Reader question: Financial returns often look almost unpredictable in direction, yet large moves tend to cluster near other large moves. How can an algorithm forecast that changing uncertainty without pretending it can forecast whether the next return will be positive or negative?
The key idea is to model the conditional variance separately from the conditional mean. ARCH, GARCH and EWMA algorithms do not need tomorrow’s return sign to be predictable. They ask a different question: given the shocks observed so far, how large is the next return likely to be in magnitude?
This page explains the recursion that makes those models work, how persistence and half-life emerge mathematically, why EWMA is a special limiting case, how multi-step forecasts mean-revert under stationary GARCH, why heavy tails and asymmetric responses matter, and how to diagnose a volatility model that is numerically converged but economically untrustworthy.
What this page owns — and what it does not
This page owns the algorithmic transformation:
return history + variance model + fitted parameters → conditional variance path and volatility forecasts.
It does not replace implied-volatility inversion, which extracts a model parameter from option prices; Heston calibration, which fits a stochastic-volatility option model; or market-risk VaR and expected-shortfall algorithms, which consume volatility and dependence estimates inside a broader loss framework.
This is statistical and mathematical education. It is not a forecast for any specific security and not personalized financial advice.
Why constant variance is often a poor model for financial returns
A simple return model might write:
rt = μ + εt,
with a constant variance:
Var(εt) = σ².
That says the uncertainty tomorrow is the same after a quiet month as it is after a market shock. Financial time series often contradict that simplification: large absolute or squared returns cluster together, while the signed returns themselves can have little linear serial correlation.
Robert Engle’s 1982 ARCH model changed the question from “what is the one constant variance?” to “how does variance depend on information available through time?” Tim Bollerslev’s 1986 GARCH extension then allowed previous conditional variance itself to become part of the recursion.
ARCH: let recent shocks determine current variance
An ARCH(1) model can be written:
σt² = ω + α εt−1².
A large shock yesterday raises today’s forecast variance whether the shock was positive or negative because it enters squared.
Higher-order ARCH models include several lagged squared shocks:
σt² = ω + α₁εt−1² + α₂εt−2² + … + αqεt−q².
The weakness is that volatility persistence can require many lag terms. GARCH compresses that long memory-like behaviour into a much smaller recursion.
GARCH(1,1): the workhorse recursion
The standard GARCH(1,1) variance equation is:
σt² = ω + α εt−1² + β σt−1².
The three terms have distinct jobs:
- ω supplies a positive long-run variance anchor;
- α measures the immediate response to the latest squared shock;
- β carries yesterday’s conditional variance into today.
The model is recursive. Once an initial variance and fitted parameters exist, the entire conditional-variance history can be updated one observation at a time.
The mechanism in one numerical step
Suppose:
- ω = 0.000002;
- α = 0.08;
- β = 0.90;
- yesterday’s variance = 0.0001;
- yesterday’s shock = 2%, so ε² = 0.0004.
Then:
σt² = 0.000002 + 0.08(0.0004) + 0.90(0.0001) = 0.000124.
The conditional standard deviation is:
σt ≈ √0.000124 ≈ 1.114%.
The shock raised the next conditional volatility above the previous 1% level, but the model also carries substantial persistence through β.
Persistence: α + β
For the standard stationary GARCH(1,1), the quantity:
p = α + β
controls how slowly a volatility shock decays.
If p is near zero, variance forgets shocks rapidly. If p is near one, shocks remain influential for many periods.
Under the usual finite-variance stationary condition:
α + β < 1.
When that condition holds, the unconditional variance is:
σ̄² = ω / (1 − α − β).
This formula is useful both economically and diagnostically. A fitted model with α + β extremely close to one can imply a very large or unstable long-run variance unless ω is correspondingly small.
Half-life of a variance shock
When 0 < α + β < 1, the expected deviation from long-run variance decays approximately geometrically. A common half-life diagnostic is:
h1/2 = ln(0.5) / ln(α + β).
If α + β = 0.98:
h1/2 ≈ 34.3 periods.
This does not mean volatility becomes normal after exactly 34 days. It means the expected excess variance created by a shock has fallen to roughly half its starting deviation under the fitted recursion.
EWMA: a parameter-light recursive alternative
The exponentially weighted moving average variance recursion is:
σt² = λσt−1² + (1 − λ)rt−1².
J.P. Morgan’s RiskMetrics framework popularised this approach, with the 1996 Technical Document becoming a standard historical reference for market-risk implementation.
EWMA gives geometrically declining weight to older squared returns:
weight k periods ago ∝ (1 − λ)λk−1.
A larger λ remembers history longer and reacts more slowly. A smaller λ reacts sharply to recent observations but produces a noisier volatility path.
EWMA as an IGARCH-like special case
Compare EWMA with GARCH(1,1):
GARCH: σt² = ω + αεt−1² + βσt−1².
EWMA: σt² = (1−λ)rt−1² + λσt−1².
EWMA resembles a GARCH recursion with:
- ω = 0;
- α = 1 − λ;
- β = λ;
- α + β = 1.
That means it has no stationary long-run variance anchor inside the recursion. A shock decays from the current estimate but the multi-step forecast does not mean-revert toward a separately estimated unconditional variance in the way stationary GARCH does.
EWMA half-life
The weight half-life is:
h1/2 = ln(0.5) / ln(λ).
For λ = 0.94, the half-life is about 11 observations. This gives a more intuitive interpretation of the decay factor than the raw number 0.94 alone.
One-step versus multi-step forecasting
A volatility model should distinguish a one-period forecast from an h-period horizon.
For a stationary GARCH(1,1), the expected future variance can be written:
Et[σt+h²] = σ̄² + (α+β)h−1(σt+1² − σ̄²).
As h grows, the conditional forecast approaches the long-run variance.
Under the simple EWMA/IGARCH recursion, the expected future conditional variance remains anchored to the latest estimate rather than mean-reverting to a separate σ̄².
Why square-root-of-time scaling can fail
A common shortcut takes a one-day volatility and multiplies it by √10 to estimate 10-day volatility.
That scaling is exact only under restrictive assumptions such as independent returns with constant variance. GARCH explicitly says conditional variance changes through time.
If conditional shocks are uncorrelated and the mean contribution is negligible, an h-period conditional return variance is better built by summing the forecast one-period variances:
Vart(Rt,t+h) ≈ Σj=1hEt[σt+j²].
When the first-day variance is far above long-run variance, mean reversion can make this sum materially different from h times the first-day forecast.
The mean model matters even when variance is the focus
GARCH operates on shocks:
εt = rt − Et−1[rt].
If the return mean contains autocorrelation, seasonality or another predictable component and the system simply treats raw returns as shocks, some mean structure can leak into the variance estimate.
For many liquid daily return series the mean is small relative to volatility, but that empirical convenience is not a mathematical right to ignore the mean in every asset, horizon or frequency.
Maximum likelihood and quasi-maximum likelihood
A common estimation method assumes standardized innovations:
zt = εt/σt
follow a standard normal distribution, then maximizes the conditional likelihood over ω, α and β subject to constraints.
Financial returns are often more heavy-tailed than normal. Gaussian quasi-maximum likelihood can still have useful consistency properties under regularity conditions even when normality is not literally true, but inference should use the appropriate robust theory rather than pretending the conditional density is correct.
An alternative is to fit a Student-t or other heavy-tailed innovation distribution directly.
Heavy tails: variance dynamics are not the whole distribution
Two models can produce the same σt² path and very different probabilities of extreme standardized shocks.
If:
εt = σtzt,
the GARCH recursion models σt, while the chosen distribution for zt models the conditional shape around that variance.
A normal innovation assumes thin Gaussian tails. Student-t innovations can allow more probability in extremes. Skewed distributions can address asymmetry.
This boundary matters if the variance forecast is later used in VaR or expected shortfall: volatility and tail shape are separate pieces of the loss model.
Symmetric GARCH cannot distinguish bad shocks from good shocks
Standard GARCH(1,1) uses ε², so +5% and −5% shocks have the same immediate variance impact.
Equity markets often exhibit asymmetric volatility response: negative returns can be followed by larger volatility increases than positive returns of the same magnitude.
Extensions include:
- GJR-GARCH, which adds an indicator term for negative shocks;
- EGARCH, which models log variance and can incorporate sign asymmetry without the same non-negativity parameter restrictions.
An asymmetry extension should be added because the data and use case justify it, not because a more complicated acronym automatically improves forecasts.
Inputs and outputs
A production volatility-forecasting engine can require:
- price or return series;
- return definition: simple or log;
- observation frequency and trading calendar;
- mean specification;
- variance specification: EWMA, GARCH, GJR, EGARCH or other;
- innovation distribution;
- estimation window;
- initial variance;
- parameter constraints;
- optimization method and tolerances.
Outputs should include:
- fitted parameters and standard errors where appropriate;
- conditional variance and volatility history;
- one-step and multi-step forecasts;
- long-run variance if defined;
- persistence and half-life;
- standardized residuals;
- likelihood and forecast diagnostics;
- convergence and constraint status.
Evidence polarity: what supports confidence?
Evidence for a useful GARCH model includes meaningful volatility clustering in the data, parameters inside the admissible region, standardized residuals with little remaining serial dependence in their squares, stable forecasts across nearby estimation windows, improved out-of-sample loss relative to simpler benchmarks, and sensible decay after known volatility shocks.
Evidence against confidence includes α+β pinned almost exactly at one because of an unmodelled structural break, repeated optimizer convergence to different local solutions, standardized residuals retaining strong ARCH effects, extreme sensitivity to a handful of returns, or a model that is beaten consistently by a much simpler EWMA or realized-volatility benchmark out of sample.
Residual diagnostics
After fitting:
zt = εt/σt.
If the model has captured conditional variance well, the squared standardized residuals should show far less serial structure than the original squared returns.
Useful diagnostics include:
- Ljung–Box tests or correlograms on zt for leftover mean dependence;
- Ljung–Box tests on zt² for leftover variance dependence;
- ARCH-LM tests on standardized residuals;
- QQ plots or tail diagnostics for innovation-distribution mismatch;
- rolling parameter stability and forecast-error plots.
Forecast evaluation needs a volatility proxy
True conditional variance is latent: we do not observe it directly after the fact.
Squared daily return is a noisy proxy. High-frequency realized variance can be more informative when reliable intraday data exist. Range-based estimators provide another proxy.
Andrew Patton’s work on volatility forecast comparison shows why the choice of loss function matters when the realized volatility proxy is itself noisy. QLIKE is widely used because it has useful robustness properties for comparing variance forecasts under imperfect proxies.
Counterexample: the highest likelihood model need not forecast best
A richer GARCH specification can improve in-sample likelihood by fitting historical quirks. That does not guarantee lower out-of-sample forecast loss.
Model selection should separate:
- in-sample fit;
- diagnostic adequacy;
- out-of-sample forecast performance;
- operational stability.
A model that wins only in-sample can be overfit.
Counterexample: a huge α+β can be a structural-break artifact
Suppose the true process had low volatility for five years and permanently shifted to a higher regime. A single stationary GARCH fitted across both periods may imitate that permanent break with very slow mean reversion, producing α+β close to one.
The model appears to discover extreme persistence when the underlying problem is a missing regime change.
This is why volatility persistence must be tested against structural breaks and rolling estimates.
Counterexample: volatility forecast and implied volatility answer different questions
A GARCH forecast estimates conditional volatility from the historical/statistical return process under a chosen physical-probability model.
Implied volatility is the parameter that makes an option-pricing model reproduce an observed option price under its pricing assumptions.
The two can differ because of risk premia, model differences, horizon, jumps, skew and supply/demand effects. One should not be treated as an error merely because it differs from the other.
Counterexample: annualized daily GARCH is not automatically a one-year forecast
Multiplying today’s daily σ by √252 assumes the current conditional variance persists unchanged across the entire year. Stationary GARCH says it mean-reverts.
A proper long-horizon forecast should aggregate the path of expected future conditional variances rather than blindly scaling the first-day value.
Weak links in implementation
Price cleaning. Bad corporate-action adjustments create artificial return shocks.
Timezone mismatch. Assets sampled at inconsistent closes create spurious volatility.
Stale prices. Zero-return runs followed by catch-up moves distort shock timing.
Optimizer constraints. An unconstrained routine can propose negative variance parameters.
Initialization. Poor initial variance treatment can matter in short samples.
Distribution mismatch. Normal residual assumptions can understate tails.
Window choice. Too short is noisy; too long can average across obsolete regimes.
Hidden re-estimation timing. Using future returns in a historical forecast backtest creates look-ahead bias.
Diagnostics: how to test the algorithm
- constant series test: near-zero returns should not create exploding variance.
- single shock test: inject one large return and verify geometric decay consistent with fitted persistence.
- EWMA identity test: set ω=0, α=1−λ, β=λ and confirm the recursion matches EWMA.
- stationarity test: verify the intended model satisfies or explicitly relaxes α+β<1.
- half-life test: compare simulated decay with the analytical persistence half-life.
- multi-step test: confirm stationary GARCH forecasts converge toward σ̄².
- residual ARCH test: standardized squared residuals should contain little remaining conditional heteroskedasticity.
- out-of-sample test: compare against EWMA, historical variance and realized-volatility benchmarks.
- stress test: replay crisis-sized shocks and verify no numerical overflow or negative variance.
- break test: split the sample around major regime changes and compare parameter stability.
What would falsify confidence?
Confidence should be withdrawn if variance becomes negative; if the same dataset and specification converge to materially different solutions without explanation; if standardized residuals retain strong ARCH structure; if multi-step forecasts violate the model’s own analytical mean-reversion property; if out-of-sample forecasts are persistently inferior to a simple benchmark; or if apparent persistence disappears once obvious structural breaks are modelled.
Alternatives and when they are better
EWMA is simple, transparent and inexpensive when a fast adaptive risk estimate matters more than explicit long-run mean reversion.
Realized-volatility models can exploit intraday data when microstructure noise is controlled.
Stochastic-volatility models such as Heston treat variance as its own latent stochastic process rather than a deterministic function of past squared returns.
Regime-switching models can represent discrete calm/stress states.
Option-implied measures use forward-looking market prices rather than historical return recursion.
The correct choice depends on whether the problem is historical risk measurement, short-horizon forecasting, derivative pricing, stress analysis or structural interpretation.
How this connects to the surrounding knowledge estate
GARCH supplies a time-varying variance estimate that can feed VaR and expected-shortfall algorithms. It contrasts with implied volatility, which inverts option prices, and with Heston calibration, where variance is stochastic and option-market information helps identify parameters. For multivariate risk, conditional-volatility estimates must eventually connect to a valid dependence structure such as the correlation-matrix layer.
Verification and update triggers
Preserve the return-definition version, price-cleaning rules, observation timezone, estimation window, distribution assumption, optimizer and constraints, parameter estimates, initial variance and forecast-loss metrics. Revalidate after market-structure changes, major volatility-regime shifts, data-vendor changes, corporate-action processing changes, persistent residual ARCH effects or a sustained deterioration versus simpler forecast benchmarks.
Primary and high-quality references
- Robert F. Engle, Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica, 1982.
- Tim Bollerslev, Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics, 1986.
- MSCI, 1996 RiskMetrics Technical Document.
- Andrew J. Patton, Volatility Forecast Comparison Using Imperfect Volatility Proxies, Journal of Econometrics, 2011.
- Robert F. Engle, GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics, Journal of Economic Perspectives, 2001.
Educational boundary: This article explains statistical volatility forecasting. It does not predict the future return or recommend any position, security or risk limit.