A $100 stock has 20% annual volatility and zero rates. The one-year at-the-money straddle costs 16. What is the risk-neutral probability the stock finishes outside the break-even points, and what does that say about buying straddles?

A $100 stock has 20% annual volatility and zero rates. The one-year at-the-money straddle costs 16. What is the risk-neutral probability the stock finishes outside the break-even points, and what does that say about buying straddles?

Approach: Find the two break-even prices, convert each to a standardised log return under the risk-neutral drift, and add the two tail probabilities.

42%. The break-even points are 84 and 116. Under the risk-neutral lognormal law the log return has mean -sigma^2/2 = -0.02 and standard deviation 0.20, so the upper tail is N(-(ln(1.16) + 0.02)/0.20), or N(-0.842), worth 0.200, and the lower tail is N((ln(0.84) + 0.02)/0.20), or N(-0.772), worth 0.220, giving 0.42 in total. So a straddle buyer at fair value loses money 58% of the time and still breaks even on average, because the 42% of paths that pay include the large moves. That asymmetry is why volatility buying feels like a losing strategy over most sample periods and why a seller who measures success by hit rate will size the position wrongly. The expected payoff equals the 16 premium exactly, which is what fair value means.

Follow-up: What is the probability the stock touches either break-even at some point during the year rather than finishing outside them?

Key concepts: break-even, lognormal distribution, risk-neutral probability, straddle.