How do volatility and variance differ in finance and what do both imply about the movement of an underlying?
Volatility is typically unobservable, and as such estimated --- for example via the (sample) variance of returns, or more frequently, its square root yielding the standard deviation of returns as a volatility estimate.
As for forecasts of the movement: well, that is a different topic as movement is the first moment (mean, location) whereas volatility is a second moment (dispersion, variance, volatility). So in a certain sense, volatility estimates do not give you estimates of future direction but of future ranges of movement.
The main underlying difference is in their definition. Variance has a fixed mathematical definition, however volatility does not as such. Volatility is said to be the measure of fluctuations of a process.
Volatility is a subjective term, whereas variance is an objective term i.e. given the data you can definitely find the variance, while you can't find volatility just having the data. Volatility is associated with the process, and not with the data.
In order to know the volatility you need to have an idea of the process i.e you need to have an observation of the dispersion of the process. All the different processes will have different methods to compute volatilities based on the underlying assumptions of the process.
By volatility people usually refer to to annualized standard deviation of an asset. For an asset it's usually quoted as a percentage of the asset price (i.e. the return volatility). For a portfolio, it is often quoted in currency units. Variance is the square of the standard deviation. It is usually not quoted directly because it doesn't have an intuitive unit of measure. Instead, it is used in variance decomposition, e.g. the idiosyncratic variance of a portfolio is 6% of the total portfolio variance.
Suppose X is a random variable representing the returns of an asset having finite mean $\mu$ and variance $\sigma^2>0$.
- Variance $\sigma^2$ represents the expected squared deviation of $X$ from $\mu$. Intuitively, this is a measure of how dispersed returns are about the mean. If returns are measured in $\%$, then the units of variance are $\%^2$. However, for many people $\%^2$ is difficult to interpret.
- Volatility $\sigma$ is the square root of variance, and has units $\%$. This change in units makes volatility more interpretable, furthermore a better tool for analysis. If we further assume $X$ follows a Gaussian distribution, then $\sigma$ provides many more additional insights.
Volatility is a tool commonly used in univariate cases, e.g. when speaking of returns of one stock, one bond, or one portfolio.
In the multivariate setting, variance is used, e.g. a covariance matrix, because taking the square root of a matrix is an unecessary additional layer of complexity.
Volatility is essentially quadratic variation. It is a property of sample paths, not probability measures. In other words, it can be calculated given a single historical path and doesn't depended upon the probability you assign to that path.
Variance, and standard deviation, are functions of the probability you assign to events.
Variance is a measure of the dispersion and is not bound by any time period. On the other hand, volatility captures the degree of variation of a time series over time. In finance, volatility is a measure of the standard deviation over a certain time horizon (typically annual).
In quant environments, there are many different things that we call volatility (this is one thing I am quite unhappy, and think we should do better):
- The statistical definition, as the standard deviation of the returns (usually logarithmic returns) of a stochastic process
- The number you have to put in the Black Scholes formula to get the price you get in the market for a given option (that is the implied volatility)
- A parameter of a model (that can be constant, or a function of time, prices, etc) that you have calibrated (from historical or market prices) to describe the dynamics of a process. You might have different volatilities in a single model (in a stochastic volatility model, for example, you have a volatility price, which is a stochastic process itself, and a volatility of volatility). So volatility is model dependent: the same process, describe by different models, will have different volatilities (the one I descibed before, the Black-Scholes implied volatility, is also the simplest example of this)
And probably more I cannot think about right now. As someone said before, there a fixed mathematical definition for variance, but the meaning of volatility is quite subjective