Implied vol from a 3-month ATM call price of 2.3

Question

An at-the-money European call on a stock priced at $12$ expires in three months and trades at $2.3$ . Using a standard at-the-money approximation, estimate the annualised implied volatility to three decimals.

Short answer

About $0.961$ . For an ATM option $C\approx\sigma S\sqrt{T/(2\pi)}$ . Solving $2.3=\sigma\cdot12\sqrt{0.25/(2\pi)}$ gives $\sigma\approx2.3/2.394\approx0.961$ .

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Lessons behind this question

Reading the Implied Vol Surface — Moneyness, Term Structure, and Quoting Conventions
Taylor and Maclaurin Series
Option Mechanics: Calls, Puts, Moneyness, Intrinsic Value, and Exchange vs. OTC Markets

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