Options and volatility · O4 · 8 min
Delta and hedging
Work out how much an option moves when the share moves, and how many shares make a hedge.
In the news: “Dealers had to buy shares as the market rose”: the hedging behind lines like that.
The essentials
- Delta
- The change in an option’s value for a 1-point move in the share. A call’s runs from 0 to 1 and a put’s from 0 to −1; an at-the-money call is about 0.5, so it moves about half as much as the share.
- Delta and moneyness
- An option far in the money moves almost one for one with the share, because it is almost sure to pay; one far out of the money barely moves. Near expiry, deltas head to these extremes.
- Position delta
- Each contract covers 100 shares, so 30 bought calls at 0.55 act like 1,650 shares: a £1.20 rise makes about £1,980. Sold options flip the sign, and adding every line gives the book’s delta.
- Delta hedging
- Holding the opposite delta in shares, so small moves cancel: 20 bought calls at 0.45 are hedged by selling 900 shares. Delta shifts as the share moves, at a rate called , so the hedge needs topping up.
- Hedging bought and sold options
- Hedging bought options means selling after rises and buying after falls. Hedging sold options is the reverse, so dealers who have sold many options can push the market further the way it is already going.
Delta hedge
Shares to trade = −Δ × contracts × 100; P&L ≈ Δ × ΔS
- Δ
- delta of one option: 0 to 1 for calls, 0 to −1 for puts (sign flipped if sold)
- 100
- shares per contract
- ΔS
- change in the share price
How it works
1. Options move less than the share
A share rises from 100 to 101. A 100-strike call on it might rise from 8.00 to 8.54: about half the move. A call far in the money moves almost one for one with the share, because it is almost sure to pay. One far out of the money barely moves, because it is almost sure not to.
2. Delta
That ratio is : the change in an option’s value for a 1-point move in the share. A call’s delta runs from 0 to 1, and an at-the-money call is about 0.5. A put’s delta runs from 0 to −1, because a put gains when the share falls.
3. Delta of a position
A contract covers 100 shares, so multiply: 20 calls with a delta of 0.45 act like 20 × 0.45 × 100 = 900 shares. Selling flips the sign, so sold calls and bought puts have negative delta. Add up every line, counting each share you own as 1, to get the delta of the whole book.
4. Hedging delta
To stop small moves in the share changing your P&L, hold the opposite delta in shares: shares to trade = −delta × contracts × 100. Bought 20 calls at 0.45 is +900 shares of exposure, so you sell 900 shares. Being flat to small moves like this is .
5. P&L from delta
Delta also gives a quick P&L: change in value ≈ delta × change in the share price. Bought 30 calls at a delta of 0.55 act like 1,650 shares, so a £1.20 rise makes about 1,650 × £1.20 = £1,980.
6. Delta moves, so hedges move
Delta isn’t fixed. As the share rises a call’s delta climbs towards 1, and as it falls delta drops towards 0. So a hedge must be topped up as the price moves. How fast delta changes is called , the subject of the next lesson.
7. Bought and sold options hedge differently
Hedging bought options means selling shares after rises and buying after falls: selling high and buying low. Hedging sold options is the reverse, buying after rises and selling after falls. That is why dealers who have sold many options can push the market further the way it is already going.
8. Near expiry
With little time left, delta heads to the extremes: options in the money towards 1 (or −1 for puts), options out of the money towards 0. At the money, a small move can swing delta from 0.2 to 0.8, so hedges need constant topping up.
Try it
The Delta figure is how steep the coloured line is at today’s price. Cut the time to expiry and the line bends sharply at the strike.
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 sold 50 call contracts on a share at 40, 100 shares each, with a delta of 0.40, and want to be delta-neutral. The share then rises to 41.50 and delta climbs to 0.52.
- Position delta = −50 × 0.40 × 100 = −2,000 shares.
- To cancel it, buy 2,000 shares.
- After the rise: −50 × 0.52 × 100 = −2,600 shares.
- You hold 2,000, so buy another 600.
- You bought after a rise: hedging sold options means chasing the market.
Buy 2,000 shares, then 600 more after the rise; hedging sold options buys high and sells low.
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.
What’s delta, and how do you hedge it?
What’s the delta of an at-the-money call, and why?
How does delta change as expiry approaches?
Remember
Hedge = −Δ × contracts × multiplier
Delta is the hedge ratio: hold the opposite delta in stock and small moves cancel, then rebalance as delta moves.