Standard deviation

The standard deviation is a measure of how spread out a set of numbers are.

The standard deviation of a set of numbers is the square root of their variance. Variance is usually denoted by σ² and the standard deviation by σ, and:

σ² = 1/n Σ(x_i - μ)²

where x_i is one of n numbers and
μ is the arithmetic mean all n numbers x.

The most common use of the standard deviation in finance is to measure the risk of holding a security or portfolio. We first need the expected price:

E[S] =ΣS_ip(S_i)

where S is a price
and p(S_i) is the probability that S will be the actual price.

Denoting the variance of S, Var(S):

Var(S) = Σ(S_i - E[S])²p(S_i)

Var(S) is a measure of volatility. Its square root (the standard deviation) is the most widely used measure of volatility.

To use continuous times and prices replace the sums above with integrals.