# Historical Volatility Calculator

Estimate the historical volatility of a financial asset using our online calculator. By inputting a series of historical price data, such as daily or weekly closing prices, you can calculate the historical volatility of the asset.

## What Is Historical Volatility?

Historical statistical volatility provides an indication of how the stock price has changed over a given period of time. While some analysts may use historical volatility as a means of predicting future stock performance, it may not necessarily be a correct indication as historical influences may have driven price changes. Furthermore, the stock price is directly influenced by major news items.

Historical volatility can be measured in a myriad of ways. This calculator computes historical volatility using two different approaches:

- The standard deviation of logarithmic returns, which is also referred to as centered historical volatility.
- The zero-mean method, which is also referred to as non-centered historical volatility.

## Calculating Logarithmic Returns

To calculate the stock volatility from a set of historical stock price data, you start by determining the daily logarithmic returns, which is known as the continuously compounded return. This is computed as follows:

R_{i} = ln ( C_{i} / C_{i-1} )

Where:

**R _{i}** is the return of a given stock over the period

**i**,

**ln** is the natural log function,

**C _{i}** is the closing price at the end of the period

**i**,

**C _{i-1}** is the closing price at the end of the period immediately before period

**i**.

## Standard Deviation of Logarithmic Returns

The standard deviation of logarithmic returns is the most commonly employed method of determining historical volatility. To compute historical volatility using this approach, simply determine the sample standard deviation of **n** last returns as follows:

Where:

**R _{i}** is the return of a given stock over the period

**i**,

**R _{avg}** is the average of all the daily returns, R

_{avg}= (1/n) Σ

_{i=1...n}R

_{i}

The formula above is applicable for 1-period historical volatility. Volatility is usually computed and cited in annualized form. To annualize 1-period of volatility, simply multiply it by the square root of the number of periods per year (**N**). For instance, if there were 252 trading days in the year, the annualized volatility will be computed as the 1-day volatility multiplied by the square root of 252. Annualized historical volatility is thus determined as follows:

## The Zero-Mean Approach

The zero-mean approach represents a modified form of the standard deviation method. Historical volatility is the standard deviation of returns; however, the average return (**R _{avg}**) is assumed to be zero. As such, the formula is modified as follows:

The annualized historical volatility is computed as follows:

## Historical Volatility Calculator example

Certainly! Historical volatility is a measure of the price volatility of a financial instrument over a specific period of time. Let's consider an example of calculating historical volatility for a stock using historical price data.

Assume we have collected the daily closing prices of a stock for the last 30 trading days. Here are the closing prices (in arbitrary units):

120, 125, 122, 130, 128, 126, 135, 132, 140, 138, 142, 145, 150, 155, 151, 148, 152, 155, 160, 159, 162, 165, 163, 160, 158, 156, 153, 158, 160

Step 1: Calculate logarithmic returns.

First, we need to calculate the logarithmic returns, which measure the rate of change in the stock price relative to the previous day's closing price. We can calculate the logarithmic returns using the formula:

Logarithmic Return = ln(Pt/Pt-1)

where Pt is the closing price at day t and Pt-1 is the closing price on the previous day.

For example, the logarithmic return for the second day would be:

ln(125/120) ≈ 0.041

Similarly, we calculate the logarithmic returns for all the days in the data set.

Step 2: Calculate the standard deviation of logarithmic returns.

Next, we calculate the standard deviation of the logarithmic returns. This gives us a measure of the volatility of the stock over the given period.

Using the formula for standard deviation, we calculate the standard deviation of the logarithmic returns.

In our example, let's say the standard deviation of the logarithmic returns is calculated to be approximately 0.029.

Step 3: Calculate historical volatility.

To calculate the historical volatility, we annualize the standard deviation by multiplying it by the square root of the number of trading days in a year. Let's assume there are 252 trading days in a year.

Historical Volatility = Standard Deviation * sqrt(252)

Using the example values, the historical volatility would be:

0.029 * sqrt(252) ≈ 0.459

Therefore, the historical volatility for this stock, based on the given data, is approximately 45.9%.

Please note that this is a simplified example for illustrative purposes. In practice, historical volatility is typically calculated using a larger sample size and more precise techniques, and there may be variations and refinements to the calculation method depending on the context and requirements.