Background:
Annual volatility in finance is defined as the standard deviation of one year log-normal asset return (using log-normal due to asset price cannot go below zero). Therefore volatility “σ” sigma translates to “s” standard deviation in statistical jargon, which is the square root of the variance. Note variance is defined as the squared sum of average distance to the sample mean or expectation. If we are measuring of spread of asset return in dollars, variance would be in the unit of dollar².
Similar to dealing with discrete random variables, variance is defined as below for continuous random variables. f(x) being the probability density function of standard normal.
Remember volatility is a such an important piece in finance for pricing, portfolio optimization, arbitrage, etc. Should we be just happy with the above?
Mean Deviation:
The most common question asked is whether we just come up with the definition of volatility out of thin air? The short answer is yes. And luckily it turned out to be not as influential due to volatility not being physically observable.
Is it possible to use the absolute mean deviation to get rid of the squared root of the variance to define volatility?
The answer is again yes; with even more benefits over commonly accepted standard deviation.
Mean Deviation vs. Standard Deviation:
- Standard deviation is strongly based on the idea of standard normal distribution like the 68–95–99.7 rule. However our stock market is very negatively skewed with larger fat tail downward risk. With large outliers, square-rooting the sum of squares enlarges the variation exponentially and undesirably. Studies show that mean deviation has lower standard deviation when compared to standard deviation not just in above equation, but also empirically proven using astronomical data. https://www.leeds.ac.uk/educol/documents/00003759.htm
- Mean deviation has more straightforward meaning than standard deviation. The use of statistics should make things more intuitive. Especially in finance, we are never trying to accurately estimate the entire population from existing samples.
With above drawback, eventually standard deviation is still widely adopted to build numerous theorems in finance and statistics due to mathematical and statistical reasons:
- The absolute value sign in the mean deviation equation is relatively harder to work with. For example, we need to break down in positive and negative regions when integrating the pdf to get corresponding cdf. In finance it is a common practice to minimize the portfolio variance for beta neutral strategies. In fact, we even have quantum annealing right now specifically for doing this kind of computing tasks.
2. Since the log normal return of all financial assets is additive, according to the Pythagorean theorem of statistics, variance is also made additive. It is very common to combine assets and use this feature to calculate total portfolio variance of risks. Not coincidentally, Fermat's last theorem states that only squared version has rational solutions. This was proved by Andrew Wiles in 1993. This leaves us the privilege of working with rational volatility numbers, which is more commonly expressed in percentage.
What this all means:
Square-rooting the sum of squared asset return deviation has created this phenomenon this the following graph.
As time goes by, volatility increases less and less. This is if we assume the population being normally distributed, however it is not. While this could be a reasonable estimate for reverse asymmetric volatility asset like the safe heaven yen with lightly positively skewed asset return, but it’s certainly a bad model for risky asset as we should expect more volatility during asset down turn. Square-rooting the sum of squared just exponentially amplifies the negative fat tails in an incorrect way.
