The formula, a finite-difference check, the conversion into shares and currency, and why gamma explodes as expiry approaches.
Gamma is the rate at which delta changes with spot, and under Black-Scholes it is phi(d1) divided by spot times volatility times the square root of time. A one-year at-the-money call on a spot of 100 at 24 per cent volatility and 4 per cent rates has gamma 0.015953: on 10,000 calls, a one-point rally adds 159.5 shares of delta to hedge, and a 1 per cent move changes the cash delta by 15,953, which is the dollar gamma. Gamma is the same for the call and the put, and it rises steeply into expiry.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The option is the same illustrative contract used across this site's option articles, so every figure can be checked against the price and the other Greeks.
| Input | Value |
|---|---|
| Spot | 100.00 |
| Strike | 100.00 |
| Risk-free rate, continuous | 4.0% |
| Volatility | 24.0% |
| Time to expiry, years | 1 |
| Position, calls | 10,000 |
d1 = [ln(S/K) + (r + vol²/2) T] / (vol × root T) = (0 + 0.0688) / 0.24 = 0.2867
phi(d1), the standard normal density = e−d1²/2 / root(2π) = 0.382882
Gamma = phi(d1) / (S × vol × root T) = 0.382882 / 24.00 = 0.015953
In Excel, with spot in B1, volatility in B4, time in B5 and d1 in B7: =NORM.S.DIST(B7,FALSE)/(B1*B4*SQRT(B5)), where the FALSE returns the density rather than the cumulative probability.
Check it numerically by repricing the option at spot plus and minus a small step and taking the second difference: [C(S + h) − 2C(S) + C(S − h)] / h². With h = 0.01 the finite difference gives 0.015953, the analytic value to the sixth decimal. Repricing at 101 confirms the meaning: delta moves from 0.612816 to 0.628592, a change of 0.015776, close to gamma because gamma itself falls slightly over the move.
A gamma of 0.015953 per option is rarely what a desk looks at. It is converted into the position's terms.
| Measure | Formula | Value |
|---|---|---|
| Shares of delta per one-point move | gamma × n | 159.5 |
| Cash delta change per 1% move (dollar gamma) | gamma × S² / 100 × n | 15,953.4 |
| Gamma P&L from a 1% move | ½ gamma × (S × 1%)² × n | 79.77 |
| Gamma P&L from a 3-point move | ½ gamma × 3² × n | 718 |
The P&L line is the reason gamma matters. A delta-hedged long option earns half gamma times the move squared whatever the direction, and pays theta for it; the move at which the two cancel is worked in the gamma-theta break-even.
Gamma depends on time far more than on anything else. Holding the option at the money and shortening its life:
| Time to expiry | Gamma | Multiple of one-year gamma |
|---|---|---|
| 1 year | 0.0160 | 1.0 |
| 3 months | 0.0329 | 2.1 |
| 1 month | 0.0574 | 3.6 |
| 1 week | 0.1199 | 7.5 |
| 1 day | 0.3175 | 19.9 |
Gamma scales roughly with one over the square root of time, so a one-day at-the-money option carries almost twenty times the gamma of a one-year one. Across strikes the shape changes with maturity:
| Strike | 1 year | 1 month |
|---|---|---|
| 80 | 0.0079 | 0.0002 |
| 90 | 0.0128 | 0.0159 |
| 100 | 0.0160 | 0.0574 |
| 110 | 0.0165 | 0.0250 |
| 120 | 0.0149 | 0.0022 |
A one-year book has its gamma spread across a wide band of strikes, and the peak is not at the money: for a fixed spot it sits at the strike where d1 is zero, 100 × e(r + vol²/2) T = 107.12, because of the drift and volatility terms in d1. A one-month book has it concentrated in a narrow band around spot. That is why short-dated books are hedged more often, and why a position that looked flat in gamma a month ago can be the largest risk on the desk this week.
The frequent error is adding per-point gamma across underlyings. A one-year at-the-money call on a 400 stock has gamma 0.003988, a quarter of the 100 stock's, so 2,500 of them, the same 1,000,000 of notional as the 10,000 calls, show a position gamma of 10.0 shares per point against 159.5. The risk is identical: both positions carry 15,953.4 of dollar gamma, and a 1 per cent move changes the cash delta of each by the same amount. Sum dollar gamma, not gamma, and say which convention the report uses, because some systems quote gamma per 1 per cent move in shares and others in currency.
Gamma is local. The table values describe the next small move; after a large one, delta has to be recomputed, not projected with today's gamma.
Gamma is phi(d1) / (S × vol × root T): 0.015953 here, 159.5 shares per point and 15,953 of dollar gamma on 10,000 calls. Check it with a second finite difference, report it in currency per 1 per cent, and watch the expiry calendar, because the same option has 19.9 times the gamma on its last day. The book's sensitivities workbook, which sets each Greek beside its finite-difference estimate, is in the free workbooks for this book; for the first derivative and the hedge it implies, see how to calculate delta and the shares to hedge.
Yes. The put's delta is the call's delta minus one, and the constant disappears when you differentiate again, so a call and a put with the same strike and expiry have identical gamma. In the illustrative case both carry 0.015953, and a long position in either gains delta as spot rises.
Near the money and close to expiry. For a fixed strike, Black-Scholes gamma peaks where spot equals the strike times e^-(r + 1.5 vol squared) T, which for the illustrative one-year option is 88.13, with gamma 0.018327. As expiry approaches the peak moves to the strike and grows: 0.3175 at one day against 0.0160 at one year.
Gamma is the change in delta per one point of spot, so it depends on the price level. Dollar gamma, gamma times spot squared divided by 100, is the change in cash delta for a 1 per cent move and can be added across underlyings. Ten thousand calls on a 100 stock and 2,500 on a 400 stock both carry 15,953.
This article is one calculation from 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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