Stock Volatility Calculation Formula

Stock volatility is a statistical measure of the dispersion of returns for a given stock or market index. It represents the extent of variation in a stock's price over time and can be an important factor in assessing the risk of an investment. There are several methods to calculate volatility, but one of the most common is the standard deviation of returns. Here’s how you can calculate it:

1. Collect Historical Prices: Gather the historical prices of the stock for a specified period. Typically, daily closing prices are used.

2. Calculate Returns: Compute the returns for each period. The return can be calculated using the formula:

   $\text{Return}=\frac{\text{Current Price}-\text{Previous Price}}{\text{Previous Price}}$Return=Previous PriceCurrent Price−Previous Price​

3. Calculate the Mean Return: Find the average of all the returns. This is done by summing all the returns and dividing by the number of returns.

4. Calculate Deviations from Mean: Subtract the mean return from each individual return to get the deviations.

5. Square the Deviations: Square each of the deviations obtained in the previous step.

6. Find the Average of Squared Deviations: Compute the average of these squared deviations. This value is known as the variance.

7. Calculate Standard Deviation: The standard deviation is the square root of the variance.

   The formula for the standard deviation is:

   $\text{Standard Deviation}=\sqrt{\frac{\sum (\text{Return}-\text{Mean Return}{)}^2}{N-1}}$Standard Deviation=N−1∑(Return−Mean Return)2​​

   where $N$N is the number of returns.

Example Calculation:

Suppose you have the following daily closing prices of a stock for 5 days: $100, $102, $101, $105, and $104.

1. Calculate Returns:

   • Day 2 Return = $\frac{102-100}{100}=0.02$100102−100​=0.02 or 2%
   • Day 3 Return = $\frac{101-102}{102}=-0.0098$102101−102​=−0.0098 or -0.98%
   • Day 4 Return = $\frac{105-101}{101}=0.0396$101105−101​=0.0396 or 3.96%
   • Day 5 Return = $\frac{104-105}{105}=-0.0095$105104−105​=−0.0095 or -0.95%

2. Mean Return:

   • Mean Return = $\frac{0.02-0.0098+0.0396-0.0095}{4}=0.010575$40.02−0.0098+0.0396−0.0095​=0.010575

3. Calculate Deviations:

   • Deviations: $0.020-0.010575=0.009425$0.020−0.010575=0.009425, $-0.0098-0.010575=-0.020375$−0.0098−0.010575=−0.020375, $0.0396-0.010575=0.029025$0.0396−0.010575=0.029025, $-0.0095-0.010575=-0.020075$−0.0095−0.010575=−0.020075

4. Square the Deviations:

   • Squared Deviations: $0.009425^2=0.000089$0.0094252=0.000089, $(-0.020375{)}^2=0.000415$(−0.020375)2=0.000415, $0.029025^2=0.000843$0.0290252=0.000843, $(-0.020075{)}^2=0.000403$(−0.020075)2=0.000403

5. Average of Squared Deviations:

   • Variance = $\frac{0.000089+0.000415+0.000843+0.000403}{3}=0.000575$30.000089+0.000415+0.000843+0.000403​=0.000575

6. Standard Deviation:

   • Standard Deviation = $\sqrt{0.000575}\approx 0.024$0.000575​≈0.024 or 2.4%

This standard deviation tells you how much the returns of the stock vary from the average return. A higher standard deviation indicates greater volatility and hence higher risk.

Other methods for calculating volatility include using the Average True Range (ATR) or Historical Volatility (HV), but the standard deviation method is widely used due to its simplicity and effectiveness.

Volatility is a crucial measure in financial markets as it helps investors understand the potential price fluctuations and the associated risks with investing in a stock. It also plays a key role in portfolio management, risk assessment, and in the valuation of financial derivatives.