Question

What is the concept of the square root of time rule?

Answer 1

Volatility is the standard deviation of returns and the standard deviation is the square root of the variance.
The square root of time rule is generally used when the risk is time clustered. It is applied in the time scaling of volatilities, like int the Black-Scholes model.

In order to annualize or de-annualize or to transform volatility to any other form of the time period, it is multiplied by the square root of the time ratio instead of the time itself. volatility, in theory, scales with the square root of time.
For example: if daily volatility is calculated as 0.75%, then, in theory, volatility, when calculated for 3 days, is daily volatility times the square root of the time period: 3, i.e., (0.75% * 1.73 = 1.29%).

As per this rule, the volatility will increase by the square root of the time if the fluctuation in the stochastic process is independent of each other and if the volatilities are based on lognormal returns then the following rule gives out exact volatility values.

This rule holds true due to the assumption used in the common option pricing and volatility models that price take a random walk and the variance of each increment of this walk is proportional to the time over which the price as moving.
Therefore, it can be concluded that standard deviation is proportional to the square root of the time as it is the square root of variance and volatility is also proportional to the square root of the time as it equals the standard deviation.