Historical Volatility

Q: What is historical volatility?

Historical volatility, also called realised volatility, is the standard deviation of an underlying's daily returns over a past window, scaled to a yearly figure. A NIFTY historical volatility of 12% means that over the window measured, daily moves had the dispersion you would expect from an index that moves about 12% a year, or about 0.76% a day. It is a backward-looking measurement, not a forecast.

Q: How is historical volatility calculated?

Take the closing prices for the window, compute each day's log return as ln(today's close divided by yesterday's close), compute the standard deviation of those returns, and multiply by the square root of 252, the approximate number of trading days in a year. A 20-day window uses 21 closes and gives 20 returns; a 30-day or 60-day window is also common.

Q: What is the difference between historical and implied volatility?

Historical volatility measures what the underlying did; implied volatility is what option prices say the market expects it to do until expiry. Implied volatility on indices usually sits above historical, because option sellers charge a premium for bearing the risk of a jump. The gap between the two, the volatility risk premium, is what premium sellers earn on average and what buyers pay for protection.

Q: Which lookback window should I use?

Match the window to the option's life. For a weekly option a 10- or 20-day historical volatility is more relevant; for a monthly a 20- or 30-day figure. Shorter windows react fast but are noisy; longer windows are stable but slow to reflect a change in regime. Many traders watch two windows and note when the short one crosses the long one.

Q: If implied volatility is above historical, are options expensive?

Not necessarily. Implied is normally above historical on indices, so a gap is the base case, not a signal. The question is whether the gap is unusually wide or narrow compared with its own recent history, and whether an event inside the option's life justifies it. A narrow or negative gap before a known event means the market expects a move that the past window did not contain.

What the index actually did, measured as one number. The reference point against which every implied volatility is judged, and a calculation you can do in a spreadsheet in a minute.

In short: historical volatility (HV, also realised volatility) is the annualised standard deviation of an underlying's past daily returns. It measures how much the market moved over a chosen window, typically 10, 20 or 30 trading days. It is the backward-looking twin of implied volatility: HV is what happened, IV is what options are priced for. On NIFTY, IV usually sits a few points above HV, and the size of that gap is the main input to any judgement about whether premium is rich or cheap.

The intuition

An index that closes within half a percent of the previous close day after day is quiet; one that swings two percent a day is not. Historical volatility turns that impression into a number by measuring the spread of daily returns around their average. The bigger the typical daily move, the higher the figure. Scaling to a year is a convention that makes the number comparable with implied volatility, which is also quoted annualised, and with India VIX.

It is a measurement, not a forecast. Nothing in a 20-day HV says anything about the next 20 days except by the assumption that the recent past continues. That assumption is often reasonable and sometimes badly wrong, and knowing which is the whole craft of comparing it with IV.

The formula

For a window of n daily returns:

Daily log return: r_t = ln(C_t / C_t−1), where C is the closing level.
Standard deviation of the returns: σ_daily = √[ Σ(r_t − r̄)² / (n − 1) ].
Annualise: HV = σ_daily × √252, since variance adds across independent days and there are about 252 trading days in a year.

Log returns are used rather than simple percentage changes because they add up across days and treat a rise and the fall that reverses it symmetrically; for daily moves on an index the difference from simple returns is small.

Worked example (illustrative numbers)

Suppose NIFTY is at 24,000 and the last 20 daily log returns, in percent, were:

+0.4, −0.6, +1.2, −0.3, +0.1, −0.9, +0.7, +0.2, −1.1, +0.5, +0.3, −0.2, +0.8, −0.4, +0.6, −0.7, +0.1, +0.9, −0.5, +0.2

Mean daily return	+0.065%
Standard deviation of daily returns	0.64%
Annualising factor √252	15.87
20-day historical volatility	10.1%

If the 27-day NIFTY at-the-money option is trading at 14% IV, the market is pricing the coming month about four points more volatile than the last one was. That is a typical gap. If IV were 10%, the market would be saying the next month looks like the last, which on an index with an event inside the window would be worth questioning. If IV were 20%, something specific is expected, or fear has risen; in either case the past window no longer describes the future the market sees.

In rupee terms, 10.1% annual volatility on a 24,000 index is a one-standard-deviation daily move of about 153 points; 14% is about 212 points. The option market, at 14%, is charging for a wider daily range than the index has recently delivered.

Implied volatility usually runs above historical. The gap compresses when realised moves catch up with expectations and widens when the market calms faster than option prices do.

Why it matters to an options trader

It anchors "expensive" and "cheap". An IV of 14% means nothing alone; against a 10% HV it is a healthy premium, against an 18% HV it is a discount.
It is what a short-premium trade is paid for. A seller profits on average when realised volatility over the option's life comes in below the implied volatility they sold. The HV-IV gap is the expected edge; the realised gap is the outcome.
It sizes a hedge. Delta-hedging a position costs money in proportion to realised volatility; the HV tells you the recent cost of carrying it.
It flags regime change. A 10-day HV crossing above the 30-day says the market has started moving more than its recent norm, usually before IV fully catches up.

Common mistakes

Treating any IV above HV as a sell signal. IV is above HV most of the time on indices. Judge the gap against its own history.
Using a long window for a short option. A 60-day HV says little about a six-day weekly; use a window close to the option's life.
Ignoring the mean. A trending market with small daily changes has low HV; a one-way move is not the same as volatility, and an option on it can still lose on direction.
Mixing close-to-close with intraday ranges. Close-to-close HV misses large intraday swings that reverse; range-based estimators give higher figures and are not comparable without saying so.
Forgetting that HV is backward-looking. The day after a shock, 20-day HV jumps and stays high for twenty sessions, long after the market may have settled.

Indian-market specifics

NSE publishes daily volatility figures for every F&O stock and index as part of its margin computation; those use an exponentially weighted method rather than a simple window, so they differ from a plain 20-day HV.
NIFTY's historical volatility in calm periods sits in the low teens or single digits, and India VIX, a forward measure, has generally sat above it; the gap narrowed sharply only in stress periods.
Trading days per year on NSE are close to 250 after holidays, so √252 and √250 give almost the same answer; what matters is using the same convention for comparison.
For Bank Nifty and stock options the appropriate HV is that of the underlying itself, not of NIFTY; the gaps to their own IVs behave differently, with stocks carrying larger event premiums around results.

FAQ

What is historical volatility?

The annualised standard deviation of past daily returns over a chosen window, a measurement of what the underlying actually did.

How is historical volatility calculated?

Daily log returns, their standard deviation, multiplied by the square root of 252.

What is the difference between historical and implied volatility?

Historical measures the past; implied is the market's forward estimate embedded in option prices, and it usually sits above historical on indices.

Which lookback window should I use?

One close to the option's remaining life: 10 to 20 days for weeklies, 20 to 30 for monthlies.

If implied volatility is above historical, are options expensive?

Only if the gap is wide relative to its own history and no event inside the option's life explains it.

HV against IV, from the latest close

The NIFTY option-chain page shows the current implied volatility at every strike, and the IV page tracks India VIX, the forward measure, over time. In the app, the option chain's analytics strip shows the spread between implied volatility and 30-day historical volatility for the selected expiry.

IV and India VIX Historical vs implied guide

Related terms

Implied Volatility, the forward-looking twin
India VIX, the market's 30-day volatility index
Vega, how option prices respond when IV moves
Term Structure, implied volatility across expiries