Quantitative Methods & RiskProbability & Distributions for Traders
Volatility Clustering & Regimes
Volatility has a memory
If returns themselves are close to unpredictable, their magnitudes are not. Large moves tend to be followed by large moves, of either sign, and quiet periods by quiet periods. This is volatility clustering, first documented by Mandelbrot in 1963, and it is one of the most robust empirical facts in all of finance. It is why a single daily σ estimate is misleading: volatility is a moving target with calm regimes and stormy regimes.
Volatility clustering
The tendency of high-volatility days to bunch together and low-volatility days to bunch together. Returns are roughly uncorrelated, but absolute or squared returns are strongly positively autocorrelated. Models like GARCH exist precisely to capture this.
Square-root-of-time scaling (and its fine print)
σ_T = σ_1 · √T
EWMA: weight recent vol more heavily
A simple way to track a moving volatility target is an exponentially weighted moving average of squared returns: σ²_t = λ·σ²_{t−1} + (1−λ)·r²_{t−1}. RiskMetrics popularised λ = 0.94 for daily data, which gives recent days more influence and adapts quickly when a calm regime breaks. GARCH(1,1) generalises this by adding a long-run mean the vol reverts to.
EWMA volatility update
σ²_t = λ · σ²_{t−1} + (1 − λ) · r²_{t−1}At λ = 0.94, the effective lookback is ~17 days and the half-life (where a day's weight has halved) is ~11.2 days. Lower λ reacts faster to a new volatility regime but is noisier; higher λ is smoother but slower to recognise that the market has changed.
Clustering breaks √T and breaks VaR
Because volatility clusters, a 10-day VaR scaled from a calm-period 1-day VaR using √10 will badly understate risk if the next 10 days land in a storm. The 2008 and 2020 risk-model failures were, in large part, failures to recognise that the volatility regime had shifted under the model's feet.
Terminal distribution
Knowledge check
Daily volatility is 1.5%. Under the standard i.i.d. assumption, what is the approximate annualized volatility?