Black-Scholes Calculator & Greeks
Theoretical price for European calls and puts, full Greek table, implied volatility solver, and interactive sensitivity curves.
Inputs
$
$
decimal, e.g. 0.045 = 4.5%
decimal, e.g. 0.25 = 25%
Solve implied volatility
$
Output
Theoretical price
—
Delta
—
Gamma
—
Theta /day
—
Vega /1% IV
—
Rho /1% r
—
Prob. ITM (Δ-adj)
—
Intrinsic / Extrinsic
—
Sensitivities
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What is Black-Scholes?
The Black-Scholes model is the foundational closed-form pricing equation for European options. Given spot, strike, time-to-expiry, risk-free rate, dividend yield, and volatility, it returns a theoretical fair value plus the partial derivatives traders call the Greeks: delta, gamma, theta, vega, and rho. The classic 1973 paper assumes lognormal returns, constant volatility, no transaction costs, and continuous trading — the standard textbook caveats. American-style options (most U.S. equity options) can differ from Black-Scholes prices because of early-exercise optionality, especially around dividends.
Why the Greeks matter
- Delta: change in option price for a $1 move in the underlying. Roughly the probability the option finishes ITM.
- Gamma: how fast delta changes. Highest at-the-money near expiry.
- Theta: time decay. Generally negative for long positions, positive for short.
- Vega: sensitivity to a 1-vol-point change in IV.
- Rho: sensitivity to a 1% change in the risk-free rate. Matters more on long-dated options.
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