Black-Scholes Option Calculator

Estimate European call and put values plus option Greeks from stock price, strike price, time to expiry, risk-free rate, volatility, and dividend yield. This is an educational model benchmark, not a trading recommendation.

How to use this calculator

Enter the current stock price, strike price, time to expiry in years, risk-free rate, volatility, and dividend yield. The output includes call value, put value, call delta, put delta, gamma, vega, daily theta, and rho. For related return projections, compare assumptions with the compound growth calculator or future value of annuity calculator.

Black-Scholes Formula

The standard Black-Scholes formulas for a non-dividend-paying European call and put are:

C = S₀N(d₁) - Ke⁻ʳᵗN(d₂)
P = Ke⁻ʳᵗN(-d₂) - S₀N(-d₁)

Where:
d₁ = [ln(S₀/K) + (r + σ²/2)t] / (σ√t)
d₂ = d₁ - σ√t

In these formulas, S₀ is the current stock price, K is the strike price, r is the continuously compounded risk-free rate, σ is volatility, t is time to expiration in years, N() is the cumulative standard normal distribution, and N'() is the standard normal probability density function.

Greeks Formulas

Delta (Δ):
Call: N(d₁)
Put: N(d₁) - 1

Gamma (Γ):
N'(d₁) / (S₀σ√t)

Vega (ν):
S₀√t × N'(d₁)

Theta (Θ):
Call: -S₀N'(d₁)σ/(2√t) - rKe⁻ʳᵗN(d₂)
Put: -S₀N'(d₁)σ/(2√t) + rKe⁻ʳᵗN(-d₂)

Rho (ρ):
Call: Kte⁻ʳᵗN(d₂)
Put: -Kte⁻ʳᵗN(-d₂)

Example and limits

With stock and strike both at 100 dollars, one year to expiry, 5 percent risk-free rate, 20 percent volatility, and no dividend yield, the model estimates a call near 10.45 dollars and a put near 5.57 dollars. Market prices can differ because of bid-ask spreads, early exercise features, discrete dividends, and changing implied volatility. For simpler rate math, see the interest calculator.