# Historical Volatility Calculation

*This page is a step-by-step guide how to calculate historical volatility. Examples and Excel formulas are available in the Historical Volatility Calculator and Guide.*

Although you hear about the concept of historical volatility often, there is confusion regarding how exactly historical volatility is calculated. If you are using several different charting programs, it is quite likely that you will get slightly different historical volatility values for the same security with the same settings with different software. The following is the most common approach – **calculating historical volatility as standard deviation of logarithmic returns**, based on daily closing prices.

## What Historical Volatility Is Mathematically

When talking about historical volatility of securities or security prices, we actually mean historical volatility of returns. It looks like a negligible distinction, but it is very important for the calculation and interpretation of historical volatility. **Mathematically, historical volatility is the (usually annualized) standard deviation of returns**. If you know how to calculate return in a particular period and how to calculate standard deviation, you already know how to calculate historical volatility. If you’re still not sure, detailed step-by-step guide follows.

## Deciding the Parameters

There are 3 parameters we need to set:

- The basic period (for which we calculate returns in the beginning) – often 1 day is used
- How many periods enter the calculation (we’ll refer to this as
*n*) – often 20 or 21 days (the number of trading days and therefore the number of basic periods in one month) - How many periods there are in a year (this is used for annualizing volatility in the end)

In Macroption research I mostly use 1 day (day-to-day returns), 21 or 63 days (representing 1 month or 3 months), and 252 (as there are 252 trading days per year on average).

It is not as important whether you use 20 or 21 days, or 252 or 262 days. Much more important is that you use the same parameters consistently, so your results will be comparable.

## Step 1: Calculating Returns

First we need to calculate the *continuously compounded* return of each period. In our case, we will calculate the day-to-day returns for each of the 21 days (our n=21):

ln = natural log

C_{n} = closing price

C_{n-1} = previous day closing price

## Step 2: Standard Deviation of the Returns

Next we need to calculate the standard deviation of the returns we got in step 1. Standard deviation is the square root of variance, which is the average squared deviation from the mean (if you are not familiar with it, here you can see a detailed explanation of variance and standard deviation calculation).

First, calculate the average of the returns we got in step 1:

Then, calculate the squared deviation from the average for each of the returns:

Calculate the average of the squared deviations by summing them up and dividing by n-1 (in our case 21 – 1 = 20). We are dividing by *n-1* rather than *n*, as we are calculating sample standard deviation (we are estimating the standard deviation from a sample – if not familiar, see the difference between population and sample standard deviation).

Note: This is the variance of the returns.

Calculate standard deviation = square root of variance. The whole formula therefore is:

Note: It may look scary, but we have just added a square root to the previous formula.

The number we got now (σ) is 1-day historical volatility.

## Annualizing Historical Volatility

The only thing left is to annualize the volatility. We do that by multiplying the 1-day volatility by the square root of the number of (trading) days in a year – in our case square root of 252. The result is the annualized volatility.

## Calculating Historical Volatility in Excel

In practice, calculating historical volatility manually would be very lengthy (and prone to errors). But it is very easy in Excel. In fact, you do the whole step 2 with the standard deviation function (use STDEV.S for sample standard deviation).

## Historical Volatility Calculator

You can download the Historical Volatility Excel Calculator from Macroption. You can use it for your own calculations, either using your own market data, or automatically downloading data from Yahoo Finance for a symbol you select. The Calculator can also provide results for **another very popular historical volatility calculation method** – the zero mean (or non-centered) method, which slightly differs from the one described above. There is a PDF Guide that comes with the calculator. It explains all the calculations and functions in detail.