Educational tool, not financial advice. Option trading carries substantial risk.
What the model does
The Black-Scholes model gives the theoretical fair value of a European option — one exercisable only at expiry — from five inputs: the spot price, the strike, the risk-free rate, the volatility, and the time remaining. It assumes the underlying follows a lognormal random walk with constant volatility, no dividends, and frictionless trading. Real markets violate every one of those assumptions, most visibly through the volatility smile: options far from the money trade at higher implied volatilities than the model says they should. Even so, Black-Scholes is the shared language of options pricing, and its Greeks are how every desk measures risk.
The Greeks
| Greek | Measures the change in option value per… |
|---|---|
| Delta | 1 unit move in the underlying (also ≈ the hedge ratio) |
| Gamma | 1 unit move in delta itself — how fast the hedge drifts |
| Vega | 1 percentage point of volatility |
| Theta | one day of time passing (usually negative — decay) |
| Rho | 1 percentage point of interest rate |
This calculator quotes theta per day and vega and rho per 1% because that is how they are used at the desk — a raw per-year theta or per-unit-vol vega is awkward to reason about.
Worked example
A one-year at-the-money option with spot and strike both 100, a 5% rate, and 20% volatility is worth 10.45 for the call and 5.57 for the put. The call delta is 0.64 — it moves about 64 cents per dollar of the underlying — and gamma is 0.0188, so that delta itself shifts as the stock moves. Vega is 0.375, meaning each extra point of implied volatility adds about 37 cents to the option.
Implied volatility
Run the model backwards and it answers a different question: given the price the market is actually paying, what volatility does that imply? There is no closed form, so the calculator solves it by bisection — narrowing a range until the model price matches the market price. A price outside the no-arbitrage band (below intrinsic value, or above the underlying) has no solution, and the tool says so rather than returning a meaningless number.
Related tools: the Kelly criterion calculator handles position sizing once you have an edge, and the APR/APY calculator covers the continuous compounding behind the risk-free rate.