# What is Volatility?

Volatility of a security or an index means the magnitude of changes in its value *(price)* over time. In more scientific terms, it can be called ‘dispersion’. High volatility usually indicates higher potential returns because investors can make more money with each deal but, at the same time, it implies higher risks because the direction of future price changes is unknown.

## Table of Contents

## Understanding Volatility - Example

Volatility shows how ‘uncertain’ a particular stock *(index, or other security)* is. If its price changes by a lot, this stock has a high volatility. Let’s imagine an asset that fluctuates in price in the interval of +-5% per day which makes it quite volatile.

To understand what volatility is, you can imagine a person with an unstable mood, maybe you even have one or two friends like that. One day, he is super cheerful, active and productive, but on the next day this person is in a deep depression and he can’t do anything. His Bipolar mood can be called ‘volatile’. Imagine that you are working with this person and you have to finish an important project soon. In this case, your friend might do the job quickly and efficiently if you are lucky and if he is in a good mood, but at the same time there is a chance that it will take him ages to do the same job properly. So, as we said before, it’s a case of high risks / high returns. In a more casual case, you wouldn’t want to work with such an unstable person but those people can manage critical situations efficiently if they are in a good mood.

Financial market volatility is kind of the same thing: volatile securities are riskier but they offer a lot of opportunities to make money. Investors with a conservative investment strategy would avoid those volatile securities and would opt for something more stable.

## Measuring Volatility With ‘Beta’

A beta value shows overall volatility compared to a relevant benchmark such as a general index. If a stock’s beta is 0.5, it means that it moved 50% for every 100% move of the benchmark. It’s quite similar to the elasticity concept in classic economic theory.

For instance, if Amazon, a stock which is included in S&P 500 index, has a beta of 1.05, this means that it moves 5% more than its benchmark: the S&P 500 index. In this case, this stock is more volatile than the average, but not by a lot.

The stocks with abnormal beta values are watched carefully by the market because they are behaving in a strange way so either something is wrong or it might be a very interesting investment opportunity.

Beta is often used together with alpha, a coefficient that measures the success of a particular portfolio.

## Average True Range *(ATR)* Indicator

Another way to determine stock’s volatility is by calculating its ATR *(also called ‘Average True Range’)*, which was developed by J. Welles Wilder Jr. It’s quite easy to calculate an ATR for day trading, but it’s a bit harder to do when it comes to other time frames. Luckily, this is a very common indicator that is automatically calculated in the most of financial software products.

The daily range of any asset can be found by subtracting the lowest price from the highest price.

In order to get the True Range *(TR)*, you have to add yesterday’s closing price if it was outside of today’s range. Here is a simple example:

- Suppose the highest price of this day was $100
- Suppose the lowest price of this day was $90

The daily range here would equal $10 *($100 - $90)*. This ten dollars is basically the volatility for the day, which is the same as the distance between the maximum and the minimum price and it can be seen on the Japanese candlestick chart.

For a longer period of time you have to add the following steps to your calculations:

- Multiply the previous N-days ATR by N-1, where ‘N’ is the number of periods
*(days for example)* - Add the most recent day’s TR value
*(which was $10 in our example)* - Divide the total by N

Let’s do it with some numbers.

If today’s TR was $10 and the average TR *(ATR)* of the last 20 days was $5, our current ATR would be:

“(($5 * 19) + $10) / 20 = 5.25”

What does this 5.25 tell you?

Visually you would see how this indicator went up a little bit recently, but it won’t go up by a lot as you might have expected, because ATR binds the curve to an average. It tells you that there are some changes in price’s behavior of this asset but this recent volatility of $10 should stay for a long time in order to indicate any significant and important changes.

## How to Calculate Variance and Standard Deviation?

Standard deviation is another common way to measure price volatility.

Imagine that the price of a certain asset has been rising by $5 daily for 19 days, but on the 20th day, it went up by $10. The price went up from $5 to $105 *(5+5+5… +10)* for 20 days in total.

How would we find variance and standard deviation in this case? Let’s dive deeper into math and statistics.

**First step** is to find the mean of the data.

To calculate the mean just add up all numbers together cumulatively *(5+10+15…)* and then divide the result by the number of periods *(20)*. You’ll get $1055 / 20 = $52.75, this is the mean *(average)* price.

**Second step** is to calculate a ‘deviation’, a difference between each particular value and the mean.

We’ll get -47.75 *(5 - 52.75)* for the first day, then -42.75 *(10 - 52.75)* and so on.

You’ll see that the deviations get lower when they come to the middle *(if you do it in Excel’s spreadsheet as a table)*. On 12th day we’d have the minimum deviation of just 2.25.

**Third step** would be to raise each deviation to the power of 2 in order to eliminate negative values.

You’d get 2280 something for the first day, 1827 something for the seconds day, etc.

**Step four** is to add all squared deviations together so you’d get 17123 in our case.

And finally **the last step five** is to divide the sum of the squared deviations by the number of data values *(20)*. 17123 / 20 = 856. Congratulations, you’ve got the variance!

Hold on though, we are looking for a standard deviation here, so we have to extract the square root of this number, which is 29.25.

Ok, so we’ve got **the final number** of 29.25, but what does it mean and how is it related to volatility?

## Explaining Standard Deviation

In our case, the standard deviation of our imaginary asset is ~30 *(29.25, to be precise)*.

This tells us that the price changes of more than $30 would be statistically significant and should be looked at. +-$30 from our mean *(average)* of $50 *(52.75 rounded up)* is one standard deviation so it has a range of $20 to $80. So, for our imaginary asset, if we’re going to track the price later and if it’s going to stop its constant growth and go back to $50-ish, we’ll know that historically a price of $20 would be a ’trend breaker’ as well as the price of $80.

Wait a minute… if the ’normal’ deviation is +-30 from 50 in our case but the current price is already $105, what does it mean? It means that each day our asset is breaking the standard deviation and has to be watched carefully because when the price goes up for 20 days in a row from $5 to $105, statistics tell us that these movements are unpredictable and unsustainable. Sure, it is not impossible, some stocks tend to grow very fast right after their IPO, but the point of this example is to show that, statistically, such behavior is abnormal and should be looked at.

What would be a normal price behavior in our case? If the price is $105 right now, a “normal” move for it would be to drop below $80, closer to the mean of $50 and stay in this range *($20 - $80)*.

Any data points which are above or below the standard deviation are called the ’tail risk moves’.

In simple words, in a set of data such as ours which starts from $5 and goes up to $105 shortly, being at the point of $105 is extremely unlikely and we were getting new low probability values daily for several days in a row. It’s quite easy to see on a chart and you don’t actually need to calculate standard deviation to figure this out, although this number might come handy when you need to decide what assets to include to your portfolio and in what proportion.

## Understanding Standard Deviation

A standard deviation is just a number that shows the magnitude of future changes. Of course, it doesn’t guarantee anything, but it works in 68 cases out of 100 *(34% for un upside + 34% for a downside = one standard deviation)*. In our example, with the current price of $105, you have a 70% chance that the price would go down the next day to a price from $20 to $80. Also, we have a 16% chance that the price drops to a number from $0 to $20, and the same chance of a less significant fall to a price from $80 to $104.

## What is Implied Volatility?

Now that we know what standard deviation means in finance, it’s easy to explain implied volatility. It’s simply one standard deviation for the one-year period. It’s the same thing we just calculated but with the number of days equal to 365 days instead of 20.

## Is Standard Deviation Reliable?

This model is nice when it comes to classic assets such as blue chips and many old conservative financial instruments with relatively low volatility. The key here is that this method stands on the historical data, but what happens in a case when a stock breaks its historical records?

The math and statistics can’t help us here because as soon as the price goes out of the previous range, even if we had a very low probability of such occurrence, then the model has no previous data to support further movements. That is why the SD is used mainly to see the speed with which a security or a stock moves and to figure out the chances of breaking a trend.

It’s important to understand that the distribution of price movements is close to the normal distribution, but it breaks it when it comes to probability of extreme events. The normal distribution assigns a very low probability for significant gains or losses but, in the real world, it happens far more often than the normal distribution would predict.

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