Black-Scholes Options Pricer
Price European call and put options and calculate the full set of Greeks.
How this is calculated
Black-Scholes-Merton prices a European option as the present value of its expected payoff under a risk-neutral probability measure. d₁ and d₂ are intermediate values derived from the ratio of spot to strike, the risk-free rate, volatility, and time to expiry; the standard normal CDF applied to them gives the probability-weighted terms that make up the call and put prices.
The risk-free rate input is pre-filled from the Yield Curve Calculator if you arrive via that tool. For the required return on equity used in option pricing models, see the CAPM Calculator.
Frequently asked questions
- What is the Black-Scholes model?
- A mathematical model, published by Fischer Black and Myron Scholes in 1973 (with contributions from Robert Merton), for pricing European-style options from five inputs: spot price, strike price, time to expiry, risk-free rate, and volatility. It remains the industry-standard starting point for options pricing.
- What are the Greeks?
- Sensitivities of the option price to each input: delta (price vs. spot), gamma (delta vs. spot), vega (price vs. volatility), theta (price vs. time decay), and rho (price vs. interest rates). Traders use them to understand and hedge risk in an options position.
- What does implied volatility mean?
- The volatility input above is your estimate. "Implied volatility" is the reverse — the volatility figure that, plugged into this same formula, would make it output the option's actual observed market price. This calculator prices forward from a volatility estimate; it does not back out implied volatility from a market price.
- Can I use this for American options?
- Not directly. Black-Scholes prices European options, which can only be exercised at expiry. American options, which can be exercised any time before expiry, require a different model (e.g. a binomial tree) to properly value the early-exercise premium — though for non-dividend-paying stocks, American and European call prices are theoretically identical.
- Can I use this in my own app?
- Yes — every calculator on Stupidly Clever has a matching REST API and MCP tool that runs the same underlying logic.