How to use the black-scholes calculator
- Enter the stock price and the strike price of the option you want to price.
- Enter the time to expiry and pick years, months or days. The tool converts it to years for the formula.
- Enter the yearly volatility and the risk-free rate. Both shown here are examples, so replace them with the figures you want to test. Open More options to add a dividend yield.
- Read the call and put prices, then the table of Greeks that shows how each price changes when an input moves.
Formula
d1 = (ln(S / K) + (r - q + vol^2 / 2) x T) / (vol x sqrt(T))
d2 = d1 - vol x sqrt(T)
Call = S x e^(-qT) x N(d1) - K x e^(-rT) x N(d2)
Put = K x e^(-rT) x N(-d2) - S x e^(-qT) x N(-d1)
S is the stock price, K the strike, T the time in years, r the continuously compounded risk-free rate, q the continuous dividend yield and N the standard normal cumulative distribution. The calculator evaluates N with a high-precision series, then checks the result against put-call parity.
A one-year at-the-money option
Take a $100 stock, a $100 strike, one year to expiry, 20% volatility and a 5% rate. Then d1 = (0 + (0.05 + 0.02) x 1) / 0.2 = 0.35 and d2 = 0.15. The normal values are N(0.35) = 0.6368 and N(0.15) = 0.5596, so the call is 100 x 0.6368 - 100 x e^(-0.05) x 0.5596 = $10.45. The put comes out at $5.57.
Parity ties them together: call minus put should equal S minus K x e^(-rT), which is 100 - 95.12 = $4.88. Here 10.4506 - 5.5735 = 4.8771, so the two prices are consistent. A second case, a $50 stock with a $55 strike, 6 months, 35% volatility, a 3% rate and a 2% dividend yield, gives a call of $3.10 and a put of $7.78.
What the Greeks show
Delta is how much the option price moves for a $1 move in the stock, so a call delta of 0.64 means about 64 cents. Gamma is how fast delta itself changes. Vega is shown per one percentage point of volatility, theta per day of time passing, and rho per one percentage point of the interest rate. Theta is negative for most options because time value drains away as expiry gets closer.
Where the model stops being true
The model prices European options, which can be exercised only at expiry. It assumes volatility and interest rates stay constant, prices move without jumps and there are no trading costs or taxes. Real markets break each of these, which is why market prices imply different volatilities at different strikes. American options, which can be exercised early, are worth at least as much as the European price and need a different method.
Use the page to learn how the five inputs interact. To see how an option position pays off at expiry, use the Options Profit Calculator. To value the stock itself from dividends, try the Dividend Discount Model Calculator.
Assumptions and limits
- The option is European, the stock follows geometric Brownian motion and volatility is constant.
- The interest rate and dividend yield are continuously compounded and constant until expiry.
- Time is converted with 12 months or 365 days to a year.
- Nothing is uploaded. All arithmetic runs in your browser, and the result is for learning, not trading advice.
Frequently asked questions
What volatility should I enter?
Enter the yearly standard deviation of the stock's returns that you want to assume. Historical volatility measures what happened, and implied volatility is what option prices suggest. The model is very sensitive to this one input, so try a range of values.
Why is the put cheaper than the call here?
With a positive interest rate and no dividends, the stock is expected to drift upward, which raises the call and lowers the put. Put-call parity ties them: call minus put equals the stock price minus the discounted strike.
Can I use this for American options?
Not exactly. This is the European formula. For a call on a stock with no dividends the early exercise right adds nothing, but for puts and for dividend-paying stocks an American option can be worth more.
What does a negative theta mean?
It is the amount the option loses in value each day if nothing else changes. The page shows theta per calendar day, found by dividing the yearly figure by 365.