Glossary

Volatility (Finance) — Definition, Formula, and Limitations

Volatility is the standard deviation of periodic returns, usually annualised. Definition, formula, worked example, and why it is an incomplete measure of risk.

By NakedPnL Research·May 7, 2026·4 min read

TL;DR

Volatility is the standard deviation of periodic returns, conventionally annualised by multiplying by the square root of the number of periods per year.
It treats upside and downside variance symmetrically and assumes returns are approximately normal.
NakedPnL doesn't display volatility on profile pages because it is sample-window-sensitive and easy to suppress with negative-skew strategies.

Definition

In finance, volatility refers to the standard deviation of a portfolio's periodic returns, usually expressed on an annualised basis. It is the most common single-figure proxy for risk in modern portfolio theory and underpins the denominator of the Sharpe ratio. Realised volatility uses observed historical returns; implied volatility, used in options pricing, is a forward-looking quantity inferred from market prices.

Formula

sigma = sqrt( (1 / (N - 1)) * sum( (R_t - mean(R))^2 ) )

Annualisation:
  sigma_annual = sigma_periodic * sqrt(periods_per_year)

For daily returns, periods_per_year is conventionally 252 for equities or 365 for crypto.

The (N - 1) divisor is the unbiased sample estimator. Some implementations use N for the population estimator.

Worked example

A daily return series over 252 trading days has a mean of 0.05% and a sample standard deviation of 0.9%. The annualised volatility is 0.9% × √252 ≈ 14.3%. A second strategy with the same mean but with returns drawn from a fatter-tailed distribution may report a lower computed volatility over the same window simply because no extreme observation has yet appeared in the sample. Both numbers are correct given the data; neither is a complete description of risk.

Why NakedPnL doesn't display this on profile pages

Volatility has three structural weaknesses as a public summary statistic. It is symmetric (a 5% up day and a 5% down day count equally), it assumes the underlying distribution is approximately normal (real return distributions are typically fat-tailed and negatively skewed), and it is sensitive to the lookback window — a manager who systematically writes short volatility through negatively-skewed positions can produce a low realised-vol number until the rare tail event arrives. On a public registry of verified performance, displaying a metric whose value can be suppressed by deliberate tail-risk concentration runs counter to the platform's quality goals. NakedPnL therefore restricts profile-level metrics to TWR, total PnL, and trade count. The underlying daily-return series is exposed through the API for allocators who want to compute volatility, downside deviation, or higher-moment statistics with their own assumptions.

Standard deviation
Sharpe ratio — uses volatility as the denominator
Downside deviation — used in the Sortino ratio
Implied volatility — option-implied future volatility

Frequently asked questions

Why annualise daily volatility by the square root of time?

Under the assumption that daily returns are independent and identically distributed with constant variance, the variance of the sum scales linearly with the number of periods, so the standard deviation scales with the square root. This 'square-root rule' is a useful approximation, not a law; in practice returns exhibit autocorrelation and volatility clustering that violate the assumption.

Is high volatility always bad?

No. Volatility is a measure of dispersion, not of expected loss. A strategy with high volatility but high expected return can be a better holding than a low-volatility strategy with negligible return. The Sharpe and Sortino ratios attempt to combine the two ideas into a single figure.

What is the difference between historical and implied volatility?

Historical (realised) volatility is computed from observed past returns. Implied volatility is the volatility number that, plugged into an options pricing model such as Black-Scholes, reproduces the current market price of an option. They typically differ; the difference is a source of trading strategies.

References

Volatility (finance) — Wikipedia