Implied volatility

Implied volatility is the value of the volatility parameter σ that makes the Black-Scholes model match the observed market price of an option. It is not the historical volatility of the underlying asset, nor is it a forecast of future volatility. It is a market-determined number that encodes everything the Black-Scholes model leaves out: the market's expectation of future volatility, its assessment of tail risk, its demand for options as insurance, and its pricing of liquidity and uncertainty.

The implied volatility surface — a three-dimensional plot of implied volatility against strike price and time to maturity — typically exhibits a volatility smile or volatility skew: implied volatility is higher for deep out-of-the-money puts than for at-the-money options. This pattern directly contradicts the Black-Scholes assumption that volatility is constant. The smile is the market's way of saying that the model's assumptions are wrong, and that the price of insurance against catastrophic events is higher than the model predicts.

Implied volatility has become the standard language for quoting options: traders do not quote prices; they quote volatilities. This convention reveals that the Black-Scholes model, despite its simplifying assumptions, has become the infrastructure of options markets. The model is not used because it is true; it is used because it provides a common coordinate system — a lingua franca that makes markets possible.