Black-Scholes Option Pricing Calculator
Price European calls and puts with the Black-Scholes model, get all five Greeks, and solve for implied volatility from a market price.
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Solve for implied volatility
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A closed form answer to an option price
The Black-Scholes model, published in 1973, gives an exact price for a European option under a specific set of assumptions: the underlying follows a geometric Brownian motion with constant volatility, there are no transaction costs, the risk free rate is constant, and the option can only be exercised at expiry. Those assumptions are all wrong in detail and the model is still the foundation of derivatives pricing, because it provides a common language and because its errors are understood.
The Greeks are the practical output
The price matters less than the sensitivities. Delta is how much the option price moves per unit move in the underlying, and it doubles as a hedge ratio. Gamma is how fast delta changes, which is why a hedged position needs rebalancing. Theta is the daily cost of holding an option as time decays, which is the single most underestimated number by new option buyers. Vega is sensitivity to volatility, and rho to interest rates. Traders manage positions by the Greeks, not by the price.
Implied volatility runs the model backwards
Every input except volatility is observable. Given a market price, the model can be inverted to find the volatility that would produce it, which is implied volatility. It is the market's forecast of future movement and it is the number options are actually quoted on. That implied volatility differs across strikes, the volatility smile, is direct evidence that the model's constant volatility assumption is false, and every options desk adjusts for it.
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Frequently Asked Questions
Does this work for American options?
Not exactly. Black-Scholes prices European options, exercisable only at expiry. For American calls on a non dividend paying stock the price is the same because early exercise is never optimal. American puts carry an early exercise premium this model does not capture.
What volatility should I enter?
Implied volatility from the option chain if you are pricing to market, or historical realised volatility if you are forming your own view. They differ, and the difference is the trade. Enter an annualised figure, so 30 means 30 percent a year.
Why is my calculated price different from the market?
Because the market is pricing in a different volatility, and because of dividends, borrow costs, bid-ask spread and the early exercise premium on American options. Solve for implied volatility to see what the market is actually assuming.
What does theta mean day to day?
It is the option value lost per calendar day, holding everything else constant. It accelerates as expiry approaches and is largest for at the money options, which is why buying short dated at the money options requires the move to happen quickly.
How are dividends handled?
Through a continuous dividend yield, which reduces the effective forward price of the underlying. That is a reasonable approximation for an index and less so for a single stock with a few discrete payments, where a discrete dividend model is more accurate.
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How to Use
Enter spot, strike, time to expiry, rate and volatility to price the option and see its Greeks.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
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