Options and volatility · O5 · 9 min
Gamma, theta and the hedger’s P&L
Work out what a hedged option position makes or loses in a day, and how big a move it needs to break even.
In the news: “Dealers are short gamma, so moves could be bigger”: what that line means and why it matters.
The essentials
- Gamma
- How much delta changes for a 1-point move in the share. Bought options have positive and sold options negative; it is highest at the money with little time left, where delta can jump from near 0 to near 1.
- Theta
- The value an option loses as a day passes, other things equal: negative for bought options, positive for sold ones. speeds up in the final weeks for options at the money.
- Gamma against theta
- A hedged bought option earns from big moves either way, locked in by re-hedging (), and pays theta daily. With gamma of 200 shares per £1 and theta of −£300, a £2 move makes £100 and a £1 move loses £200.
- Breakeven move
- The daily move at which gamma gains pay the theta: about £1.73 for gamma of 200 and theta of −£300. It is the move the option’s volatility priced, so the hedged buyer wins if the share moves more.
- Dealers’ gamma
- Dealers who are long gamma sell rallies and buy dips, which calms the market and near expiry can a share to a strike. Dealers who are short gamma sell falls and buy rallies, which can make moves bigger.
Delta-hedged P&L over a day
P&L ≈ ½ × Γ × (ΔS)² + Θ; breakeven move = √(2 × |Θ| ÷ Γ)
- Γ
- gamma: the change in delta per 1-point move (in shares, for a book)
- ΔS
- the day’s move in the underlying
- Θ
- theta: the day’s time decay (negative for long options)
How it works
1. A hedged option still makes or loses money
Buy a call and hedge its delta, and small moves in the share cancel out. Big moves don’t. As the share rises, the call’s delta grows, so you end up holding more exposure just as it goes up; as it falls, you hold less. So a hedged bought option gains from big moves, whichever way.
2. Gamma
How much delta changes for a 1-point move in the share is . Bought options have positive gamma and sold options negative gamma. Gamma is highest at the money with little time left, where delta can jump from near 0 to near 1 on a small move.
3. Theta: the daily cost
That gain from big moves isn’t free. An option loses a little value each day as expiry approaches, other things equal. That daily decay is : negative for bought options and positive for sold ones. It speeds up in the final weeks for options at the money.
4. A day’s P&L
For a hedged option, a day’s P&L ≈ ½ × gamma × (move)² + theta. With gamma of 200 shares per £1 and theta of −£300: a £2 move makes ½ × 200 × 4 − 300 = +£100, and a £1 move makes 100 − 300 = −£200.
5. The breakeven move
Set the gamma gains equal to the theta and you get the daily move a hedged option needs to break even: √(2 × |theta| ÷ gamma). With gamma 200 and theta −£300: √(600 ÷ 200) = √3 ≈ £1.73. Bigger moves than that make money; smaller ones lose.
6. What the breakeven means
An option’s price is set with a volatility in mind, and that volatility implies a typical daily move. The breakeven move is that daily move. So a hedged option buyer wins if the share moves more than the option priced, and the seller wins if it moves less.
7. Gamma scalping
Re-hedging bought options means selling shares after rises and buying after falls. Each round trip locks in a small gain, and those gains pay the theta. Traders call this .
8. When dealers’ gamma moves markets
Dealers who are long gamma sell rallies and buy dips, which calms the market. Near expiry, large holdings at one strike can hold a share close to it, which is called . Dealers who are short gamma, often because clients bought puts from them, must sell into falls and buy into rallies, which can make moves bigger.
Try it
Set time to expiry to 1 day, then 1 year: gamma and theta are far bigger near expiry. Switch to Short and their signs flip.
P&L per share against the stock priceAt expiryToday, 1 year to expiry
- Premium paid
- 7.97
- time value 7.97
- Breakeven
- 107.97
- Delta
- +0.54
- Gamma
- +0.020
- per 1 point
- Theta
- −0.011
- per day
- Vega
- +0.40
- per vol point
| Stock price | P&L at expiry | P&L today |
|---|---|---|
| 70 | −7.97 | −7.72 |
| 80 | −7.97 | −6.78 |
| 90 | −7.97 | −4.38 |
| 100 | −7.97 | 0.00 |
| 110 | 2.03 | 6.33 |
| 120 | 12.03 | 14.18 |
| 130 | 22.03 | 23.04 |
Worked example
You run a delta-hedged book of bought options on a share at £50: gamma 400 shares per £1 move and theta −£200 a day. On Monday the share moves £2; on Tuesday it moves £0.50.
- Monday’s gamma gain: ½ × 400 × 2² = £800.
- Less theta: £800 − £200 = +£600.
- Tuesday’s gamma gain: ½ × 400 × 0.5² = £50.
- Less theta: £50 − £200 = −£150.
- Breakeven move: √(2 × 200 ÷ 400) = √1 = £1 a day.
- Days that move more than £1 make money; quieter days lose it.
+£600 on the £2 day and −£150 on the quiet one; the book needs moves of about £1 (2%) a day.
Check yourself
8 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
If you’re long gamma, what do you want the market to do?
Why is gamma highest for at-the-money options near expiry?
How can dealer hedging pin a stock to a strike?
Remember
Hedged P&L ≈ ½ × Γ × (ΔS)² + Θ
Long gamma earns from big moves and pays theta every day; it wins when the market moves more than the option priced.