Volatility

There are several ways that volatility is expressed.

Historical volatility, also called realized volatility, represents how much an asset's prices have fluctuated in the past and is calculated from historical prices over a specified period. This is usually computed by taking the standard deviation of returns or the square root of summed squared returns. Simple, logarithmic, or dollar returns may be used.

Implied volatility (IV) is a forward-looking measure that reflects the market's expectations of future variation in an asset's prices based on the current prices of derivatives on that underlying asset. This is usually computed from the prices of options on the asset by plugging those prices into the solution of an option pricing model. Indices like the Cboe Volatility Index (VIX) and MIAX SPIKES index are based on a weighted measure of implied volatility.

Closed-form solutions of the Black-Scholes model can be used to estimate IV of instruments with European options such as SPX index options. Variations of this, such as the Black-76 model, are used to estimate IV of index futures like the E-mini S&P 500 futures.

Tree-based models provide generalized numerical methods for estimating implied volatility. Examples of tree-based models include the binomial options pricing model, popularized by Cox, Ross, and Rubinstein, and the trinomial options pricing model. Tree-based models provide additional flexibility for handling instruments with American-style options, which can be exercised at any time before expiration. Such instruments with American-style options include US single-name stocks and ETFs like SPY. Tree-based models are also useful for instruments with Bermuda-style options, which can be exercised at specific times before expiration.

Volatility smile or volatility skew refers to the stylized observation that the plot of IV against strike prices produces a curve instead of a flat line or that the IV surface, which plots IV against strike and maturity, is not a flat surface. Volatility smile and skew are terms often used interchangeably; however, "volatility smile" is usually used in the context of equity index options, where the surface is usually curved upwards, whereas "volatility skew" is more often used in the context of equity options, where the surface is usually curved downwards.

Vol-of-vol refers to the volatility of volatility, which is a measure of tail risk.

Stochastic volatility models such as the Heston, SABR, SVI, and parameterized Carr-Wu models are more often used by dealers for pricing exotic derivatives that are materially affected by volatility smile or vol-of-vol effects. On the other hand, most traders and electronic market makers in liquid options markets use Black-Scholes or tree-based models, as well as greeks and IV derived from these.

Relative measures of volatility also exist, such as an asset's market beta, which expresses the asset's correlation with overall market returns. In this case, market returns may be those of a benchmark index like the S&P 500, those of an arbitrary universe of stocks in a portfolio, or those obtained from a vendor's factor risk model like Barra or Axioma.

There are also various models for volatility forecasting, such as the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) and Autoregressive Conditional Heteroskedasticity (ARCH) models.

A common use of historical volatility is in computing the Sharpe ratio of a portfolio or strategy. When doing so, it's typical to annualize the volatility with a constant scaling factor (volatility scaling). There's no harm in doing so where the Sharpe ratio is merely used for business decisions, e.g., deciding a trader's compensation. However, annualized volatility is biased in the presence of non-IID returns, e.g., returns that exhibit autocorrelation, and it may be necessary to account for this to make accurate inferences, e.g., in comparing performances of two funds. See also Lo (2002), The Statistics of Sharpe Ratios, and Riondato (2018), Sharpe Ratio: Confidence Intervals, and Hypothesis Testing.

Features used to express volatility in prediction models, e.g., as split variables in a random forest, do not necessarily have to follow any of the above formulations or definitions strictly. For example, max range (high minus low), messaging rate, matching engine latencies, or even packet sizes, may be useful "volatility" features.