# Historical Volatility

## Introduction

“Volatility” is an important term used in the stock markets and by many market professionals. Many investors use this term to manage their risk. “Volatility”, in simpler terms, means the fluctuation in a stock’s or any other underlying’s price, over a specific period of time. Volatility is divided in to two categories. The first one is “Historical volatility” and the other one is “Implied volatility”. “Historical volatility” is defined as the historical fluctuation in a particular underlying’s price, whereas “Implied volatility” is the estimation of volatility in an asset’s price. Generally, “Implied volatility” finds extensive use in “Options”.

Understanding “Historical volatility” is important for many investors as well as traders. For investors, this term is important, because it helps in estimating or calculating their risk. Traders, generally use “Historical volatility” to know how volatile a stock or an index will be in the future.

## Calculation

Before attempting to calculate “Historical volatility” of any particular asset, one needs to keep in mind, the time frame in which the volatility is calculated. According to the preferences of a trader or an investor, the time frames change. More specifically, the time frame defined, is, in most cases, different for an investor and a trader. An investor mostly looks out for the future volatility in an underlying over a longer period of time, whereas a trader feels better with shorter time frames.

A time frame of around 30 days has around 21 trading days in it. Similarly, for all the other time frames, the trading days would be less than the actual number. So, this point also should be kept in mind, while calculating Historical volatility.

The calculation would be easy if we divide this whole process in to two parts. The first one would include calculating periodic returns of a stock. The second one is choosing the time frame, as specified above. Calculating periodic returns, in turn, can be grouped in to calculating simple returns, and continuously compounded returns. Most preferably, investors look out for the continuously compounded return because this return has more accurate information associated with it, and will be able to provide more accurate volatility in a stock.
As stated before, there are two types of returns, and these are ‘simple’ and ‘continuously compounded’. A ‘return’ is defined as the movement of a share’s price in a specific period. So, a simple daily return is the percentage change of a stock’s price from yesterday to today. Mathematically, it can be shown as:

R = ( St - Sy ) / Sy

Here,
R = Simple daily return,
St = Today’s share price,
Sy = Yesterday’s share price.

On the other side, a continuously compounded return is the logarithm of the ‘ratio between today’s share price and yesterday’s share price’. Mathematically,

R = Log ( S/ S)

Now, with both, time frame and periodic returns available, it becomes easy to calculate “Historical volatility”. Using the time period and continuously compounded returns, variance is calculated by the following formula:

Here, s^2 = Variance,
n = Time period or number of days,
R = Average of daily returns.

Now, that the variance is known, we can calculate the standard deviation by taking the square root of it. This standard deviation is called “Historical volatility”.

## Example

Let us take the example of “Dow Jones Industrial Average”. Below is a table showing necessary calculations of the index in a time period of 7 days.

Using the above data, formulae and assuming the time period to be 7, we get the value of the historical volatility over a span of 7 days to be 0.93 %.

## Theory

“Historical volatility” is used in the estimation of future fluctuations in an asset. The risk associated with each asset could be measured by “Historical volatility”. A stock which has a lesser historical volatility than the other tends to be less volatile and more stable in the coming future. The goals of an investor’s portfolio would be simplified, and the main thing is, risk could be managed and estimated. We always can’t criticize assets with high historical volatility. Some traders as well as investors use high historical volatility to buy a stock cheap and sell high, because the particular stock is exhibiting higher volatility and hence, the range for the stock is higher than the other ones.

Calculating “Historical volatility” is a little complicated. The time frame must be defined in this case. Many professionals do waste good amount of time in defining their time frames, and that is definitely essential. The returns for the selected stock must be calculated. Continuously compounded returns should be used to generate accurate results.

Volatility of an index is measured using “VIX”, for a particular country. Generally, we see that the VIX increases in value, when the markets fall. This is due to the fact, that risk and volatility, both increase when markets fall. The VIX remains stable in an uptrend.