Options Calculator (Black-Scholes)
Price a European call or put with the Black-Scholes-Merton model and get every Greek — delta, gamma, vega, theta and rho — instantly. Enter the contract details below and the numbers update as you type.
| Greek | Call | Put |
|---|---|---|
| Delta per $1 move in S | 0 | 0 |
| Gamma delta change per $1 | 0 | |
| Vega per +1% volatility | 0 | |
| Theta per day, time decay | 0 | 0 |
| Rho per +1% rate | 0 | 0 |
How it works
This calculator uses the Black-Scholes-Merton formula, the closed-form solution for pricing European options. With dividend yield q, risk-free rate r, volatility σ and time to expiry T (in years), it computes two probability-style terms:
d₂ = d₁ − σ√T
Call = S·e−qT·N(d₁) − K·e−rT·N(d₂) · Put = K·e−rT·N(−d₂) − S·e−qT·N(−d₁)
N(x) is the cumulative standard normal distribution (we approximate it with the Abramowitz & Stegun 7.1.26 rational formula). An option's value splits into intrinsic value — what it would be worth if exercised today — and time value, the extra worth paid for the chance it moves further into the money before expiry.
The model assumes constant volatility, lognormal price moves and exercise only at expiry. Real markets show volatility "smiles" and American-style early exercise, so treat these as fair-value estimates, not guaranteed market quotes.
The Greeks, briefly
Calls profit when the underlying rises; puts profit when it falls. The Greeks tell you how the price reacts:
Delta
Price change per $1 move in the underlying. Call delta runs 0–1, put delta −1–0. Roughly the option's "share equivalent."
Gamma
How fast delta itself changes. Highest at the money and near expiry — that is where hedges need the most rebalancing.
Vega
Price change per 1 percentage-point change in volatility. Long options are long vega: they gain when volatility rises.
Theta
Daily time decay. Usually negative for buyers — each day that passes, all else equal, the option loses a little value.
Rho
Price change per 1 percentage-point change in interest rates. Matters most for long-dated options; small for short ones.