volatilities

Volatility refers to the degree of variation in the price of a financial asset over time, typically measured by the standard deviation of returns. It is a statistical measure of dispersion and a common proxy for risk: higher volatility implies larger price movements and greater uncertainty, while lower volatility implies more stable prices. In finance, volatility is usually annualized and expressed as a percentage.

Historical, or realized, volatility uses past price data to estimate dispersion. Implied volatility is derived from option prices and reflects the market’s expectation of future volatility over a given horizon. Realized and implied volatility can diverge, and both are used in risk management, forecasting, and derivative pricing. Additional concepts include forward volatility, local volatility, stochastic volatility, and rough volatility. Local volatility models treat volatility as a function of price and time; stochastic volatility models treat volatility itself as a random process (for example, the Heston model).

Volatility estimates feed into several tools and models. The volatility surface or smile shows how implied volatility varies with strike and maturity, and the term structure shows how it changes with expiration. Volatility clustering is the tendency for high-volatility periods to follow high-volatility periods, and mean reversion describes a tendency for volatility to revert toward a long-run average. The leverage effect notes a negative relationship between asset returns and volatility changes.

Applications include pricing options, risk management, portfolio optimization, and stress testing. Limitations include that volatility measures do not capture tail risk or the direction of price moves, and that models rely on assumptions that may fail in stressed markets.