# Historical volatility

Historical volatility allows predictions of volatility in the short and medium term through statistical inference.

In other words, past volatilities serve as a basis for estimating and quantifying the probabilities of obtaining a certain future volatility.

The famous phrase "history repeats itself" is the main reason why we use historical variables, in this case, volatility, to make a meaningful forecast in the short and medium term.

Mathematically, we can understand the repetition of the value of a variable at different moments of time as the seasonality of the value. If the expected value of the past variable follows a stationary process, it means that that expected value will be the same as the past one. That is, the value is constant, not trend, and therefore does not change over time.

## Procedure

1. We obtain listing prices and calculate ongoing returns and volatility (standard deviation).
2. We choose the time horizon in which we want to estimate volatility.
3. Estimation by Ordinary Least Squares and contrast of the results by hypothesis testing.

### Drawbacks

Estimating future volatility is not easy and this simple process presents problems such as the optimal number of elements in the sample or using the OLS estimation as a prediction tool.

Using OLS as a prediction tool implies assuming that:

• There is a linear relationship between the volatilities of different time periods.
• The expected volatility will be the average of the past volatilities.

## Practical example

We assume that we want to conduct a study on the volatility of the AlpineSki share price for 18 months (one year and a half) in order to inform investors of the future volatility of the stock.

### Procedure

 Months Returns Dec-17 2.75% Jan-18 4.00% Feb-18 7.00% Mar-18 9.00% Apr-18 7.00% May-18 -0.40% Jun-18 -2.00% Jul-18 -4.00% Aug-18 -20.00% Sep-18 1.50% Oct-18 2.00% Nov-18 4.50% Dec-18 3.75% Jan-19 9.00% Feb-19 7.00% Mar-19 9.00% Apr-19 -1.50% May-19 -2.00% Volatility 6.89% Annualized volatility 23.85%
• We represent returns.
• We plot the highs and lows of the channel that form the returns closest to the linear estimate.

### Interpretation

Based on historical volatility, volatility is expected to be 6.89% for the following months.

Should we take into account the -20% of August 2018 in the calculation of historical volatility?

This "rare" observation is called an outlier because it is outside the common observations ( insiders ).

If we don’t take that -20% into account, volatility drops to 4.27%.

We can see that the price is completely cyclical: the share price rises when there is snow on the ski slopes and falls when there is no snow. So, in addition to financial and economic risks, the stock is also linked to time risk. In other words, a season with little snow on the slopes can be terrible for the price. The variability of the weather makes it impossible to consider all the months in the same way since not all winters are equally cold.

In addition, we have the temporal effect: the further observations should be considered less relevant than the closer observations.

Then, we should not consider all the observations equal due to the effect of the nature of the variable (time risk) and the temporal effect.

One solution would be to weight historical volatility with different weights. In other words, apply weighted historical volatility to future volatility estimates.