Black-Scholes Calculator

A mathematical model for pricing European-style options (exercisable only at expiration), developed by Fischer Black, Myron Scholes, and Robert Merton. It calculates a theoretical fair value based on five inputs: the underlying asset's price, the strike price, time to expiration, the risk-free interest rate, and the asset's volatility.

Call price = S·N(d1) − K·e^(−rT)·N(d2). Put price = K·e^(−rT)·N(−d2) − S·N(−d1). Where d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T), and d2 = d1 − σ√T. S is the stock price, K the strike, T time to expiration in years, r the risk-free rate, σ (sigma) the volatility, and N() the cumulative standard normal distribution.

The annualized standard deviation of the underlying asset's returns — a measure of how much the price is expected to fluctuate. Higher volatility increases both call and put option values, since it raises the chance of a large favorable price move before expiration.

Black-Scholes assumes constant volatility and interest rates, no dividends, European-style exercise only (not American-style, which allows early exercise), and frictionless markets with no transaction costs — real option prices often deviate from the model, especially for American options or assets with significant dividends.

No — this is the standard non-dividend-paying Black-Scholes formula. For dividend-paying stocks, the actual theoretical price is somewhat lower than this calculator shows for a call (and somewhat higher for a put), since expected dividends reduce the stock's expected future price.