How to use the option Greeks calculator
The page has three tools that share the same inputs.
- Choose the option and position. Pick call or put, long (bought) or short (sold), and the number of contracts. US equity options control 100 shares each, so position figures multiply the per-share Greeks by 100 × contracts, with the sign flipped for a short position.
- Enter the stock price, strike and days to expiry. Count calendar days to the expiration date.
- Enter the implied volatility, the risk-free rate and the dividend yield. Use the option’s implied volatility from your broker’s chain; the Greeks then describe the option at its market price. Our implied volatility calculator solves for it from a price.
Greeks shows all five sensitivities per share and per position, dollar delta and gamma exposure, plus a table of Greeks for strikes from 90% to 110% of the stock price. Greeks vs price charts the Greek you pick across stock prices, now and with a quarter of the time left. Scenario estimates the P&L of a price move, elapsed days and a volatility change from the Greeks, and compares it with an exact repricing.
The Greek formulas
All five come from differentiating the Black-Scholes-Merton price with a continuous dividend yield q.
Gamma: Γ = e−qT·n(d₁) ÷ (S·σ·√T)
Vega: ν = S·e−qT·n(d₁)·√T ÷ 100 (per 1 vol point)n(·) is the standard normal density, N(·) its cumulative distribution, S the stock price, σ the volatility, T the time to expiry in years (days ÷ 365), and d₁ = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T), d₂ = d₁ − σ√T. Gamma and vega are the same for a call and a put with the same strike and expiry.
Thetaput = [ −S·e−qT·n(d₁)·σ ÷ (2√T) + r·K·e−rT·N(−d₂) − q·S·e−qT·N(−d₁) ] ÷ 365
Rhocall = K·T·e−rT·N(d₂) ÷ 100 Rhoput = −K·T·e−rT·N(−d₂) ÷ 100Theta is per calendar day and rho per percentage point of the risk-free rate.
Dollar delta = position delta × S Gamma exposure = position gamma × S² × 1%Dollar delta is the size of the stock position the options behave like. Gamma exposure is how much that dollar delta changes when the stock moves 1%.
A worked example
You buy 5 contracts calls with a $52 strike on a $50 stock, 45 days to expiry, at 30% implied volatility and a 4% rate. Each call is worth $1.38.
- Delta 0.3925: the position behaves like 196.2 shares, a dollar delta of $9,812.10.
- Gamma 0.0730: each $1 rise in the stock adds 36.49 shares of delta, and a 1% move changes dollar delta by $912.23.
- Theta -$0.0245 per share per day: the position loses $12.25 a day from time decay alone.
- Vega $0.0675 per vol point: a rise in implied volatility from 30% to 31% adds about $33.74.
- Rho $0.0225 per rate point: a 1-point rise in rates adds $11.25.
The put with the same strike has a delta of -0.6075 but the same gamma (0.0730) and vega ($0.0675).
What each Greek tells you
Delta is the option’s exposure to the stock: the change in its value for a $1 move. It also hedges: selling 196 shares would make the example position roughly neutral to small moves. Call deltas run from 0 to 1 and put deltas from −1 to 0, and the call delta minus the put delta for the same strike equals e−qT, which is 1 here because the stock pays no dividend.
Gamma is the rate at which delta changes. Long options have positive gamma: delta rises as the stock rises and falls as it falls, so the position gets longer into rallies and shorter into declines. Short options have the opposite profile, which is why short gamma positions can lose quickly in sharp moves.
Theta is the cost of holding that gamma. Gamma and theta are two sides of one trade-off: for a delta-hedged option with no dividend and rates near zero, theta ≈ −½·Γ·S²·σ² per year. A position cannot collect time decay without being short gamma, or own gamma without paying theta.
Vega measures exposure to implied volatility, which moves with fear and supply and demand for options, not only with the stock. It is largest for at-the-money and long-dated options.
Rho measures exposure to interest rates. Higher rates raise call values and lower put values, because they reduce the present value of the strike.
Time to expiry reshapes all of them. For the at-the-money call:
| Days to expiry | Value | Delta | Gamma | Theta/day | Vega/pt |
|---|---|---|---|---|---|
| 7 | $0.85 | 0.5156 | 0.1919 | -$0.0619 | $0.0276 |
| 45 | $2.22 | 0.5396 | 0.0754 | -$0.0259 | $0.0697 |
| 180 | $4.66 | 0.5789 | 0.0371 | -$0.0141 | $0.1373 |
Short-dated options are gamma and theta trades; long-dated options are vega trades. The Greeks vs price chart shows the same pattern across stock prices: the gamma peak around the strike gets taller and narrower as expiry nears.
Estimating P&L from the Greeks
The Greeks are the terms of a Taylor expansion of the option value. Over a short horizon:
For the example position, a 5% rise to $52.50 over 7 days gives an estimate of $518.91 against $507.09 from repricing the option exactly, an error of $11.81. The gap grows with the size of the move:
| Stock move | Greeks estimate | Full repricing | Error |
|---|---|---|---|
| 2% | $128.76 | $119.80 | $8.97 |
| 5% | $518.91 | $507.09 | $11.81 |
| 20% | $3,701.15 | $3,494.86 | $206.29 |
| -15% | -$531.29 | -$671.18 | $139.89 |
The estimate ignores higher-order effects: delta drifts as time passes (charm), gamma changes with the stock price (speed) and with time (color), and vega changes with the stock price (vanna) and with volatility (volga). For large moves the quadratic also has the wrong shape. A long call cannot lose more than its premium, but the parabola turns back up, so on a big fall the estimate understates the loss; on a big rise it overstates the gain, because delta cannot climb above 1 while the parabola keeps steepening. The vega term handles volatility changes the same way: a 5% fall with IV up 4 points gives -$327.34 against -$354.20.
Risk systems use this expansion for quick, aggregated estimates across many positions and switch to full repricing for stress tests.
What this calculator does not include
- American exercise. The Greeks are European. For American puts and for calls before an ex-dividend date, early exercise changes delta and theta, most noticeably deep in the money.
- One volatility per option. Each strike’s Greeks use the volatility you enter. In practice implied volatility differs by strike (the skew), and when the stock moves the IV of a given strike usually moves too, so real deltas differ from the flat-volatility figure.
- Discrete dividends and rate curves. Dividends are a continuous yield and the rate is flat.
- Multi-leg strategies. Each run covers one option series; add the position Greeks of each leg to get a spread’s totals.
- Costs. Commissions, the bid-ask spread and margin are not included.