A stock trades at 80 with 32% implied volatility. Price the 73-day at-the-money call to the nearest cent using the standard trading-floor approximation, with zero rates.

A stock trades at 80 with 32% implied volatility. Price the 73-day at-the-money call to the nearest cent using the standard trading-floor approximation, with zero rates.

Approach: Use the at-the-money call approximation 0.4*sigma*sqrt(T)*S, converting the day count to a year fraction first.

4.58. The at-the-money approximation is C = 0.4*sigma*sqrt(T)*S, with T = 73/365 = 0.2 and sqrt(0.2) = 0.4472. That gives 0.4*0.32*0.4472*80 = 4.579, so 4.58. The 0.4 is 1/sqrt(2*pi) = 0.3989 rounded, and it comes from expanding the Black-Scholes call around a zero forward moneyness. The matching straddle is twice this, 0.8*sigma*sqrt(T)*S, or 9.16, which is the number a market maker actually quotes. Price scales with the square root of time, so doubling the maturity to 146 days multiplies it by 1.414 rather than by two. The approximation holds to about 1% while sigma*sqrt(T) stays under roughly 0.4, and it overstates the call once total volatility gets large because it ignores the second-order term.

Follow-up: What is the next term in the expansion, and how big is the error at sigma = 80% and T = 2 years?

Key concepts: at-the-money approximation, implied volatility, square root of time, straddle.