Black–Scholes

About this tool

The Black–Scholes formula gives the price of a European call or put from the stock price, strike, time to maturity, interest rate and volatility. This visualizer plots the price and the Greeks — delta, gamma, vega and theta — across stock prices and maturities, in 2D curves and a 3D surface, so you can see how sensitivities change as an option moves in or out of the money.

Frequently asked questions

Learn the theory

• The Black–Scholes formula — How the formula is obtained, what d₁ and d₂ mean, and what the model leaves out.
• Delta and hedging — Delta as a sensitivity and as a hedge ratio; gamma and the error from discrete rebalancing.
• Put–call parity — Two portfolios with the same payoff give a model-free link between call and put prices.
• Kolmogorov equations and the Feynman–Kac formula — The generator of a diffusion, the backward equation, and how an expectation becomes the solution of a PDE.
• Multi-step trees and backward induction — Recombining trees, the CRR parametrisation, pricing node by node and convergence to Black–Scholes.
• The Greeks: gamma, vega, theta and rho — How an option's value responds to the stock, volatility, time and rates; gamma and theta as two sides of hedging.
• Options on dividend-paying stocks, indices, currencies and futures — Black–Scholes with a continuous yield, the forward price as the common thread, Black-76 for futures and Garman–Kohlhagen for currencies.
• Implied volatility and the volatility smile — Inverting Black–Scholes for σ, vega, and what the smile and the term structure say about the model.
• Static replication: Breeden–Litzenberger and Carr–Madan — The risk-neutral density from call prices, and building any payoff from bonds, forwards and options.