Gamma against theta: the contest a hedged option is about
A delta-hedged long option makes half of gamma times the move squared every day, and pays theta for the privilege. The break-even daily move is the implied volatility restated in dollars.
- Compute a day's gamma P&L from gamma and the move
- Net it against theta and say who won the day
- Find the break-even daily move, and the implied volatility it corresponds to
- Explain what a long-volatility trader is really betting on
The intuition
Own an at-the-money call, delta-hedged. The stock moves $1.50, either way. The option's delta changes in your favour as it moves, so relative to the hedge you have bought low and sold high: the gain is half of gamma times the move squared. With a gamma of 0.05 that is $0.056 a share, $562.50 across 100 contracts. Direction never entered, because the move is squared.
Nothing is free. The option lost a little value through the day, theta, say six cents a share, $600 on the position. So the day netted −$37.50: the move was a little too small. The break-even move is √(2θ ÷ Γ) = $1.55, and with a zero rate that is exactly S × σ ÷ √365, the implied volatility as a daily dollar move. If the stock moves more than that on average, gamma beats theta and long vol wins. If it moves less, theta wins.
Delta-hedged daily P&L per share ≈ ½ Γ (ΔS)² − θ. Position P&L = per share × contracts × 100. With r = 0, θ per day = ½ Γ σ² S² ÷ 365, so the break-even move is √(2θ ÷ Γ) = S σ ÷ √365. Implied vol from a break-even move: move × √365 ÷ S. A move twice as big earns four times the gamma P&L; theta does not change.
Why it works
- The conventions here: an at-the-money European call on a non-dividend stock at a zero rate, delta-hedged. One contract is 100 shares. Gamma and theta per calendar day are printed to four decimals and every answer uses the printed figures.
- Why ½ Γ (ΔS)². Delta is the tangent; the option follows a curve. The gap between the curve and the tangent after a move of ΔS is the second-order term, half the curvature times the move squared. For a long option the curve is above the tangent in both directions.
- Why theta is the mirror of gamma. With no rate, the Black-Scholes equation says θ = −½ Γ σ² S². The option's daily decay is exactly what its gamma would earn on a one-sigma daily move. Long gamma, pay theta; short gamma, collect theta.
- The break-even move is implied vol in dollars. Set ½ Γ m² = θ and m = S σ ÷ √365. A long-vol trader is betting that the stock's realized daily moves will, on average, exceed that; direction does not enter.
- Quadratic gain, linear cost. A long-gamma book bleeds slowly on quiet days and makes it all back on a handful of large ones. A short-gamma book has the mirror image: steady income and occasional painful days.
- A second way to win. Implied vol can rise before expiry, marking the option up through vega, even if realized moves have been small. That is separate from the gamma-theta contest.
| Gamma P&L per share: ½ × 0.05 × 1.50² | $0.05625 |
| Across 10,000 shares | $562.50 |
| Theta: $0.06 × 10,000 | −$600 |
| Net for the day | −$37.50: the decay beat the move |
| Break-even move: √(2 × 0.06 ÷ 0.05) | $1.549 |
| A $3.00 move instead: 4 × $562.50 | $2,250 of gamma P&L, net +$1,650 |
A $1.549 break-even on a $100 stock is 1.549% a day; times √365 that is 29.6% annualized, the implied volatility these Greeks correspond to.
The formulas
Gamma gain, squared in the move, less the day's decay.
Scale by the shares under the options.
Theta is what gamma would earn on a one-sigma day.
The break-even move, read back as a volatility.
Worked example
Half of gamma times the move squared, per share, then scale by the shares. The follow-up asks why the direction does not matter.
See it move
Same hedged position. Change the size of the day's move, the implied volatility, the days to expiry and the number of contracts.
- Make the move bigger. The gamma P&L rises with the square of the move, theta does not change, and the net rises.
- Raise the implied volatility. Theta rises, gamma falls, and the break-even move rises: dearer options need bigger days.
- Add days to expiry. Gamma and theta both fall, so the day's gain and cost both shrink; the break-even move barely moves, because in theory it does not depend on days at all, and the small wobble is the rounding of the printed gamma and theta.
- Add contracts. Gain and cost scale together.
Run it backwards
Gamma and theta are known. How large a daily move does the position need just to break even?
Set ½ Γ m² equal to θ and solve: m = √(2θ ÷ Γ). Divide by the stock price for the move as a percentage of the price.
The check question runs it one step further: a break-even move times √365 over the stock is the implied volatility the option is priced at.
Traps
Say it in the interview
“You are long options and delta-hedged. When do you make money?”
Check yourself
5 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Daily P&L ≈ ½ Γ (ΔS)² − θ. Squared in the move, so direction does not matter.
- θ = ½ Γ σ² S² ÷ 365 at a zero rate: theta is the price of gamma.
- Break-even move = √(2θ ÷ Γ) = S σ ÷ √365: implied vol in dollars.
- Long gamma bleeds on quiet days and wins on big ones; short gamma is the mirror.