You sell a one-year at-the-money straddle on a $100 stock at 20% implied and delta hedge it daily. Realised volatility over the year is 16%. Approximately what do you make per share, and name the single largest reason the actual number differs.

You sell a one-year at-the-money straddle on a $100 stock at 20% implied and delta hedge it daily. Realised volatility over the year is 16%. Approximately what do you make per share, and name the single largest reason the actual number differs.

Approach: Value the position change through vega times the volatility difference, then ask what makes the realised gamma weighting differ from the constant one that approximation assumes.

$3.20. Straddle vega at the money is roughly 0.8*S*sqrt(T) per unit of volatility, which is $0.80 per volatility point on a $100 one-year straddle, so capturing four points of variance risk premium pays about 0.80*4 = $3.20 per share. The straddle sold for 0.8*0.20*100 = $16.00 and the hedging cost tracked 16% realised, so roughly $12.80 of it was paid back out. The number that actually lands differs because the delta-hedged profit and loss weights each day's volatility by that day's dollar gamma, so a period of low realised volatility while the option is at the money and gamma is large is worth far more than the same quiet period once the stock has drifted 20% away and gamma has collapsed. Path dependence, not the average volatility, decides the outcome.

Follow-up: Which position would make the payoff depend on average variance rather than on the path, and what does that cost you?

Key concepts: variance risk premium, delta hedging, vega, path dependence, gamma.