Black-Scholes model

The Black-Scholes model is a mathematical formula used to estimate what a fair price for an option should be. It was developed in the early 1970s and remains the reference point that options traders, market makers, and risk managers use to think about pricing, even when they rely on more refined variations of it in practice.

The model takes five main inputs, the price of the underlying asset, the option's strike price, the time remaining until expiration, the risk-free interest rate, and the expected volatility of the underlying asset, and combines them into a theoretical price for a call or put. Volatility is the input that matters most and the hardest to know in advance, since it is the only one of the five that has to be estimated rather than observed directly in the market. In practice, traders often run the model in reverse, taking the market price of an option as given and solving for the volatility figure that would justify it, which is how implied volatility is calculated.

The model rests on a set of simplifying assumptions, constant volatility, no dividends, frictionless trading, and a stock price that moves in a particular statistical pattern, none of which hold perfectly in real markets. That gap between theory and reality is well understood and shows up most clearly in the volatility smile, a pattern where options at different strike prices imply different volatility levels even though the model assumes volatility should be the same across all of them. Traders use Black-Scholes as a starting framework and a common language for quoting prices, not as an infallible predictor of where an option should trade.