Overview
The Black-Scholes model, developed by Fischer Black, Myron Scholes, and Robert Merton, is a mathematical framework for pricing European-style options. This interactive tool calculates option prices and the Greeks (Delta, Gamma, Theta, Vega, Rho) in real-time as you adjust the input parameters.
Options Pricing Calculator
Input Parameters
Results
Black-Scholes Formula
The Black-Scholes formula for European options is based on several key assumptions:
- Constant risk-free interest rate
- Constant volatility of the underlying asset
- No dividends during the option's life
- European exercise (only at expiration)
- No transaction costs
Call Option Formula
C = S₀ × N(d₁) - K × e^(-rT) × N(d₂)
Put Option Formula
P = K × e^(-rT) × N(-d₂) - S₀ × N(-d₁)
Where:
- d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
- d₂ = d₁ - σ√T
- N(x) = cumulative standard normal distribution function
The Greeks
The Greeks measure the sensitivity of option prices to various factors:
Delta (Δ)
Measures the rate of change of option price with respect to the underlying asset price. For calls: N(d₁), for puts: N(d₁) - 1.
Gamma (Γ)
Measures the rate of change of delta with respect to the underlying asset price. φ(d₁) / (S₀ × σ × √T)
Theta (Θ)
Measures the rate of change of option price with respect to time decay (time to expiration).
Vega (ν)
Measures the rate of change of option price with respect to volatility. S₀ × φ(d₁) × √T
Rho (ρ)
Measures the rate of change of option price with respect to the risk-free interest rate.