Black-Scholes Option Pricing Calculator
Compute theoretical European call and put prices, d1, d2 and the main Greeks, with a continuous dividend yield.
Quick answer
The Black-Scholes-Merton model prices European options from spot, strike, time to expiry, volatility, risk-free rate and dividend yield. Call and put values come from d1, d2 and the normal distribution N(d1), N(d2).
Black-Scholes Option Pricing Calculator
Call Price
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Put Price
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d1
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d2
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Put-Call Parity Difference
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Educational only, not trading or investment advice. Theoretical estimates, not market quotes.
Greeks
| Greek | Call | Put |
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Assumptions & method
- d1 = [ln(S/K) + (r − q + σ²/2) T] ÷ (σ√T); d2 = d1 − σ√T. T = days ÷ 365.
- Call = S e^(−qT) N(d1) − K e^(−rT) N(d2). Put = K e^(−rT) N(−d2) − S e^(−qT) N(−d1).
- Delta (call) = e^(−qT) N(d1); delta (put) = −e^(−qT) N(−d1). Gamma = e^(−qT) φ(d1) ÷ (Sσ√T), the same for both.
- Theta is per calendar day (annual theta ÷ 365). Vega and rho are per 1 percentage point change.
- N is the standard normal CDF (a close numerical approximation) and φ is the normal density.
- European exercise, constant volatility and rates, continuous dividend yield. Not included: early exercise, discrete dividends, skew, fees and taxes.
- Educational only; this is not trading or investment advice.
Results are estimates based on the figures you enter. They are not guarantees and not financial advice.
How Black-Scholes pricing works
The Black-Scholes-Merton formula prices a European option from five things: the underlying price, the strike, the time left, the volatility of the underlying and the risk-free rate, plus a continuous dividend yield. The call is the discounted expected payoff in a risk-neutral world: spot × e^(−qT) × N(d1) − strike × e^(−rT) × N(d2). The put follows the same logic. Volatility is the one input you cannot observe directly, so it is the main source of error. The Greeks show how the value changes when each input moves a little.