Standard Deviation¶
Standard deviation (σ) measures how widely a set of returns disperses around their mean, and in trading it is the primary quantitative expression of volatility. A stock with a 2% daily standard deviation swings twice as hard as one with a 1% daily standard deviation. It is the foundational risk unit underlying implied volatility, Bollinger Bands, Value at Risk, and the Sharpe ratio.
HOW IT WORKS¶
Given n periodic returns with mean μ, standard deviation is calculated as:
σ = √[ Σ(rᵢ − μ)² / (n − 1) ]
Subtract the mean from each return, square the differences, average them (dividing by n − 1 for a sample), and take the square root. The result is in the same units as the returns — typically percent. Traders annualize by multiplying by the square root of 252 trading days: a 1.5% daily σ becomes approximately 23.8% annualized. Under a normal distribution, roughly 68% of returns fall within ±1σ of the mean, 95% within ±2σ, and 99.7% within ±3σ — the rule set behind Bollinger Band interpretation and options one-standard-deviation move estimates.
Key limitations: σ treats upside and downside dispersion equally, assumes approximately normal returns (equity tails are fatter in practice), and is backward-looking — trailing σ reflects past volatility, not future risk.
IN TAPEBOARD¶
Tapeboard surfaces standard deviation across the risk and options workflows. In the position sizing tools, vol-targeting logic scales exposure inversely to each position's σ, so a stock running twice the volatility receives proportionally less capital. On the options chain, historical (realized) σ is displayed alongside implied volatility, letting traders immediately assess whether options are priced rich or cheap relative to recent realized dispersion. Scanner filters accept σ thresholds so you can screen for statistically stretched moves — for example, flagging names trading beyond 2σ from a moving average as mean-reversion candidates.