Black-Scholes model

The Black-Scholes model is a mathematical formula for estimating the fair price of a European-style option — one that can only be exercised at expiry — based on the underlying asset's current price, the option's strike price, time to expiry, the risk-free interest rate, and volatility. Published by Fischer Black and Myron Scholes in 1973, with related work by Robert Merton, it remains the reference point for options pricing even where more advanced models are used in practice.

The model assumes the underlying's price follows a continuous random walk with constant volatility, that markets have no transaction costs or arbitrage opportunities, and that volatility and interest rates are known and constant over the option's life. Its output feeds directly into the Option Greeks, each a partial derivative of the pricing formula. Run in reverse, solving for the volatility that reproduces the market price instead of solving for price, it is also how implied volatility is calculated.

Black-Scholes matters because it gave options markets a common pricing language and remains a baseline every other model, including those used in quantitative trading, is compared against. Its best-known limitation is its constant-volatility assumption: real markets show volatility that varies by strike and expiry, the "volatility smile," which the model cannot explain and which practitioners work around with adjustments or alternative models.