Black-Scholes delta for a call and a put, the hedge ratio in shares, how gamma moves it, and delta across spot and time to expiry.
In Black-Scholes a call's delta is N(d1) and a put's is N(d1) − 1, and the hedge is delta times the number of options, in shares, taken the opposite way. For a one-year at-the-money call at 24 per cent volatility and a 4 per cent rate, d1 is 0.2867 and the delta is 0.6128: a holder of 10,000 calls sells 6,128 shares to be hedged. If the stock rises by 2, delta becomes 0.6440 and the hedge needs 312 more shares sold.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Delta is the first number a trader reads off a position and the one that tells you how many shares make it indifferent to small moves. It is easy to compute and easy to get wrong in three specific ways: using the wrong probability, ignoring dividends, and forgetting that it changes as the stock moves.
Kestrel 100 is the illustrative European call that runs through the book. The same inputs price it at 11.4541; see the Black-Scholes worked example for d1 and d2 in full.
| Input | Value |
|---|---|
| Spot, S | 100 |
| Strike, K | 100 |
| Maturity, T | 1 year |
| Rate, continuously compounded | 4% |
| Dividend yield | 0% |
| Volatility | 24% |
| Position | 10,000 calls |
d1 = [ln(S/K) + (r + σ²/2)T] ÷ σ√T = (0 + 0.0688) ÷ 0.24 = 0.2867
Call delta = e−qT N(d1) = N(0.2867) = 0.6128 (0.612816)
Put delta = e−qT (N(d1) − 1) = 0.6128 − 1 = −0.3872
Excel: =EXP(-q*T)*NORM.S.DIST(d1,TRUE)
Read it as a recipe: 0.6128 of a share, financed partly by borrowing, behaves like one call for small moves in the stock. That is why the same number serves as the sensitivity and as the hedge.
A check that costs nothing: bump the stock by 0.01 either way, reprice, and divide the change by 0.02. The finite-difference delta here is 0.612816, identical to six decimals. Call delta minus put delta is always e−qT, here 1.0000, which is put-call parity differentiated.
Shares to hedge = − delta × number of options
Long 10,000 calls: −0.6128 × 10,000 = sell 6,128 shares
Long 10,000 puts: −(−0.3872) × 10,000 = buy 3,872 shares
On listed equity options each contract usually covers 100 shares, so 100 contracts are 10,000 options and the same 6,128 shares apply. Getting the multiplier wrong is the quickest way to be hedged a hundred times over or not at all.
The hedged book has no exposure to a small move: a rise of 1 makes about 6,128 on the calls and loses 6,128 on the short shares.
Risk reports often state delta in money rather than shares: 6,128 shares at 100 is a cash delta of 612,816, the stock position the calls behave like for small moves. The calls cost 114,541, so each unit of premium carries 5.35 units of stock exposure. That leverage is why an unhedged option book can move far more than its premium suggests.
Delta itself changes as the stock moves. The rate of change is gamma, n(d1) ÷ (Sσ√T) = 0.015953 here. A move of 2 shifts delta by about 2 × 0.0160 = 0.0319, to an estimated 0.6447; repricing gives 0.6440 exactly.
Stock to 102: new delta 0.6440, new hedge 6,440 shares, sell 312 more
Stock to 98: new delta 0.5802, new hedge 5,802 shares, buy back 326
On the move to 102 the calls gain 12,571 and the 6,128 short shares lose 12,256: the hedged position makes 314, close to the gamma estimate of ½ × gamma × move² × 10,000 = 319. That small gain from movement, paid for by time decay, is the economics of a delta-hedged option. The break-even move at which they cancel is worked in the gamma-theta break-even article.
| Spot | Call delta | Put delta | Gamma | Shares per 10,000 calls |
|---|---|---|---|---|
| 80 | 0.2601 | −0.7399 | 0.0169 | 2,601 |
| 90 | 0.4395 | −0.5605 | 0.0183 | 4,395 |
| 100 | 0.6128 | −0.3872 | 0.0160 | 6,128 |
| 110 | 0.7529 | −0.2471 | 0.0120 | 7,529 |
| 120 | 0.8523 | −0.1477 | 0.0080 | 8,523 |
| Time left | Delta, strike 100 | Gamma, strike 100 | Delta, strike 110 |
|---|---|---|---|
| 1 year | 0.6128 | 0.0160 | 0.4560 |
| 3 months | 0.5570 | 0.0329 | 0.2575 |
| 1 month | 0.5330 | 0.0574 | 0.0980 |
| 1 week | 0.5159 | 0.1198 | 0.0024 |
As expiry approaches, an at-the-money delta drifts towards 0.5 while gamma rises sharply: a week out, gamma is 0.1198, seven and a half times the one-year figure, so the hedge has to be adjusted far more often. An out-of-the-money option's delta collapses towards zero.
Delta is a model output: it depends on the volatility you put in. A different volatility gives a different hedge, which is why a hedged book's result depends on realised volatility.
Call delta = e−qTN(d1), put delta = call delta − e−qT; hedge with minus delta times the position: sell 6,128 shares against 10,000 Kestrel calls. Then watch gamma, because the hedge goes stale with every move. All five Greeks with a finite-difference check beside each are in the free sensitivities workbook for this case.
Not exactly 0.5. With a positive rate and volatility, d1 is above zero, so an at-the-money call's delta is above a half: 0.6128 for one year at 24 per cent volatility and a 4 per cent rate. It drifts towards 0.5 as expiry nears, 0.5570 at three months and 0.5159 at one week.
Buy the absolute value of put delta times the number of options. A one-year at-the-money put at 24 per cent volatility has a delta of -0.3872, so 10,000 puts are hedged by buying 3,872 shares. Put delta equals call delta minus one when there is no dividend.
No. The risk-neutral probability of finishing in the money is N(d2), 0.5186 for this option, while delta is N(d1), 0.6128. Using N(d2) as a hedge ratio would sell 5,186 shares against 10,000 calls instead of 6,128, leaving 942 shares of exposure.
All five Greeks, each with a finite-difference check, are computed in the sensitivities workbook for chapter 8 of Quantitative Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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