Black-Scholes Options Calculator

Pricing follows the original Black‑Scholes–Merton closed‑form solution under the assumptions of constant volatility, lognormal underlying returns, continuous dividend yield, and no early exercise (European options). Intermediate terms d1 and d2 are computed and passed into the standard normal cumulative and density functions.

Implied volatility is found by numerically solving for sigma that makes the Black‑Scholes model price equal to the observed market price. The solver uses bracketed root‑finding with optional Newton updates and reports status codes for convergence, out‑of‑bounds results, or invalid input.

Operational controls and accuracy practices: validate put-call parity, verify forward price consistency, and check implied vol against typical ranges. For implementation and numerical stability, follow established secure development and numeric testing practices per NIST and IEEE guidance.

This calculator provides analytic Black‑Scholes prices and Greeks and a solver for implied volatility with explicit solver status and model price validation.

Use sanity checks (put‑call parity, forward price, implied vol bounds) and follow cited standards for numerical reliability and secure implementation. Be mindful of the model's assumptions and limitations before using results for trading or regulatory reporting.

Example 1: S=100, K=100, T=0.5, r=3%, q=0, sigma=20% → computes call and put price and Greeks. Verify put-call parity: C - P = S*e^{-qT} - K*e^{-rT}.

Example 2: Given market call price 2.50 with same market inputs, use implied volatility mode to recover sigma. Check solver status; if the status reports nonconvergence, verify input price is within no-arbitrage bounds (between intrinsic value and forward price discounted).