# Lecture 7 Note

*2021-07-28*

# Chapter 1 Volatility Modeling

Basic Definition: Annualized standard deviation of the change in price or value of a financial security.

## 1.1 What Is Volatility?

Volatility is a statistical measure of the **dispersion**^{1} of returns for a given security or market index. In most cases, the higher the volatility, the riskier the security. Volatility is often measured as either the standard deviation or variance between returns from that same security or market index.

In the securities markets, volatility is often associated with big swings in either direction. For example, when the stock market rises and falls more than one percent over a sustained period of time, it is called a “volatile” market. An asset’s volatility is a key factor when pricing options contracts.

Volatility represents how large an asset’s prices swing around the mean price - it is a statistical measure of its dispersion of returns.

There are several ways to measure volatility, including beta coefficients, option pricing models, and standard deviations of returns.

Volatile assets are often considered riskier than less volatile assets because the price is expected to be less predictable.

Volatility is an important variable for calculating options prices.

## 1.2 Understanding Volatility

Volatility often refers to the amount of uncertainty or risk related to the size of changes in a security’s value. A higher volatility means that a security’s value can potentially be spread out over a larger range of values. This means that the price of the security can change dramatically over a short time period in either direction. A lower volatility means that a security’s value does not fluctuate dramatically, and tends to be more steady.

One way to measure an asset’s variation is to quantify the daily returns (percent move on a daily basis) of the asset. Historical volatility is based on historical prices and represents the degree of variability in the returns of an asset. This number is without a unit and is expressed as a percentage.

While variance captures the dispersion of returns around the mean of an asset in general, volatility is a measure of that variance bounded by a specific period of time. Thus, we can report daily volatility, weekly, monthly, or annualized volatility. It is, therefore, useful to think of volatility as the annualized standard deviation.

## 1.3 How to Calculate Volatility

Volatility is often calculated using variance and standard deviation. The standard deviation is the square root of the variance.

For simplicity, let’s assume we have monthly stock closing prices of $1 through $10. For example, month one is $1, month two is $2, and so on. To calculate variance, follow the five steps below.

Find the mean of the data set. This means adding each value and then dividing it by the number of values. If we add, $1, plus $2, plus $3, all the way to up to $10, we get $55. This is divided by 10 because we have 10 numbers in our data set. This provides a mean, or average price, of $5.50.

Calculate the difference between each data value and the mean. This is often called deviation. For example, we take $10 - $5.50 = $4.50, then $9 - $5.50 = $3.50. This continues all the way down to the first data value of $1. Negative numbers are allowed. Since we need each value, these calculations are frequently done in a spreadsheet.

Square the deviations. This will eliminate negative values.

Add the squared deviations together. In our example, this equals 82.5.

Divide the sum of the squared deviations (82.5) by the number of data values.

In this case, the resulting variance is $8.25. The square root is taken to get the standard deviation. This equals $2.87. This is a measure of risk and shows how values are spread out around the average price. It gives traders an idea of how far the price may deviate from the average.

If prices are randomly sampled from a normal distribution, then about 68% of all data values will fall within one standard deviation. Ninety-five percent of data values will fall within two standard deviations (\(2\times 2.87\) in our example), and 99.7% of all values will fall within three standard deviations (\(3\times2.87\)). In this case, the values of $1 to $10 are not randomly distributed on a bell curve; rather. they are uniformly distributed. Therefore, the expected \(68\%-95\%-99.7\%\) percentages do not hold. Despite this limitation, traders frequently use standard deviation, as price returns data sets often resemble more of a normal (bell curve) distribution than in the given example.

What Is Dispersion? Dispersion is a statistical term that describes the size of the distribution of values expected for a particular variable and can be measured by several different statistics, such as range, variance, and standard deviation. In finance and investing, dispersion usually refers to the range of possible returns on an investment. It can also be used to measure the risk inherent in a particular security or investment portfolio.↩︎