Volatility is a term used to describe how much the security prices, market indices, interest rates, etc., move up and down around their mean. It is measured by the standard deviation of the random variable that represents the financial quantity we are interested in. Most investors prefer low volatility to high volatility and therefore expect to be rewarded with higher long-term returns for holding higher volatility securities.

Many financial computations require volatility estimates. Mean-variance optimization trades off the expected return and volatility of a portfolio of securities. The celebrated option valuation formulas of Black, Scholes, and Merton (BSM) involve the volatility of the underlying security. Risk management revolves around the volatility of the current positions. Therefore, accurate estimation of the volatilities of security returns, interest rates, exchange rates, and other financial quantities is crucial to many quantitative techniques in financial analysis and management.

Most volatility estimation techniques can be classified as either a historical or an implied method. One either uses historical time series to infer patterns and estimates the volatility using a statistical technique, or considers the known prices of related securities such as options that may reveal the market sentiment on the volatility of the security in question. GARCH models exemplify the first approach while the implied volatilities calculated from the BSM formulas are the best-known examples of the second approach. Both types of techniques can benefit from the use of optimization formulations to obtain more accurate volatility estimates with desirable characteristics such as smoothness.