Quote the 30-day at-the-money straddle on a $200 stock with 25% implied volatility and zero rates, using 365 days.

Quote the 30-day at-the-money straddle on a $200 stock with 25% implied volatility and zero rates, using 365 days.

Approach: Use the straddle form of the at-the-money approximation, 0.8*sigma*sqrt(T)*S, after converting the days to a year fraction.

11.47. The straddle at-the-money approximation is 0.8*sigma*sqrt(T)*S, with T = 30/365 = 0.08219 and sqrt(T) = 0.2867. That gives 0.8*0.25*0.2867*200 = 11.47. The break-even points sit at 188.53 and 211.47, so the buyer needs a 5.7% move in a month to make anything, and that is the number to compare against the stock's typical monthly range. The same approximation says the expected absolute move over the month is 0.8*sigma*sqrt(T), or 5.7%, so the straddle is priced at exactly the expected absolute move, which is what it pays. Any premium a seller earns over that comes from realised movement falling short of implied, and nothing else.

Follow-up: Earnings fall inside the 30 days and the market prices an 8% move on the print. What should the straddle be worth now?

Key concepts: straddle, at-the-money approximation, implied volatility, break-even.