Pick the call price from the terminal spread

Question

A one-year at-the-money call is written on a stock at $100$ ; rates are zero and the standard deviation of the stock price in one year is $10. Which is the closest value for the call: $1, $5 or $10?

Short answer

$5. If $S_T$ is roughly normal with standard deviation $10$ around $100$ , then $\mathbb{E}[\max(S_T-100,0)] = 10/\sqrt{2\pi} \approx 4$ , which is nearest to $5. This is the ATM rule $C \approx 0.4 \times$ (dollar standard deviation).

Adapted from Crack, Heard on the Street, Q2.22, p.27; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

The Black-Scholes PDE, Formula and Implied Volatility
Covariance and Correlation
Taylor and Maclaurin Series

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