What is the difference between volatility and dispersion in finance?

Question

I am confuse whether the volatility and dispersion is same or not because are use to measure the risk associated with asset. Even if they are different than what is the relationship between, if exist.

Answer 1

Dispersion (also called variability, scatter, or spread) is the extent to which a distribution varies (to the left and right) from its central tendency. Sample variance, $\sigma^2$ is the most common measure of dispersion. The square root of variance, $\sqrt{\sigma^2}$, is standard deviation, $\sigma$.

In finance, risk is proxied with volatility, which is measured using the standard deviation, $\sigma$.

Answer 2

Regarding terminology, dispersion is sometimes used to refer to a trade which is short index correlation, i.e. you are betting that the constituents of a certain index (usually an equity one like the S&P 500) will be decorrelated. This involves being (1) long individual component volatilities, and (2) short overall index volatility.

For an index $I$ with constituent prices $S_1,\dots,S_n$ and weights $w_1,\dots,w_n$ the volatility $\sigma_I$ of index returns is: \begin{align} \sigma_I =\sqrt{\sum_{i,j}w_iw_j\sigma_i\sigma_j\rho_{ij}} \end{align} where $\sigma_i$ are constituents' price volatilities and $\rho_{ij}$ their pairwise correlations. When you are long dispersion you are betting that these pairwise correlations will go down; you short index options but you hedge any change in the component vols $\sigma_1,\dots,\sigma_n$ by being long options on the individual components.

Hence in this context dispersion relates to correlation - and not volatility per se.