# Volatility

Earlier, we learned about the option-greek Vega that measures the change in option price per unit change in volatility. But what exactly is volatility?

Volatility is a rate at which the price of a security increases or decreases for a given set of returns. It is measured by calculating the standard deviation of the annualized returns over a given period of time. It shows the range to which the price of a security may increase or decrease.

Let us consider an example to understand volatility better.

There are 2 students A and B and the table below shows the marks they have obtained in their last 7 semesters.

I am supposed to choose a student for a competition, who will be able to score at least 27 points.

**Situation 1**

Let us add up all the scores and see who scored the most

A=35+36+47+42+28+32+30= 250

B=55+25+34+27+40+43+28= 252

So, in this situation we can choose B as the total score is 252 (greater than A’s score)

**Situation 2**

Let us consider the mean or average for both the students

A=250/7= 35.7

B= 252/7= 36

So from this perspective also, I will choose B over A because his average is higher.

But we are not sure who will be able to score at least 27 points.

So let’s calculate the deviation from mean for each student and each semester.

Now let’s further calculate another variable called variance. It is the sum of the squares of the deviation divided by the total number of observations.

So, variance= [(-0.7)^2 + (3.3)^2 + (11.3)^2 + (14.3)^2 + (-7.7)^2 + (-3.7)^2 + (-5.7)^2] / 7

=449/7

=64

Further, we will calculate Standard Deviation as √ of variance

= SQRT (64)

= 8

Similarly, B’s standard deviation comes to 10.75

Standard Deviation or the SD represents the deviation from the average. To forecast how much A and B will score in the next semester, we need to add and subtract SD from their average.

From these calculations, we can easily interpret that A is likely to score from 27.7 to 43.7 while B is likely to score 25.25 to 46.75. B has a wide range of scores because of which it is difficult to interpret whether he’ll be able to score at least 27. He can score either 25.25 or 46.75 or anything in between.

However A seems to be better placed because of consistency. His range is comparatively smaller and lies between 27.7 to 43.7.

This means A will score either 27.7 or 43.7 or anything in between. This fulfills our criteria of at least being able to score 27. We can also say selecting B over A for the next semester will be more risky.

Thus we can say that **Standard deviation represents risk.In stock market, the riskiness of a stock or index is defined as Volatility. It is expressed in % terms.**

If 2 stocks X and Y have volatility of 18% and 30% respectively, then it shows that stock Y has riskier price movements when compared to X.

As we computed SD for A and B, we can do the same for stocks and index. This will help us to compute the range of the stock price and will thus help in identifying options that are likely to expire OTM. We can sell those options and lock the premiums. Larger the range of stock, higher will be its volatility or risk.

We followed the following steps to calculate the Standard Deviation for A and B.

- Calculate the average
- Calculate deviation (subtract the average from the actual data)
- Calculate variance
- Calculate SD

### How to calculate volatility of stocks?

This entire process can be instantly done on MS Excel, but the idea was to explain the whole mechanism behind.

For MS Excel, the steps to follow are as given below:

- Download the historical closing price data for a stock (This can be taken from NSE India website which is the most reliable source)
- Calculate the daily returns
- Use the STDEV function directly to get the Final output

I have downloaded the historical closing price data for SBIN from NSE , for the past one year (31st Dec 2021- 1st Jan 2022). Let’s look at the excel below. I have used the function LN to find the daily returns.

Now using the **STDEV** function, let me compute the daily volatility:

Remember that 3% is the daily volatility that we have computed. To get the annual volatility, we have to multiply it with the square root of time (SQRT is the function that can be used to find the square root of a number).

**Daily volatility** = 2%

**Annual volatility** = 3% * SQRT(252) = 33%

Likewise, if we have annualized volatility of a stock, we can divide it by square root of time to get the daily volatility.