Volatility

Volatility starts from a series of prices \(P_0, P_1, \ldots, P_n\) observed at regular intervals (daily closes, say). The return over each interval is usually taken as the log return, \(r_t = \ln(P_t / P_{t-1})\), rather than the simple percentage change, because log returns compound additively over time and are approximately symmetric for gains and losses of the same magnitude.

Variance is the average squared deviation of these returns from their mean:

\[\text{Var}(r) = \frac{1}{n-1}\sum_{t=1}^{n} (r_t - \bar{r})^2\]

Volatility is simply the square root of variance — \(\sigma = \sqrt{\text{Var}(r)}\) — expressed in the same units as the return itself (e.g. "1.2% per day"). Variance is the more natural object mathematically (it adds under independence, which is why GARCH models below are formulated in terms of variance), but volatility is the conventional unit quoted in markets because it is directly comparable to a return.

A daily volatility figure is not directly comparable to a monthly or yearly one, so volatility is conventionally annualised by scaling with the square root of the number of periods in a year — the \(\sqrt{t}\) scaling of a random walk's standard deviation. For daily returns, the convention is \(\sqrt{252}\) (the typical number of trading days in a year), not \(\sqrt{365}\): only days the market actually trades contribute return observations. A daily vol of 1% therefore annualises to roughly \(1\% \times \sqrt{252} \approx 15.9\%\). Some markets quote vol on calendar days (\(\sqrt{365}\)) instead — see "Normalised Vol" in the FX Volatility Surface page's volatility-kinds table for how this ambiguity is handled explicitly rather than assumed.

Historical volatility is computed directly from a window of past returns using the variance formula above — it is a backward-looking statistic, no different in kind from any other sample standard deviation. The main design choice is the window: how many past observations to include (30 days? 252 days?) and whether older observations are weighted equally (a simple rolling window) or decayed exponentially (an EWMA — exponentially weighted moving average — which reacts faster to recent moves). Volatility clustering and GARCH models covers a more sophisticated family of models for how realised volatility itself evolves and clusters through time (calm periods followed by turbulent ones), rather than being constant.

Implied volatility is not measured from a price history at all — it is inferred from a single observed market price. The Black-Scholes option pricing formula takes five inputs (spot, strike, time to expiry, risk-free rate, and volatility \(\sigma\)) and produces a theoretical option price. Every one of those inputs except \(\sigma\) is directly observable in the market; \(\sigma\) is the one degree of freedom the model leaves free. Given an actual traded option price, implied volatility is the value of \(\sigma\) that, plugged back into Black-Scholes, reproduces that price exactly — found numerically (there is no closed-form inverse), typically via Newton-Raphson or bisection on the pricing formula.

Because it is backed out from where the market is actually trading an option right now, implied volatility is inherently forward-looking and reflects the market's collective expectation of future volatility over the option's remaining life — as distinct from historical volatility, which only reflects the past. The two can and routinely do diverge, sometimes sharply (e.g. implied vol spiking ahead of an anticipated event while historical vol, measured from the calm period before it, stays low).

If the Black-Scholes model's constant-\(\sigma\) assumption held exactly, every option on the same underlying and expiry — regardless of strike — would imply the identical \(\sigma\) when inverted. In practice it does not: implied volatility varies systematically by strike (and by expiry), tracing out a curve conventionally called the volatility smile (or skew, when asymmetric). This is the market pricing in fatter tails and directional bias than the log-normal Black-Scholes model assumes, rather than a flaw in the inversion itself. FX Volatility Surface covers how this smile is quoted, parameterised (ATM/RR/STR pillars), and modelled (SABR, spline) in the FX market specifically, and how the smile is stitched across strike and expiry into a full two-dimensional surface; Vol surface no-arbitrage conditions covers the constraints that surface must satisfy to avoid implying a static arbitrage.

Every section above treats volatility as if it were the natural measure of risk — the bigger the swings, the riskier the position. This is the standard desk assumption, but it is worth stating precisely where it can mislead: volatility is a property of what a system's path has looked like; risk, in the sense of how close a system is to losing stability altogether, is a property of the law generating that path, and the two are not interchangeable.

Take the canonical mean-reverting process (the same Ornstein-Uhlenbeck process behind Vasicek/Hull-White/CIR short-rate models elsewhere in this knowledge base):

\[dx_t = -\theta (x_t - \mu)\, dt + \sigma\, dW_t\]

Two parameters, opposite roles: \(\sigma\) is the size of the random shocks; \(\theta\) is the speed at which the system pulls back toward its equilibrium \(\mu\) — equivalently, the eigenvalue \(-\theta\) of the linearised system, governing its stability. The process's stationary variance — the quantity a volatility estimator actually recovers from a sample path — is

\[\text{Var}(x) = \frac{\sigma^2}{2\theta}\]

Observed volatility therefore only pins down the ratio \(\sigma^2 / \theta\), never \(\sigma\) and \(\theta\) separately. Hold that ratio fixed while letting \(\theta \to 0\): the restoring force vanishes, the process drifts toward a non-stationary random walk and edges toward instability, yet its measured volatility stays flat — or can even fall. A system moving toward instability can display calm, constant volatility the entire way there, because the parameter that actually governs the risk, \(\theta\), is invisible to volatility alone; no amount of historical data recovers it without explicitly modelling the dynamics — an identification problem, not a sample-size one. This is the content of "critical slowing down" near a transition: what changes is not the amplitude of the moves (what volatility measures) but the time the system takes to recover from them — a different observable entirely.

Every QUANT desk regards LOW VOLATILITY as low risk. At Quantis, this is the assumption we oppose most strongly. Volatility is a statistical measure that reflects what is observed. But what about risk? Risk is a property of the way the system evolves. They are not the same thing, and one does not determine the other. […] Look at what this says. Observed volatility fixes only the ratio \(\sigma^2/\theta\), not the two parameters separately. […] A system marching toward instability can display calm, constant volatility the whole way there. […] Volatility tells you how much the system moved. It does not tell you how stable the law that moved it is.

— Alessio Fratini, Quantis Research LinkedIn, source