Options Pricing Methodology

The Black-Scholes Model

Our options pricing calculators are built on the Black-Scholes-Merton model, the industry standard for European-style options valuation. The model calculates the theoretical fair value of an option based on five key inputs.

Key Inputs

• S - Current stock price
• K - Strike price of the option
• T - Time to expiration (in years)
• r - Risk-free interest rate
• σ - Volatility of the underlying asset

The Greeks

Our calculators derive all major Greeks from the Black-Scholes framework:

Delta (Δ)

Measures the rate of change in the option price relative to a one-unit change in the underlying asset price. For calls, delta ranges from 0 to 1; for puts, from -1 to 0.

Gamma (Γ)

Measures the rate of change in delta relative to the underlying asset price. Gamma is highest for at-the-money options and decreases for deep in-the-money or out-of-the-money options.

Theta (Θ)

Measures the rate of time decay in the option price. As expiration approaches, theta typically accelerates, eroding option value more quickly.

Vega (v)

Measures the sensitivity of the option price to changes in implied volatility. A higher vega means the option price is more sensitive to volatility changes.

Rho (ρ)

Measures the sensitivity of the option price to changes in the risk-free interest rate. Generally has less impact than other Greeks.

Implied Volatility

Our implied volatility calculator uses an iterative Newton-Raphson method to back-solve the Black-Scholes equation, finding the volatility level that produces a theoretical price matching the observed market price.

Payoff Diagrams

All options strategy payoff diagrams show profit and loss at expiration based on the net premium paid or received. These diagrams assume options are held to expiration and do not account for early exercise, dividends, or transaction costs.

Limitations

The Black-Scholes model assumes constant volatility, log-normal distribution of returns, no transaction costs, continuous trading, and no dividends. In practice, these assumptions do not perfectly hold, which is why actual option prices may differ from theoretical values.