How option Greeks change: a worked guide to delta, gamma, theta and…

Option Greeks measure how an option's value responds when one input moves and the others are held fixed. Delta is the change per $1 of the underlying, gamma is how fast delta itself changes, theta is the value lost per day, and vega is the change per volatility point. Each one is a slope, so each one is only accurate for small moves.

Most explanations stop at the definitions. This guide prices one option, computes all five Greeks, and then checks them against a full repricing. That check is where the useful intuition lives: it shows when a Greek is a good guide to your profit and when it misleads you.

Every number below comes from the Black–Scholes formula for a European call with no dividends. You can reproduce each one in the free Black–Scholes option pricing calculator, and see whole positions in the option strategy builder. Short definitions of each term are in the glossary under option Greeks.

What are the Greeks for one at-the-money call?

Take a three-month call: spot

S = 100

, strike 100, a 4% rate and 20% volatility. Black–Scholes values it at about 4.49 per share. The matching put is worth about 3.49, and the difference of 1.00 is exactly what put–call parity requires:

S - Ke^{-rT} = 100 - 99.00

Greek	Value	Read it as
Delta	0.560	+ $0.56 per$ 1 rise in spot
Gamma	0.0394	delta rises by 0.039 per $1 rise in spot
Theta	−0.027	loses about 2.7 cents per calendar day
Vega	0.197	+$0.20 per 1 point of volatility (20% → 21%)
Rho	0.129	+$0.13 per 1 point of interest rate

For one contract of 100 shares, multiply each line by 100. Delta of 0.56 becomes a position that behaves like 56 shares; theta becomes about $2.73 a day.

Two of these readings deserve a second look. Delta is above 0.5 even though the option is exactly at the money. That is because the forward price of the stock sits above 100 when rates are positive, and because a lognormal model puts more weight on upward moves in price terms. And the vega figure is easy to sanity-check: raise volatility to 21% and the call reprices to 4.68, up by 0.197, just as vega predicted.

How accurate is delta for a real move?

Delta is a tangent line. Move spot from 100 to 102 and compare three answers:

Full repricing at 102: 5.682.
Delta only: 4.485 + 2 × 0.560 = 5.604, which is 7.7 cents too low.
Delta plus gamma: 5.604 + ½ × 0.0394 × 2² = 5.683, within a tenth of a cent.

The error from delta alone is the curvature of the price, and gamma measures exactly that curvature. This is why traders describe a long option position as "long gamma": the payoff curve bends upwards, so a delta-only estimate always understates gains from a move in either direction.

The same logic explains why delta hedging is never finished. After the move to 102 the call's delta is about 0.64, not 0.56. A hedger short 56 shares is now under-hedged and must sell more. A position that is long gamma buys low and sells high as it rebalances; a short-gamma position does the opposite.

Call delta and gamma across spot for 90 days and 14 days to expiry

Why does gamma spike near expiry?

The chart shows delta and gamma for the same strike at 90 days and at 14 days. With little time left, delta changes from near 0 to near 1 over a narrow range of prices around the strike. Gamma, the slope of that delta curve, becomes tall and narrow.

Away from the strike, gamma is lower but not zero. A 90-strike call (deep in the money) has delta 0.89 and gamma 0.019; a 110-strike call has delta 0.21 and gamma 0.029. The out-of-the-money option has more gamma than the in-the-money one here because spot is closer to its region of uncertainty, measured in standard deviations.

For a trader this matters in two ways. A short at-the-money option close to expiry can swing from harmless to fully exercised on a small move, so its hedge has to be adjusted often. And because hedging costs depend on how often you rebalance, high gamma means high transaction costs for the hedger.

Why does time decay speed up?

Theta is the value an option loses as one day passes with nothing else changing. For our at-the-money call it grows as expiry approaches:

Days to expiry	Call value	Theta per day	Gamma
90	4.45	−0.027	0.040
30	2.45	−0.044	0.069
7	1.14	−0.084	0.144
1	0.42	−0.214	0.381

The at-the-money value scales roughly with the square root of time, so the last few days carry a large share of the remaining value and lose it quickly.

Theta and gamma rise together, and that is not a coincidence. In the Black–Scholes setting, a delta-hedged option's daily profit is roughly ½ × gamma × (actual move)² + theta. Theta is the rent you pay for gamma. If the stock moves more than the volatility you paid for implies, the gamma term wins; if it moves less, theta wins. Every long-option position is a bet on that comparison.

At-the-money call value and daily theta from 180 days to 1 day before expiry

What does vega tell you, and what does it hide?

Vega is the change in value for a one-point change in volatility. It is largest for at-the-money options with plenty of time left: our three-month call has vega 0.197, the 110 call 0.145 and the 90 call 0.097. Longer-dated options have more vega, because a change in volatility applies to more remaining time.

Vega assumes the whole volatility input moves. In real markets, implied volatility differs by strike and expiry, and those points do not move together. A position that is vega-neutral in total can still lose when the smile steepens or the short-dated options reprice more than the long-dated ones. You can see those shapes in the volatility surface explorer.

How do the Greeks add up across a position?

Greeks are linear in position size, so a strategy's Greeks are the signed sum of its legs. Buy a 100 call and sell a 110 call (a bull call spread) and you get:

delta of 0.56 − 0.21 = 0.35;
gamma of 0.039 − 0.029 = 0.010;
vega of 0.197 − 0.145 = 0.052.

The spread keeps much of the bullish exposure while removing most of the volatility exposure. That is often the point of building a spread rather than buying a call outright.

Rho, the sensitivity to interest rates, is small for short-dated equity options: 0.129 per rate point here, so a quarter-point rate rise adds about 3 cents. It matters more for long-dated options and for positions where many legs share the same sign.

In the option strategy builder, switch the chart to delta, gamma, theta or vega to see these sums across every spot price, not just today's.

Common mistakes when reading the Greeks

Treating delta as a probability. Delta is close to, but not equal to, the chance of finishing in the money. Under Black–Scholes that risk-neutral probability is

N(d_2) = 0.520

here, while delta is

N(d_1) = 0.560

. The two diverge more for long-dated, high-volatility options.

Mixing units. Vega is often quoted per one volatility point and theta per calendar day, but some systems use per-unit volatility and per trading day. A vega of 19.7 and a vega of 0.197 can describe the same option.

Using one Greek for a large move. Delta and vega are local slopes. For moves of several standard deviations, reprice the position rather than extrapolating. The delta-only error is 7.7 cents for a

2 move but

1.70 for a $10 move, where the call reprices to 11.78 against a delta estimate of 10.08.

Forgetting that the Greeks move. Each Greek depends on spot, time and volatility, so a hedge that was neutral this morning will not be neutral this afternoon.

Where to go next

Black–Scholes: why the expected stock return leaves the price derives where these sensitivities come from.
What does an option price actually mean? separates price, payoff and profit.
The Greeks and numerical sensitivities lesson computes them by finite differences and explains where numerical Greeks go wrong.

References

Black, F. and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654.

Hull, J. C. Options, Futures, and Other Derivatives. Chapter on the Greek letters, for the delta–gamma–theta relationship and hedging practice.

All figures in this article are computed from the Black–Scholes formula with the stated inputs. They are model values, not market quotes.