Prices and Returns
Prices don’t follow a statistical distribution (they are not ‘stationary’.) There is no obvious mean price and it doesn’t make sense to talk about the standard deviation of the price. Working with such non-stationary timeseries is a hassle.
But returns, on the other hand, are distributed somewhat like a normal (Gaussian) distribution.
And there doesn’t seem to be any auto-correlation between consecutive returns.
If returns are normally distributed, then how are prices distributed? It turns out that the logarithm of the price is normally distributed. Why? Because
returns(t) = log(price(t)/price(t-1))
Now statisticians can magically transform a random time-series (prices) into something that is normally distributed (returns) and work with that instead. Almost all asset pricing models that you will come across in literature has this basic assumption at heart.
The assumption that returns are normally distributed allow mathematically precise models to be constructed. However, they are not very accurate.
In the normal distribution, events that deviate from the mean by five or more standard deviations (“5-sigma events”) have lower probability, thus meaning that in the normal distribution rare events can happen but are likely to be more mild in comparison to fat-tailed distributions. On the other hand, fat-tailed distributions have “undefined sigma” (more technically, the variance is not bounded).
For example, the Black–Scholes model of option pricing is based on a normal distribution. If the distribution is actually a fat-tailed one, then the model will under-price options that are far out of the money, since a 5- or 7-sigma event is much more likely than the normal distribution would predict.
Precision vs Accuracy
When you build models, the precision that they provide may lull you into a false sense of security. You maybe able to compute risk right down to the 8th decimal point. However, it is important to remember that the assumptions on which these models are build don’t led themselves to accuracy. At best, these models are guides to good behavior, and nothing more.