The option Greeks explained: delta, gamma, theta, vega and rho

The Greeks estimate how an option’s price changes when one input moves and the rest stay put: delta for the stock price, gamma for how fast delta itself changes, theta for the passing of time, vega for implied volatility, and rho for interest rates. As the Options Industry Council puts it, they are a theoretical guidepost, not a guarantee of exact price changes.

Updated October 2026

What each Greek measures

These conventions follow OIC’s definitions. Some tools quote theta per trading day or vega per 1.00 of volatility instead, so check the units before comparing numbers from different sources.

A worked example

Take a call and a put with the same strike and expiry, priced with the Black–Scholes model:

• stock price S = $100, strike K = $100, six months to expiry (T = 0.5 years);
• risk-free rate r = 5% a year, volatility σ = 20% a year, no dividends.

Read the call’s row like this. If the stock rises $1, the call gains about $0.60 (its delta), and its delta rises by about 0.027 (its gamma). One day passing, with nothing else changing, takes about $0.022 off its value (theta). A one-point rise in volatility, from 20% to 21%, adds about $0.27 (vega).

You can check these against the model itself. Reprice the call with the stock at $101 and it comes to $7.50; delta plus half of gamma predicts $7.50. Reprice it at 21% volatility and it comes to $7.16; vega predicts $7.16.

Reading the Greeks

• Delta runs from 0 to 1 for a call and from −1 to 0 for a put. With no dividends, a call’s delta minus the put’s delta at the same strike and expiry is exactly 1, as in the table (0.598 − (−0.402) = 1). OIC notes that as expiry nears, in-the-money options approach a delta of 1.00.
• Gamma and vega are the same for a call and a put with the same strike and expiry, and both are positive for an option you buy: owning options means delta moves your way as the stock moves, and higher volatility raises the value.
• Theta is usually negative for an option you buy, and OIC notes that it accelerates as expiration approaches.
• Greeks move. Each one depends on the stock price, time and volatility, so it changes as they do. OIC’s example: delta is not a fixed percentage.
• Greeks add up. A position’s delta, or any other Greek, is the sum of each option’s Greek times the quantity held, with sold options counted as negative. That is how a strategy’s net Greeks are found.

The formulas

For a stock that pays no dividends, Black and Scholes (1973) give the prices, and the Greeks are their derivatives. N is the standard normal cumulative distribution and φ its density.

d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ − σ√T
Call = S·N(d₁) − K·e^−rT·N(d₂)
Put = K·e^−rT·N(−d₂) − S·N(−d₁)

Delta: call N(d₁), put N(d₁) − 1
Gamma: φ(d₁) / (S·σ·√T)
Vega: S·φ(d₁)·√T, per 1.00 of σ (divide by 100 per point)
Theta, call: −S·φ(d₁)·σ / (2√T) − r·K·e^−rT·N(d₂)
Theta, put: −S·φ(d₁)·σ / (2√T) + r·K·e^−rT·N(−d₂)
Rho: call K·T·e^−rT·N(d₂), put −K·T·e^−rT·N(−d₂)

Theta here is per year; divide by 365 for a calendar day. Merton (1973) extended the model, including to stocks that pay a continuous dividend yield.

Limits

Greeks come from a model, so they are only as good as its inputs and assumptions. Black–Scholes assumes, among other things, constant volatility and interest rates and exercise only at expiry, and OIC notes that it does not perfectly describe real-world options markets, although it is still widely used. American options, dividends and exotic payoffs need other models, which give different Greeks.

The volatility input matters most of all; see implied volatility explained.

Educational only. This is not investment advice and not a recommendation to buy or sell any security or strategy. Figures are model outputs for a made-up example, not market prices.