Black-Scholes Options Pricing Calculator
The Black-Scholes model estimates the theoretical price of European-style options using five key inputs. It also computes the Greeks — sensitivities that measure how the option price changes relative to each variable. All calculations run in your browser.
The Greeks
Worked Example
Stock at $100, strike $100, 30 days, 5% rate, 20% volatility:
- T = 30/365 = 0.0822 years
- d₁ = [ln(100/100) + (0.05 + 0.2²/2) × 0.0822] / (0.2 × √0.0822) = 0.102
- d₂ = 0.102 − 0.2 × √0.0822 = 0.045
- Call = 100 × N(0.102) − 100 × e⁻⁰⋅⁰⁵ˣ⁰⋅⁰⁸²² × N(0.045) ≈ $2.28
- Put = 100 × e⁻⁰⋅⁰⁵ˣ⁰⋅⁰⁸²² × N(−0.045) − 100 × N(−0.102) ≈ $1.87
How the Formula Works
The Black-Scholes formula calculates C = S·N(d₁) − K·e⁻ʳᵀ·N(d₂) for calls, and P = K·e⁻ʳᵀ·N(−d₂) − S·N(−d₁) for puts. N() is the standard normal CDF. The model assumes European exercise (no early exercise), constant volatility, continuous trading, and no dividends.
Important Limitation
Most U.S. listed equity options are American-style and can be exercised before expiration. The Black-Scholes model prices European options only. For American options, this calculator provides a theoretical reference — actual market prices may differ, especially for in-the-money options and dividend-paying stocks.
Common Errors
- Using calendar days instead of trading days: Time to expiration should be in years (calendar days ÷ 365), not trading days.
- Confusing volatility with daily moves: A 20% implied volatility means ~20% annualized, not 20% per day.
- Forgetting dividends: The basic model assumes no dividends. For dividend-paying stocks, the forward price should be adjusted.
- Using the risk-free rate as 0%: Even a small rate affects deep in-the-money options. Use the Treasury yield matching your expiration.
Methodology: Calculation uses the standard Black-Scholes-Merton formula with the cumulative normal distribution approximated via the Abramowitz and Stegun polynomial. All math runs client-side. Last reviewed: April 2026.