Before you can say whether an option is expensive, you need a yardstick for how much the stock tends to move. The most basic yardstick is what it has already done. Take the last few weeks of daily price changes, measure how spread out they are, and you have historical volatility, also called realized volatility or HV. Everything else in this module is measured against it.
From daily moves to one number
Start with daily returns: each day's close divided by the previous close, minus one. A stock that closed at $100 and then $101.50 had a 1.5% return. Collect twenty of those, one for each trading day of the past month, and compute their standard deviation, the usual measure of spread. Say it comes out at 1.5% per day.
Traders quote volatility per year, not per day, so the daily figure is scaled up. Spread grows with the square root of time, and there are about 252 trading days in a year, so the daily number is multiplied by the square root of 252, which is 15.9. A 1.5% daily standard deviation becomes 1.5 times 15.9, or about 24% annualized. That is the number you will see on every screen: a stock with 24% historical volatility has been moving about 1.5% on a typical day.
The reverse is a handy rule of thumb. Divide an annualized volatility by 16 and you get the typical daily move. 32% volatility means roughly 2% a day; 16% means roughly 1%.
Volatility is a measure of spread, so it is easiest to see as one. The chart shows the range of prices a $100 stock could finish at 30 days from now, for a given volatility.
Where a $100 stock might finish in 30 days at 24% volatility. Drag volatility up to 48% and the darker one-standard-deviation band doubles in width; drop it to 12% and it halves. The price in the middle does not move.
Double the volatility and the band doubles in width. That is all the number says. It carries no direction and no opinion about whether the stock goes up; it only measures how widely the outcomes scatter.
The window matters
Historical volatility is always measured over a window, and the length of that window is a choice. The Terminal shows several at once:
- 10-day: two weeks. Reacts fast, so it catches a change in behavior early, but two weeks is a small sample and one wild day swings it.
- 20-day: about one month. The most common comparison point for 30-day implied volatility, because a 30 calendar-day option covers roughly 20 trading days.
- 60-day: a quarter. Smoother, and it usually includes one earnings report.
- 90-day and longer: the slow baseline. It tells you what is normal for this stock, not what is happening this week.
The windows disagree, and the pattern of disagreement is information. If the 10-day reads 45% and the 60-day reads 28%, something has changed recently: the stock has started moving far more than its own recent norm. If the 10-day is well below the 60-day, the stock has gone quiet. Either way, one number alone would have hidden the story.
Close to close, or the whole day
The classic calculation only looks at closing prices. A stock that swings 3% during the day but finishes flat registers as a quiet day. ORATS also publishes a version that takes each day's trading range into account, and the Terminal draws both side by side. When the range version reads higher than the close-to-close version, the stock is moving inside the day more than the closes suggest.
What the number cannot tell you
Historical volatility is a measurement of a sample, and samples are noisy. Twenty daily returns is not much data: a 20-day reading of 24% carries a statistical margin of roughly 4 points either way, before anything about the stock has changed. That is one reason the longer windows exist. It is also blind to the calendar. A stock with earnings next week can show a calm 20-day reading right up to the report, because the window only knows what has already happened. Treat HV as the yardstick for the options you will price in the next lessons, not as a forecast.
The maths: computing historical volatility step by step
Desks use log returns, r = ln(P_today / P_yesterday), rather than simple percentage changes. For small moves the two are almost identical, but log returns add up neatly across days, and a rise and an equal fall in log terms cancel exactly.
Then: HV = stdev(r_1 ... r_n) × √252, where stdev is the sample standard deviation (divide the sum of squared differences from the mean by n minus 1) and 252 is the number of trading days in a year.
A worked example with six closes: $100, $101, $99.50, $101.50, $100.50, $102. The five log returns are about 0.99%, minus 1.50%, 1.99%, minus 0.99% and 1.48%. Their mean is 0.40%. The squared differences from the mean add up to about 0.000958; divide by 4 and take the square root and the daily standard deviation is 1.55%. Multiply by √252, which is 15.87, and the historical volatility is about 24.6%.
Why the square root? If each day's return is independent of the last, their variances add: 252 days carry 252 times the variance of one day, so the standard deviation is √252 times as large.
Your turn
Open TSLA in Price Context and read the HV range bars, which are the figures the grader checks. Hover a bar to see its exact value. Note how far apart the 20-day and 60-day readings are before you answer; that gap is the first thing a volatility trader looks at.