The Black-Scholes model, published in 1973 by Fischer Black, Myron Scholes, and Robert Merton (who won the Nobel Prize for it), is the foundation of modern options pricing. Before Black-Scholes, there was no rigorous way to determine what an option should cost. The model takes five inputs — stock price, strike price, time to expiration, risk-free interest rate, and volatility — and produces a theoretical fair value for a European-style option. The key insight is that you can construct a risk-free hedge by continuously adjusting a portfolio of the option and the underlying stock, which means the option must be priced so that no arbitrage opportunity exists.

The Black-Scholes formula assumes that stock prices follow geometric Brownian motion with constant volatility, that markets are frictionless (no transaction costs, continuous trading), and that returns are normally distributed. Every one of these assumptions is wrong in reality. Volatility is not constant — it clusters, spikes during crises, and varies across strikes (the volatility smile). Stock returns have fat tails — extreme moves happen far more often than a normal distribution predicts. The 2008 crash was a 25-sigma event under normal distribution assumptions — essentially impossible — yet it happened. These limitations don't make Black-Scholes useless, but they define where it breaks down.

Extensions and alternatives have addressed Black-Scholes limitations. The Heston model (1993) allows volatility itself to be random (stochastic volatility), capturing the volatility smile and the mean-reverting nature of vol. Jump-diffusion models (Merton, 1976) add sudden price jumps to the continuous diffusion process, better modeling crash risk and earnings gaps. The SABR model is standard in interest rate derivatives. Local volatility models (Dupire, 1994) derive a deterministic volatility function from market prices. Each model trades off complexity for accuracy in different ways, and major derivatives desks run multiple models simultaneously.

For the practical trader, Black-Scholes matters because it's the language of options markets. When you see an option's implied volatility, that's the volatility you'd need to plug into Black-Scholes to get the option's market price. The Greeks — delta, gamma, theta, vega — are all derived from Black-Scholes partial derivatives. Understanding the model's assumptions helps you understand where options might be mispriced: if you believe volatility will be higher than implied, buying options has positive expected value. If real-world returns have fatter tails than Black-Scholes assumes, out-of-the-money options are theoretically underpriced. This is how institutional vol traders think.