From prices to returns
Volatility is not calculated from prices directly. It starts with returns, the percentage change between one observation and the next. If a price goes from 100 to 102, the return is 2 percent. If it then goes from 102 to 101, the return is about -0.98 percent, since the change of -1 is divided by 102.
Working with returns makes numbers comparable across time and across assets. A $1 move means something very different for a coin priced at $2 than for one priced at $50,000.
There are two common ways to write returns. Simple returns are the percentage change just described. Log returns take the natural logarithm of the ratio of the new price to the old one. For small moves they are almost identical, and log returns have convenient mathematical properties, such as adding up neatly over several periods. Either is acceptable as long as it is applied consistently.
Standard deviation in plain terms
Standard deviation measures how spread out a set of numbers is around its average. For returns, a higher standard deviation means individual returns have tended to land further from the average, so the price has been more erratic.
The steps are:
- Calculate the return for each period.
- Find the average return.
- Subtract the average from each return to get each deviation.
- Square each deviation, add them up, and divide by the number of observations minus one. This gives the sample variance.
- Take the square root. The result is the standard deviation.
Squaring removes the sign, so a drop and a rise of the same size count equally, and it gives extra weight to large moves.
A small worked example
These numbers are invented purely for illustration. Suppose an asset had these five daily returns, in percent: +2, -1, +3, -2, 0.
- Average: (2 - 1 + 3 - 2 + 0) / 5 = 0.4
- Deviations from the average: 1.6, -1.4, 2.6, -2.4, -0.4
- Squared deviations: 2.56, 1.96, 6.76, 5.76, 0.16
- Sum of squares: 17.2
- Sample variance: 17.2 / 4 = 4.3
- Standard deviation: the square root of 4.3, which is about 2.07 percent
So in this made-up sample the typical daily return sits roughly 2 percentage points from the average. Real calculations use many more observations, because five days say very little.
What realized volatility means
Realized volatility, sometimes called historical volatility, is simply the standard deviation of returns measured over a past window, such as the last 30 days. It is called realized because it describes what actually happened rather than what anyone expects. The window length is a choice. A short window reacts quickly to recent turbulence but is noisy. A long window is steadier but slow to notice a change in conditions.
Some analysts use an exponentially weighted version that gives more weight to recent days, which responds faster. Others compute volatility from intraday data, such as hourly returns, to capture movement that daily closing prices miss.
Annualizing
Daily volatility is often converted to an annual figure to make comparisons easier. The usual rule scales by the square root of the number of periods in a year. Crypto trades every day, so analysts commonly use 365 periods. For the example above:
Square root of 365 is about 19.1, and 2.07 percent times 19.1 is roughly 39.6 percent per year.
This is only an illustration. The square-root rule assumes returns on different days are independent of one another and that volatility is stable, assumptions that real markets often break. Annualized figures should be read as a convenient unit of comparison, not a forecast of what a year will look like.
Whenever a figure is quoted, check the window, the return frequency and the number of periods per year used. Two sources can report very different numbers for the same asset simply because they chose different settings.
Limits of standard deviation
Standard deviation is useful but incomplete.
- It treats upward and downward moves alike, although most people worry mainly about falls.
- It is based on the past and says nothing certain about the future. Volatility tends to change over time, so a calm window can precede a turbulent one.
- It can understate the chance of extreme moves. Crypto returns often show more very large moves than a bell-shaped distribution would suggest, a feature sometimes called fat tails.
- A few extreme days can dominate a short window, making the number jump when they enter or leave it.
Other measures fill some of these gaps. Maximum drawdown focuses on the worst peak-to-trough fall. Implied volatility, taken from options prices, reflects what the market expects rather than what has happened, and is derived from options pricing.
Where the figure shows up
Realized volatility appears on many data sites and charting tools, often as a single line under a price chart. Volatility envelopes such as Bollinger Bands are built on a similar idea, using the standard deviation of recent prices to draw bands around a moving average. Risk tools and position sizing methods sometimes use volatility to decide how large a position should be, so that more volatile assets get smaller positions.
None of this makes the future predictable. Cryptocurrency prices are volatile, past figures do not guarantee future behavior, and anyone can lose all they invest. This article is educational and is not financial advice.