. The at-the-money call or put should cost roughly $4, close enough to a full Black-Scholes calculation (which gives about $3.99 under standard assumptions) that the shortcut is genuinely usable, not just a rough guess.

The formula's real value is in reverse: given an observed at-the-money premium, you can back out implied volatility instantly, without a solver, which is the classic use case in a trading interview asking "the market is quoting this option at $X, what vol is that pricing in?"