A stock is at $60 and its two month at the money straddle trades at $4.20. What implied volatility is that, and how far does the stock have to move by expiry for the buyer to break even?

A stock is at $60 and its two month at the money straddle trades at $4.20. What implied volatility is that, and how far does the stock have to move by expiry for the buyer to break even?

Approach: Invert the at the money straddle approximation for sigma, then express the premium as a fraction of the one sigma move over the option's life.

21.4%. Invert the approximation that the at the money straddle is 0.8 * S * sigma * sqrt(T). With T = 2/12, sqrt(T) is 0.4082, so 0.8 * 60 * 0.4082 = 19.60 and the implied volatility is 4.20/19.60 = 0.2143. The buyer needs the stock more than $4.20 from $60 at expiry, which is a 7.0% breakeven move. Compare that with the two month one sigma move of 21.43% * 0.4082 = 8.75%: the breakeven sits at exactly 0.8 sigma, which is the same 0.8 constant seen from the other side. Read daily, the implied volatility is 21.43/16 = 1.34% a day, about $0.80 on a $60 stock, so the straddle prices roughly five and a quarter average daily moves.

Follow-up: The realised volatility over the two months turns out to be 18%. What does a delta hedged buyer of that straddle lose, to first order?

Key concepts: at the money straddle, implied volatility, breakeven move, approximation.