What is the Black-Scholes formula?

The Black-Scholes model is a closed-form solution for European option prices derived from no-arbitrage assumptions. It computes theoretical fair value from spot price, strike, time to expiration, risk-free rate, and volatility. See our methodology page for the exact formulas we use.

What is delta in options?

Delta measures how much the option price changes when the underlying stock moves $1. Call deltas range from 0 to +1; put deltas range from 0 to −1. Delta also serves as a rough proxy for the probability the option finishes in-the-money.

What is theta and how does time decay work?

Theta measures how much the option loses per day from time decay. Theta accelerates as expiration approaches, especially for at-the-money options. Premium sellers benefit from theta; premium buyers lose to it. The Black-Scholes calculator shows theta per day so you can quantify the daily decay on any contract.

What is implied volatility (IV)?

Implied volatility is the volatility figure that, when plugged into the Black-Scholes formula, returns the market price of the option. It represents the market's forecast of future volatility. IV expands and contracts independently of the underlying price.

Black-Scholes calculator and Greeks — price, delta, gamma, theta, vega, IV solver

Theoretical price for European calls and puts, full Greek table, implied volatility solver, and interactive sensitivity curves.

Inputs

Option type

Spot price (S)

Strike (K)

Days to expiry

Risk-free rate (r)

decimal, e.g. 0.045 = 4.5%

Implied volatility (σ)

decimal, e.g. 0.25 = 25%

Dividend yield (q)

Solve implied volatility

Market price

Implied volatility: —

Results & Greeks

Theoretical price

—

Delta

—

Δ stock = $1

Gamma

—

Δ of Δ

Theta

—

per day

Vega

—

per 1 vol point

Rho

—

per 1% rate

Probability ITM

—

Risk-neutral

Inputs

Solve implied volatility

Results & Greeks

Sensitivity