Option Greeks Calculator

What the Greeks of an option measure

The first order Greeks tell you how the price of an option moves: delta for the underlying price, gamma for how fast delta itself moves, vega for volatility, theta for time and rho for the interest rate. The second order Greeks describe how those numbers change. They matter once a position is held for days, or when volatility jumps: a book that is delta neutral this morning is not delta neutral after a volatility move or a weekend, and vanna and charm say by how much.

The formulas

European option, Black-Scholes-Merton with a continuous dividend yield q. With d1 = (ln(S/K) + (r − q + σ²/2)T) / (σ√T), d2 = d1 − σ√T, φ the normal density, N the normal distribution function and eq = e^−qT:

• Gamma = eq × φ(d1) / (S × σ × √T)
• Vega = S × eq × φ(d1) × √T (per unit of volatility; the page and the example show vega per volatility point, one hundredth of this)
• Vanna = −eq × φ(d1) × d2 / σ
• Vomma = Vega × d1 × d2 / σ
• Speed = −(Gamma / S) × (d1 / (σ√T) + 1)
• Zomma = Gamma × (d1 × d2 − 1) / σ
• Charm (call) = q × eq × N(d1) − eq × φ(d1) × (2(r − q)T − d2σ√T) / (2Tσ√T)
• Veta = Vega × (q + d1 × D − 1/(2T)), Color = Gamma × (q + d1 × D + 1/(2T)), with D = (2(r − q)T − d2σ√T) / (2Tσ√T)

Charm, veta and color are the change as calendar time passes, so each is the derivative with respect to time to expiry with the sign flipped. The page shows them per calendar day (a day is 1/365 year), vanna, vomma and zomma per volatility point, and speed per unit of the underlying price.

Worked example

A call on a $100 asset, strike $105, 60 days, 40% volatility, 4% rate, no dividend:

• Price $4.6681, delta 0.42889, gamma 0.024207, vega 0.15917, theta −0.057246 per day
• Vanna 0.0033506: delta rises by 0.00335 for each volatility point
• Vomma 0.00024346: vega rises by 0.00024 for each volatility point
• Charm −0.0013822: delta falls by 0.00138 as one day passes
• Veta −0.0014269: vega falls by 0.00143 as one day passes
• Speed 0.000025428, zomma −0.00056816, color 0.00018645

In plain words: if volatility goes from 40 to 45, this call gains about 5 × 0.159 = $0.80 from vega plus a little from vomma, and its delta grows from 0.429 to about 0.446 because of vanna. A week later, with nothing else changed, the delta is lower by about 7 × 0.00138 = 0.01, because the strike is out of the money and time is running out. The figures above agree with the independent reference within a relative 1e-7.

Why look past delta and gamma

Hedging with delta alone assumes that nothing else moves. In practice volatility and time move all the time, and the Greeks of the second order are the size of the hedge you will have to redo: vanna and charm change delta, vomma and veta change vega, speed, zomma and color change gamma. An out-of-the-money option is where they are largest in proportion to its price.

How it was checked

Every Greek is compared in the test suite with a Python reference that does not use these formulas: central differences of the price at 100 digits (mpmath), on 1,600 options that include expiries from about 0.01 day, volatility from about 1%, rates and dividend yields up to 100% and expiries up to 100 years. On 1,000 of them the second order Greeks are also compared with differences of QuantLib's own delta, gamma and vega. The method is described in the methodology.

Where to go next

Want the price against a live Deribit, Binance or Bybit quote? Use the Black-Scholes Calculator. Have a price and need the volatility? Use the Implied Volatility Calculator. Sizing the position that goes with the option? Run it through the Pre-Trade Check.

Use via API or MCP

This calculation is available as a deterministic API call: useful for bots or AI agents that need delta, gamma, vanna, charm and the other Greeks of an option without reimplementing the formulas.