Black-Scholes Option Pricing Model

The Black-Scholes model is a mathematical formula used to determine the theoretical price of options contracts. This tool calculates option prices and various sensitivity measures known as "Greeks" based on your input parameters.

• Option Pricing: Calculate the fair market value of call and put options.
• The Greeks: Sensitivity measures that describe how option prices change with respect to different variables:
• Delta: Measures the rate of change of option price with respect to changes in the underlying asset price. Δ = e^(-qT) × N(d₁) [Call] Δ = -e^(-qT) × N(-d₁) [Put]
• Gamma: Rate of change of Delta with respect to changes in the underlying price. Γ = e^(-qT) × φ(d₁) / (S₀ × σ × √T)
• Theta: Measures option price sensitivity to the passage of time (time decay). Θ = (1/365) × [e^(-qT)Sq×N(d₁) - e^(-rT)Kr×N(d₂) - S₀e^(-qT)σφ(d₁)/(2√T)]
• Vega: Measures sensitivity to changes in volatility. ν = S₀ × e^(-qT) × √T × φ(d₁) / 100
• Rho: Measures sensitivity to interest rate changes. ρ = K × T × e^(-rT) × N(d₂) / 100

The model can be used for example to manage the effect of time passage (Theta) on option values, understand how portfolio value will change as the underlying asset price fluctuates (Delta), construct delta-neutral portfolios.

The charts below show how option prices and Greeks change in relation to time to expiration, volatility, and strike price, giving you a visual understanding of these relationships.