Options Greeks Calculator
Black-Scholes European option pricing plus all 5 Greeks (Delta, Gamma, Theta, Vega, Rho) computed live. Dividend yield supported.
Option prices
Greeks
| Greek | Call | Put | Interpretation |
|---|---|---|---|
| Delta (Δ) | 0.5343 | -0.4657 | Price change per $1 move in underlying |
| Gamma (Γ) | 0.046213 | Delta change per $1 move (same for call/put) | |
| Theta (Θ) per day | -0.0631 | -0.0508 | Time decay per calendar day |
| Vega (ν) per 1% IV | 0.1140 | Price change per 1% IV change (same call/put) | |
| Rho (ρ) per 1% rate | 0.0409 | -0.0409 | Price change per 1% interest rate change |
Methodology
This tool uses the standard Black-Scholes-Merton formula for European option prices. The cumulative normal distribution is computed via the Abramowitz & Stegun 7.1.26 approximation (error < 7.5e-8). Theta is shown per day, Vega per 1% IV move, Rho per 1% rate move. Dividend yield assumes continuous payment.
Limits
Model assumes European exercise (expiration-only). American options (most single-stock US options) may differ, especially calls on dividend payers. For precise American pricing use a binomial tree or Monte Carlo.
FAQ
What exactly does Delta represent?
Delta measures how much the option price changes per $1 move in the underlying. Call Delta ranges 0 to 1 (deep ITM ≈ 1), put Delta ranges -1 to 0 (deep ITM ≈ -1). It also approximates the risk-neutral probability the option finishes in-the-money. A 0.50 Delta call has roughly a 50% chance of finishing ITM under Black-Scholes assumptions.
Why is Theta negative?
Theta measures how much an option loses per calendar day, all else equal. For option buyers, Theta is always negative — time decay works against you. ATM options have the largest Theta, deep ITM/OTM options have smaller Theta. This is why covered-call and cash-secured-put sellers specifically earn "time premium."
Why are Vega and Gamma the same for calls and puts?
Because IV affects both sides symmetrically — high implied volatility makes both calls and puts more expensive. The formula is Vega = S × e^(-qT) × N'(d1) × √T, identical for calls and puts. Same logic applies to Gamma (rate of Delta change). Both Greeks are insensitive to direction.
Is Black-Scholes accurate for American options?
For non-dividend-paying underlyings, BS prices are very close to American option values (difference typically <1%). For dividend-paying stocks, American calls may be worth more than BS predicts (due to early-exercise value around ex-div dates). Most US single-stock options are American; index options (SPX, NDX) are European and BS applies directly. For precise American pricing, use a binomial tree instead.
Should I use Gamma to hedge?
Gamma hedging (dynamic delta-hedging) is standard for market makers and professional portfolio traders. Retail traders should focus on portfolio-level Gamma exposure — short Gamma positions (sold options) experience non-linear loss amplification on large moves. If you're short a batch of ATM options, the Gamma number tells you how fast Delta shifts per $1 move — that's the hidden risk to watch.
Where do I get implied volatility input?
Standard sources: TradingView option chain (free, 15-min delayed), CBOE.com official, Barchart (free tier), IBKR TWS (paid, real-time). Enter annualized IV as a percentage (e.g. 30 means 30% annualized). Different brokers may show slightly different IV because they correspond to different tenor benchmarks — the differences are usually small for standard expirations.
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