Back-of-envelope value of a 3-month ATM call

Question

Without a calculator, estimate the price of an at-the-money call expiring in three months on a stock priced at $100 with implied volatility $40\%$ . Take $r=0$ and assume geometric Brownian motion.

Short answer

About $8. For an at-the-money option with negligible rates, $C\approx\frac{1}{\sqrt{2\pi}}\sigma S\sqrt{T}\approx0.4\,\sigma S\sqrt{T}$ . Here that is $0.4\times0.4\times100\times0.5=8$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Reading the Implied Vol Surface — Moneyness, Term Structure, and Quoting Conventions
Option Mechanics: Calls, Puts, Moneyness, Intrinsic Value, and Exchange vs. OTC Markets
Option Trading Strategies: Spreads, Combinations, and Payoff Engineering

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