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Black-Scholes Option Pricing Calculator
Find the theoretical value of a European call or put with the Black-Scholes-Merton model. Enter the stock price, strike, time to expiration, volatility, interest rate and dividend yield to get both prices, all five Greeks and the full working. For education only, not trading advice.
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Option
Highlight
$
Current price of the underlying.
$
Market inputs
%
Implied or historical volatility.
%
Annual, continuously compounded.
%
Annual continuous yield; 0 for none.
Enter your values to see the result.
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How to use the Black-Scholes Option Pricing Calculator
Enter the stock price and strike price of the option.
Enter the time to expiration and choose whether it is in calendar days, trading days, months or years.
Enter the annual volatility, risk-free rate and dividend yield as percentages.
Choose call or put to highlight it. Both prices are always shown.
Press Calculate to see the price, the Greeks, the working and how the value changes at other stock prices.
How the Black-Scholes-Merton model works
The model, published by Fischer Black and Myron Scholes in 1973 and extended by Robert Merton, values an option by assuming the stock follows a log-normal random walk with constant volatility. Under that assumption a portfolio of the stock and the option can be continuously hedged so it is risk-free, which leads to a closed-form price:
S is the stock price, K the strike, T the time to expiration in years, σ the annual volatility, r the continuously compounded risk-free rate and q the continuous dividend yield. N(d2) is the risk-neutral probability that a call finishes in the money; it is not a real-world forecast.
The Greeks at a glance
Greek
Call formula
Put formula
What it tells you
Delta
e−qTN(d1)
−e−qTN(−d1)
Option move per 1-unit stock move; also a rough hedge ratio
Gamma
e−qTn(d1) ÷ (Sσ√T)
How quickly delta changes; highest at the money near expiry
Vega
S e−qTn(d1)√T ÷ 100
Value change for a 1-point rise in volatility
Theta
Time decay, shown per calendar day (annual theta ÷ 365)
Usually negative for long options
Rho
K T e−rTN(d2) ÷ 100
−K T e−rTN(−d2) ÷ 100
Value change for a 1-point rise in rates
n() is the standard normal density. With S = K = 100, T = 1 year, σ = 20%, r = 5% and no dividend, the call is worth about 10.45 and the put about 5.57; the difference, 4.88, equals 100 − 100 × e−0.05, as put-call parity requires.
Tips for sensible inputs
Volatility drives the price. Doubling σ roughly doubles the value of an at-the-money option, so test a range of values rather than relying on one number.
Match the rate to the term. Use a Treasury bill or government bond yield with roughly the same maturity as the option.
Dividends lower calls and raise puts. For an index, use its current dividend yield; for a single stock with one known dividend, a yield is only an approximation.
Very short expiries are sensitive. With a few days left, small input changes can move gamma and theta a lot.
Frequently asked questions
What is the Black-Scholes formula?
For a call, C = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2), and for a put, P = K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1). Here d1 = [ln(S/K) + (r − q + σ²/2)T] ÷ (σ√T), d2 = d1 − σ√T and N() is the standard normal cumulative distribution function.
What inputs does the model need?
Six: the current stock price, the strike price, the time to expiration in years, the annualized volatility of the stock, the risk-free interest rate and the dividend yield. Volatility is the only one that cannot be read directly from the market, which is why traders often quote implied volatility instead.
Can I use it for American options?
Black-Scholes prices European options, which can only be exercised at expiration. For American calls on stocks that pay no dividend the value is the same, but American puts and calls on dividend-paying stocks can be worth slightly more because of the early-exercise right.
What do the Greeks mean?
Delta is the change in option price for a 1-unit move in the stock, gamma is how fast delta changes, theta is the value lost per day as time passes, vega is the change for a 1-point change in volatility, and rho is the change for a 1-point change in interest rates.
Should I use calendar days or trading days?
Either works as long as volatility is measured on the same basis. Annual volatility is usually computed from 252 trading days, so many traders divide trading days by 252; dividing calendar days by 365 is also common. The theta shown here is per calendar day.
Why is the market price different from the Black-Scholes price?
The model assumes constant volatility and interest rates, no transaction costs and log-normally distributed prices. Real markets have volatility smiles, jumps, bid-ask spreads and early exercise, so the quoted price reflects the market's implied volatility rather than the number you entered.
Is this calculator investment advice?
No. It shows a theoretical model value for learning and analysis. Options are risky and can expire worthless; nothing on this page is a recommendation to buy or sell anything.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.