Standard Deviation
The classic measure of dispersion that most options models use to quantify volatility and risk.
What Standard Deviation Measures
Standard deviation (denoted σ, the lower-case Greek sigma) is the most common statistical measure of dispersion or variability of a data set around its mean. Intuitively, it answers the question: "how far from the mean do the values typically sit?" A low standard deviation indicates data clustered near the mean; a high one indicates data spread widely. Mathematically, for a sample of n data points {x₁, x₂, …, xₙ} with mean x̄: σ = √[Σ(xᵢ − x̄)² / (n−1)]. The formula reads as "the square root of the average squared deviation from the mean". Squares are used instead of absolute values for mathematical reasons — they ease the calculation, have desirable analytical properties, and penalise large deviations more heavily. In trading, standard deviation is operationally synonymous with volatility: the standard deviation of an asset’s logarithmic returns, generally annualised. If an asset has an annual σ of 20%, its returns typically vary ±20% around its annual mean. Standard deviation is fundamental because it is embedded in practically every modern financial model: Black-Scholes assumes lognormal returns with constant σ, VaR uses σ to quantify risk, and the Sharpe ratio divides excess return by σ.
The 68-95-99.7 Rule
If data follows a normal distribution (the Gaussian bell curve), there is a memorable rule known as the empirical rule or the 68-95-99.7 rule: (1) roughly 68% of observations fall within ±1 standard deviation of the mean; (2) roughly 95% within ±2; (3) roughly 99.7% within ±3. This rule is extraordinarily useful for quick interpretation. Applied to an asset at $100 with 20% annualised implied volatility: over the next 365 days there is (1) about a 68% probability price finishes between $80 and $120 (±1σ, assuming lognormality); (2) about 95% between $64 and $144 (±2σ); (3) about 99.7% between $51 and $173 (±3σ). These figures are the mental reference many traders use when choosing a strike: selling a 16-delta put is roughly equivalent to selling one standard deviation out of the money, implying around an 84% probability of expiring worthless. Two caveats are worth remembering: (1) this assumes a normal distribution, which is only an approximation — financial returns have fat tails that make extreme events more frequent than predicted; (2) volatility is annualised, so for shorter horizons it must be adjusted by multiplying by √(T/365). For 30 days: σ_30d = σ_annual × √(30/365) ≈ σ_annual × 0.286.
Daily vs Annualised Volatility
Price volatility is almost always reported annualised, but in practice you need to convert it to useful horizons (daily, weekly, monthly, or to a specific expiration). The conversion assumes independence between sessions, a reasonable first approximation: total variance is the sum of individual variances, and standard deviation scales with the square root of time. The key formula: σ(T days) = σ(annual) × √(T / 252), where 252 is roughly the number of trading days in a year. Examples: (1) VIX at 20% annualised → daily σ = 20% / √252 ≈ 1.26% per day; assuming normality, there is about a 68% probability the SPX moves less than 1.26% on a given day, and about 95% less than 2.52%; (2) Bitcoin with 80% IV → daily σ ≈ 5.04%, which fits with daily moves of 3-5% being entirely normal; (3) Apple with 25% IV → monthly σ ≈ 7.2%, a reasonable figure for a large cap. That volatility scales with the square root of time is fundamental to options pricing: a 60-day option accumulates roughly 1.41 times the deviation of a 30-day one, not double. That also explains why time decay accelerates near expiration — extrinsic value depends on σ × √T, and as T shrinks, value decays at an ever-increasing rate.
How It Is Used in Implied Volatility
Implied volatility (IV) is simply the annualised standard deviation of the underlying implied by the market price of its options under a pricing model (usually Black-Scholes). It is the inverse input: instead of using a known σ to calculate the option’s theoretical price, you take the market price and solve for the σ that makes the model fit. If IV is 30% for a stock’s options, the market is "saying" it expects 30% annualised σ through expiration. IV is critical for several decisions: (1) if implied persistently exceeds realised, options are on average expensive and selling premium carries a structural edge, known as the variance risk premium; (2) IV Rank indicates where current implied sits within its annual range, which helps decide when to enter; (3) IV Percentile indicates what share of sessions over the past year closed with implied below the current level. Systematic traders sell premium with IV Rank above 50 and buy it below 20; the historical behaviour of the SPX, the VIX and most individual stocks supports that approach. Implied volatility also exhibits skew, differences between strikes, and term structure, differences between expirations: two additional dimensions full of information.
Limitations and Fat Tails
Standard deviation has significant limitations as a risk measure, some of them brutally exposed by historical crashes. (1) It assumes a normal distribution, but financial returns have fat tails, or excess kurtosis: extreme events occur far more frequently than the normal predicts. Black Monday (19 October 1987) was roughly a 22σ event by the model; under a normal distribution that should occur less than once in the age of the universe. And yet several episodes of that magnitude have happened in the four decades for which we have data. (2) It treats upside and downside deviations equally, when what concerns a trader is losses. Alternative measures such as semi-deviation, which counts only negative returns, maximum drawdown or conditional VaR capture real and psychological risk far better. (3) It assumes independent and identically distributed returns, when markets exhibit volatility clustering: agitated periods follow agitated periods, and calm follows calm. GARCH models capture that behaviour far better than a constant deviation. (4) It does not capture asymmetry: two distributions can share the same deviation with very different skew, with the long tail up or down. Market returns tend to have negative skew. Despite these limitations, σ remains the most widely used volatility measure for its simplicity and good mathematical properties, but it should always be complemented with specific tail-risk analysis.
Practical Application for Options Traders
For anyone trading options actively, standard deviation and its implied version have practical applications every day. (1) Sizing the position: if the underlying’s daily deviation is 2%, your stop should sit at least two deviations away — 4% in this example — so normal noise does not take you out; if that means risking more capital than acceptable, the position is simply too large. (2) Choosing the strike: selling a 16-delta put is roughly equivalent to selling one deviation out of the money, with around an 84% probability of expiring worthless; a 7 delta is about two deviations, around 97%. These are approximations, but they work well for deciding quickly. (3) Estimating the expected move: the ATM straddle price runs around 80% of σ × S × √T, so you can back out implied volatility from the straddle. If Apple’s pre-earnings straddle trades at $5 with the stock at $150 and one day to the event, implied volatility is around 80%. (4) Comparing strangle and straddle: a strangle placed one deviation out on each side should trade at 40-50% of the equivalent ATM straddle under a normal distribution; a large deviation from that proportion signals a tradeable skew. (5) Dynamic management: when real-time realised volatility exceeds the implied that was quoted at entry, consider hedging or closing; when it falls below, holding and expecting realised to revert toward implied is usually right.