What is 'Volatility'
Volatility is a statistical measure of the dispersion of returns for a given security or market index. Volatility can either be measured by using the standard deviation or variance between returns from that same security or market index. Commonly, the higher the volatility, the riskier the security.
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. Market volatility can be seen through the VIX or Volatility Index. The VIX was created by the Chicago Board of Options Exchange as a measure to gauge the 30day expected volatility of the U.S. stock market derived from realtime quote prices of S&P 500 call and put options. It is effectively a gauge of future bets investors and traders are making on the direction of the markets or individual securities. A high reading on the VIX implies a risky market.
A variable in option pricing formulas showing the extent to which the return of the underlying asset will fluctuate between now and the option's expiration. Volatility, as expressed as a percentage coefficient within optionpricing formulas, arises from daily trading activities. How volatility is measured will affect the value of the coefficient used.
Breaking Down 'Volatility'
Volatility 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 measure of the relative volatility of a particular stock to the market is its beta. A beta approximates the overall volatility of a security's returns against the returns of a relevant benchmark (usually the S&P 500 is used). For example, a stock with a beta value of 1.1 has historically moved 110% for every 100% move in the benchmark, based on price level. Conversely, a stock with a beta of .9 has historically moved 90% for every 100% move in the underlying index.
Calculating 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 our first data value of $1. Negative numbers are allowed. Since we need each value, these calculation 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 distributed (and often they are not), then about 68% off all data values will fall within one standard deviation. 95% of data values will fall within two standard deviations (2 x 2.87 in our example), and 99.7% of all values will fall within three standard deviations (3 x 2.87). In this case, the values of $1 to $10 are not randomly distribute on a bell curve, rather there is a significant upward bias. Therefore, all the values do not fall within three standard deviations. Despite this limitation, standard deviation is still frequently used by traders, as price data sets often contain up and down movements, which resemble more of a random distribution.
For additional reading, see A Simplified Approach to Calculating Volatility and Option Volatility.

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