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Black–Scholes European Option Calculator

Calculate European option values using the Black–Scholes–Merton model and your supplied assumptions. The model uses continuous dividend yield and does not price American exercise or discrete cash dividends.

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Your numbers

%
Annual standard deviation; not a price-growth forecast.
%
Use a rate consistent with the horizon.
%
Proportional dividend yield, not a cash dividend.
Display currency only. Convert all inputs to this currency before calculation.
Your figures stay on this device.

How the calculation works

Black–Scholes–Merton values European call and put options under a continuous-time model. The inputs describe a theoretical price distribution and discounting convention; the output is not a broker quote or a forecast of the actual payoff.

Call = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2); put follows put–call parity. Zero time and zero volatility use their exact limits.

  1. Set the option horizon

    Time is entered in years. Spot and strike must use the same price units, and the result is per underlying unit rather than automatically per contract.

  2. Weight the discounted values

    The model combines discounted underlying value and discounted strike using normal-distribution weights derived from price, volatility, rates and time.

  3. Handle limiting cases

    At expiry, only intrinsic value remains. With zero volatility, the calculation uses the deterministic discounted-value limit instead of dividing by a zero volatility term.

Worked example: Check European call value

Sample inputs demonstrate the method. They are not a forecast or product offer.

The first example sets spot and strike to 100 with one year remaining, 20% volatility, a 5% continuous rate and no dividend yield. Those assumptions give a European call value of approximately 10.450584 per unit.

Underlying spot price
100
Strike price
100
Time to expiry (years)
1
Annual volatility (%)
20
Continuously compounded risk-free rate (%)
5
Continuous dividend yield (%)
0
Currency for all amounts
Euro (EUR)

Expected result

European call value
€10.450584
Worked example: Check European put value

Sample inputs demonstrate the method. They are not a forecast or product offer.

Underlying spot price
100
Strike price
100
Time to expiry (years)
1
Annual volatility (%)
20
Continuously compounded risk-free rate (%)
5
Continuous dividend yield (%)
0
Currency for all amounts
Euro (EUR)

Expected result

European put value
€5.573526

Understanding your result

Call and put values answer different payoff questions. The difference between them is consistent with the same discounted spot and strike assumptions used in the model.

American early exercise, discrete cash dividends and contract-specific features require another model. Changing volatility cannot substitute for a missing exercise convention.

Questions you might have

What does the result include?

Estimate European call and put values from a spot price, strike, volatility and continuously compounded rates.

Which limits should I check?

European exercise only; theoretical values are not executable market prices.

Choosing the right inputs

Annual volatility
Enter annual standard deviation as a percentage, not an expected annual price gain. A 20% volatility input is distinct from 20% expected growth.
Continuously compounded risk-free rate
Use a continuously compounded annual rate consistent with the option horizon. A quoted effective rate may need conversion.
Continuous dividend yield
This is a continuous proportional dividend yield. It does not represent a specific cash dividend paid on a known date.

Common mistakes to avoid

  • Do not multiply a contract's total premium and a per-share strike in the same calculation.
  • Do not treat the model's probability weights as your forecast of the stock's future return.

Assumptions to check

  • European exercise only; theoretical values are not executable market prices.
  • All calculations run on your device. Figures are scenarios based on your inputs, not investment recommendations.

Sources and method

Read our calculation and testing approach.