Black-Scholes is a model, not the world. Three short sections: which assumptions routinely fail in real markets, what came before BS in 1973, and what came after to patch those failures.
Two violated assumptions
1. Constant volatility
The Black-Scholes model uses one σ for every option on a stock, assuming volatility is the same for all strike and expiration date combinations. However, in reality, this is not entirely accurate. For one, volatility will differ for short-term vs long-term options. For another, volatility differs for deep OTM options. Puts at those extremes are underpriced by BS compared to the market which hurts the seller in the case of a crash (which BS assumes is essentially impossible). ·
2. Continuous price paths (no jumps)
The second assumption is that price teleportation is impossible. For a stock to drop from $100 to $90, it must pass through $99, $98, $97… and every price in between. At each step, σ predicts how big the next move is — such that a large drop requires many such moves. Therefore, under BS, massive near-instant crashes are essentially impossible. In reality, these crashes do happen. Prices can teleport when news arrives between trades or when prices change overnight between market close and open. ·
A predecessor
Louis Bachelier's doctoral thesis at the Sorbonne derived the first equation for option prices. Stock prices were modeled as Brownian motion and option value was computed as the expected payoff under that distribution. His model differed from Black-Scholes in several ways, but the most crucial was the price process itself. Bachelier used arithmetic Brownian motion, where the price change () is Gaussian, which allowed the price to drift below zero. Black-Scholes switched to geometric Brownian motion and treated the log return as Gaussian, which kept the price strictly positive and matched the empirical lognormal distribution of stock returns. ·
A successor
The Heston model (1993) addresses the aforementioned Black-Scholes assumption of constant volatility. Heston treats variance itself as a random, mean-reverting process which drifts back toward a long-run level, thus giving volatility its own "volatility" while preventing it from running off to extremes. This better matches real markets, where vol spikes are temporary and equity indices show a persistent skew. The flexibility comes at the cost of five parameters to calibrate instead of BS's one, and harder hedging: because variance isn't directly tradeable, volatility exposure must be hedged via portfolios of other options rather than the underlying alone. ·