Delta hedging: right for an instant
An option's delta is how many shares it behaves like. Hedge it and you are flat, until the stock moves and the delta changes. A short-option hedger buys after rallies and sells after declines, and that is the cost the premium pays for.
- Size the share hedge for a short call position from its delta
- Compute the rebalancing trade after a move, and its direction
- Explain why a delta-neutral short position still loses on a large move
- Hedge short puts from the call delta, and recover the contracts from the hedge
The intuition
A dealer sells 20 call contracts, 100 shares each, with a delta of 0.55. Each call moves 55 cents per dollar of stock, so the position is short 2,000 × 0.55 = 1,100 shares of exposure. Buy 1,100 shares and it is flat. Then the stock rallies and the delta rises to 0.62: the dealer must buy another 140 shares, at the higher price. When it falls back, sell them lower.
Buying high and selling low on every rebalance is not a mistake; it is what being short an option means, and it is what the premium collected up front is for. Delta hedging removes the direction, not the risk: the dealer is left betting that the stock moves less than the premium assumed. That curvature, the delta changing as the stock moves, is gamma, and the next chapter prices it.
Call delta Δ = N(d₁); put delta = Δ − 1. Hedge for short calls: buy contracts × 100 × Δ shares. Rehedge after a move: contracts × 100 × (Δ_new − Δ_old), a purchase if positive. Short puts have positive delta: sell contracts × 100 × (1 − Δ) shares. Hedged P&L = hedge shares × ΔS − contracts × 100 × (C_new − C_old).
Why it works
- The conventions here: European options on a non-dividend stock at a zero rate, Black-Scholes deltas and prices. One contract is 100 shares. Deltas are printed to two decimals and prices to cents before anything is computed, so every answer can be reproduced from the printed figures. The stock move is instantaneous.
- Delta is a share count. A short call position's delta is −contracts × 100 × Δ share-equivalents. Buy that many shares and the first-order exposure is zero.
- In, at, out. An in-the-money call has a delta above 0.50, an out-of-the-money one below. At the money it sits a little above 0.50 under Black-Scholes, because the log-normal distribution skews the upside.
- The hedge drifts. As the stock rises the call behaves more like stock and Δ climbs; the hedger buys more. As it falls, Δ drops and the hedger sells. Time passing moves Δ too. Desks rebalance on bands, when Δ drifts past a threshold, rather than on a clock.
- Why the hedged position still loses. Over a large move the calls gain value faster than the shares on the way up, and lose it more slowly on the way down. Delta was the tangent; the option follows the curve. A short-option position has negative gamma and loses on big moves in either direction; the hedge shrinks the loss but cannot remove it.
- Puts have the same gamma. Parity says a call and a put at the same strike differ by a stock position, which has no gamma. Short either and the rebalancing goes the wrong way after every move.
| Shares under the options: 20 × 100 | 2,000 |
| Hedge: 2,000 × 0.55, bought | 1,100 shares |
| Stock to $105, delta 0.62: new hedge 2,000 × 0.62 | 1,240 shares: buy 140 more |
| Put delta: 0.55 − 1 | −0.45; short 20 puts would be hedged by selling 900 shares |
| Call from $4.00 to $6.95 on the $5 move: stock leg 1,100 × $5; option leg −2,000 × $2.95 | +$5,500; −$5,900 |
| Hedged P&L | −$400, against −$5,900 unhedged |
The option rose $2.95 on a $5 move: more than 0.55 × 5 = $2.75, because the delta climbed along the way. That extra $0.20 a share across 2,000 shares is the $400 the hedge could not catch.
The formulas
Shares per option; a put's is negative.
Cancel the position's negative delta.
Positive is a purchase, after a rally.
A short put is long delta.
Stock leg plus option leg. Negative on a large move: gamma.
Worked example
Contracts times 100 times delta. The follow-up asks why the delta is where it is relative to 0.50.
See it move
Same dealer and calls. Change the size of the stock move, where the strike sits, the volatility, the days to expiry and the number of contracts.
- Make the move bigger, up or down. The hedged loss grows, and faster than the move: it is the square that matters.
- Raise the strike. Delta falls, the share hedge falls, and the short-put hedge rises, because the put's delta grows as the call's shrinks.
- Add contracts. The hedge scales with them, and so does the hedged loss.
- Raise the volatility or add days. Watch the delta: it moves toward 0.50 from whichever side it started on, because more uncertainty makes in and out of the money less certain.
Run it backwards
A book holds a known number of shares as the delta hedge against short calls of known delta. How many contracts is it short?
Shares = contracts × 100 × Δ. Divide the shares by 100 × Δ.
Risk reports work this way round all day: from a share position back to the options it is hedging.
Traps
Say it in the interview
“You sold calls and hedged them. The stock rallied. What happened, and what do you do?”
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.
- Hedge = contracts × 100 × Δ shares; put Δ = call Δ − 1.
- Rehedge by the change in delta: buy after rallies, sell after declines.
- Delta-neutral is not risk-free. Short gamma loses on big moves either way.
- Same strike and expiry: same gamma for calls and puts.