A delta-hedged position is neutral only at the price where you set it. The stock moves, the delta drifts, and you trade shares to bring it back to zero. Done over and over, those trades sell after rallies and buy after dips, and each round trip banks a small profit. Traders call it gamma scalping. It is the mechanism by which a hedged long option turns movement into cash.
A worked path
Start with a stock at $100 and a 30-day at-the-money straddle: one call and one put at the $100 strike. The call's delta is about 0.50 and the put's about minus 0.50, so the straddle starts delta neutral with no shares at all. Each option has a gamma of 0.04, so the straddle's gamma is 0.08: for every $1 the stock moves, the position's delta changes by 0.08, which is 8 shares per contract.
- Stock rises to $102. Delta has grown by 2 times 0.08, to about +0.16, so the position now acts like 16 shares long. Sell 16 shares at $102.
- Stock falls back to $100. Delta is back at zero, so you are short 16 shares you no longer need. Buy them back at $100. That round trip made $2 on 16 shares, $32.
- Stock falls to $98. Delta is now about minus 0.16, 16 shares short. Buy 16 shares at $98.
- Stock returns to $100. Sell those 16 shares at $100. Another $32.
Each rebalance sells above where it bought, or buys below where it sold, because gamma always tilts the position the right way after a move. The profit from one leg of a move is about half of gamma, times the move squared, times 100. With gamma 0.08 and a $2 move: 0.5 × 0.08 × 4 × 100 = $16. Two legs, up and back, is $32, and that is exactly what the share trades locked in.
The bill
The straddle loses value every day. Say its theta is minus $0.12 per share per day, which is $12 per straddle. A day in which the stock moves $2 earned $16 of gamma and paid $12 of theta: ahead by $4. A day in which the stock barely moves earned nothing and still paid $12. The break-even move is the one where gamma and theta match: half of gamma times the move squared equals theta, so the move is the square root of 2 × 0.12 ÷ 0.08, about $1.73.
Both per share, on one axis. Gamma is a hump at the strike and theta is a dip in the same place. Drag days down and both deepen together; raise volatility and gamma flattens while theta deepens.
Gamma and theta are two views of one thing. Wherever the straddle has the most bend, it pays the most rent, and both grow together as expiration approaches. The volatility slider shows the one place they part: higher implied volatility makes the curve gentler, so gamma falls, while the daily bill rises. Paying a higher implied volatility means paying more rent for less bend.
Break-even is the one-sigma move
That $1.73 is not a coincidence. Options are priced from an implied volatility, and the one-day move that volatility implies is the annual figure scaled down by the square root of the number of days in a year. At 32% implied, $100 × 0.32 ÷ √365 is about $1.67. On a desk you will also see it divided by √252 for trading days, about $2.00; the idea is the same. Theta is set so that a hedged option breaks even if the stock moves about one implied standard deviation a day. Move more than that on average and the scalper wins; move less and theta wins.
A straddle bought at 25% implied on a simulated $100 stock that moves at 35%, re-hedged to delta zero every day. The green line is gamma banked, the red is theta paid. Press New price path a few times: the net usually finishes ahead, but not on every path.
On any single path the net line wobbles: some days pay, some cost. What decides where it ends is how the stock's actual movement compares with the implied volatility you paid. So the question a gamma scalper asks every morning is not which way the stock goes, but whether it will move more or less than the price implies.
How often to hedge
The worked path rebalanced at every $2. You could hedge every $1, every $5, once a day at the close, or once a week. The choice changes which profits you bank, but not what you expect to bank. Hedging often locks in many small moves; hedging rarely waits for bigger ones. As long as each move is independent of the last, the average squared move over a day equals the sum of the average squared moves within it, so over many days both styles earn the same gamma profit on average.
What frequency does change is the spread of outcomes. Hedge rarely and a trending day pays you far more than a choppy one, so your result wanders a long way from what the stock's volatility would suggest. Hedge often and the result tracks volatility more tightly, but you pay the bid-ask spread on the shares every time. Desks often use a band: rebalance whenever the position drifts a set number of shares from neutral.
The maths: why frequency changes the path but not the expected result
Use the straddle above, gamma 0.08. Each hedge banks ½ × Γ × (move since the last hedge)² × 100, which is $4 for every dollar-squared of move. Say a day's typical move is $2, so a half day's typical move is $2 ÷ √2, about $1.41. Two days, each made of two half-day moves of $1.41:
- Trend. Up $1.41, then up $1.41 again. Hedge once at the close: the move is $2.83, squared about 8, so $32. Hedge at midday and at the close: 2 + 2 = 4, so $16.
- Round trip. Up $1.41, then back down. Hedge once: the move is zero, so $0. Hedge twice: again 2 + 2, so $16.
Hedging once earns $32 or $0; hedging twice earns $16 either way. The averages match at $16, because the average squared move matches: ½ × (8 + 0) = 4 = 2 + 2. In general the variance of a day's move is the sum of the variances of its pieces, so the expected gamma profit ½·Γ·E[dS²] does not depend on how you slice the day. More frequent hedging shrinks the spread of results roughly with the square root of the number of rebalances: four times as many halves it.
Your turn
Open SPY, monthly expiration, and build the numbers for an at-the-money straddle from the Γ and Θ columns. Then switch to AAPL and work out the one-standard-deviation move its at-the-money implied volatility prices in to expiration.