Black-Scholes Calculator: Option Price and Greeks

European option pricing model

Black-Scholes Calculator

Estimate a European call or put price, d1, d2 and five option Greeks with continuous dividend yield.

The model does not price early exercise.
Formatting only; no currency conversion is performed.
Current price of the stock or other modeled underlying.
Exercise price in the same currency as the underlying.
Enter calendar days or years according to the next field.
Days are divided by 365 to obtain model years.
Annualized standard deviation, not a daily percentage.
Continuously compounded rate; negative values are accepted.
Continuous yield. Enter 0 for a non-dividend-paying underlying.

Black-Scholes-Merton assumes a European option, constant volatility and rates, continuous trading and lognormal prices. It is a model estimate, not a market quote.

Black-Scholes formula with dividends

d1 = [ln(S ÷ K) + (r − q + σ² ÷ 2)T] ÷ (σ√T)

d2 = d1 − σ√T

Call = S·e−qT·N(d1) − K·e−rT·N(d2)

Put = K·e−rT·N(−d2) − S·e−qT·N(−d1)

S is the current underlying price, K is the strike, T is time in years, r is the continuously compounded risk-free rate, q is the continuous dividend yield, σ is annual volatility and N is the standard normal cumulative distribution.

Verified examples

At-the-money one-year call

With S = 100, K = 100, T = 1 year, volatility 20%, risk-free rate 5% and dividend yield 0%, the European call is approximately 10.4506. The matching put is 5.5735, d1 is 0.3500 and d2 is 0.1500.

At-the-money one-year put

Using the same assumptions and selecting put gives approximately 5.5735. Call-put parity is satisfied: call − put = S − K·e−rT.

How to interpret the Greeks

  • Delta estimates the option-value change for a one-unit underlying move.
  • Gamma estimates how much delta changes for a one-unit underlying move.
  • Vega is displayed per one percentage-point increase in annual volatility.
  • Theta is displayed per calendar day passing, with other inputs unchanged.
  • Rho is displayed per one percentage-point increase in the annual rate.

Limits and assumptions

  • The model is for European options. American exercise features are not valued.
  • Dividend yield is continuous; discrete dividend dates and amounts are not modeled.
  • Volatility, interest rate and dividend yield remain constant through expiration.
  • The model excludes spreads, commissions, taxes, liquidity, assignment risk and volatility smiles.
  • Spot and strike support six decimals; percentages and time support four decimals. Time is limited to 100 years and volatility to 1,000%.
  • The risk-neutral finish probability is N(d2) for a call finishing above strike or N(−d2) for a put finishing below strike. It is model-implied, not a forecast.

Risk note: Small changes in volatility, time and rates can materially change the estimate. Real options may trade far from Black-Scholes values. This calculator is educational and is not investment advice.

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