Active Risk Budgeting: Volatility is Not Standard Deviation

We try to show the danger of confusing the concept of volatility with that of the standard deviation of a probability distribution. We work in the theoretical Black-Scholes model to give an explicit relationship between the two measures. We apply and then illustrate this relationship, firstly in a classical value at risk approach, secondly in the determination of the risk contributions of a portfolio. We see profound differences that should not lead to the rapprochement of the volatility and the standard deviation