Volatility Estimation: EWMA and GARCH Models Explained

The GARCH model is the industry standard for estimating and forecasting financial market volatility. Unlike simple historical measures that treat all past observations equally, conditional volatility models like EWMA and GARCH recognize a fundamental truth about financial markets: volatility changes over time, and recent returns carry more information about current risk than distant ones. This guide covers the full progression — from basic historical variance to the GARCH(1,1) model used by banks, risk managers, and derivatives traders worldwide.

Understanding how to estimate volatility is essential for anyone involved in risk management, option pricing, or portfolio construction. A Value at Risk (VaR) model is only as good as its volatility input. A derivatives trader who misprices volatility will misprice options. This article explains the key models, their formulas, and when to use each one — with a worked example you can follow step by step.

What Is Volatility Estimation?

Volatility measures the magnitude of price fluctuations in a financial asset. Unlike price or return, volatility is not directly observable — it must be estimated from historical return data or inferred from option prices (implied volatility). This article focuses on the first approach: estimating volatility from observed returns.

The distinction between unconditional and conditional volatility is central to this topic. Unconditional volatility is a single long-run average — treat all data equally and compute one number. Conditional volatility changes at each point in time, reflecting the information available up to that moment. For risk management, conditional estimates are far more useful because they capture the current volatility regime.

An important simplification used throughout: for daily returns, the expected return is approximately zero (the daily drift is negligible compared to daily volatility). This means we can use observed returns r_t directly in variance formulas without subtracting the mean — a standard convention in volatility modeling.

Simple Historical Volatility

The simplest approach to estimating volatility is to compute the realized variance — the equally weighted average of squared daily returns over a lookback window of n days.

This is the realized variance form under the zero-mean assumption. It differs from the textbook sample variance formula (which uses 1/(n-1) and subtracts the sample mean), but for daily data with hundreds of observations and a near-zero mean, the difference is negligible. To annualize, multiply the daily variance by 252 (trading days per year) and take the square root:

The key limitation: every observation receives equal weight, regardless of how recent it is. A return from six months ago has the same influence as yesterday’s return. Consider the March 2020 COVID crash — the S&P 500 fell over 30% in weeks, then recovered sharply. A 1-year historical volatility estimate would remain elevated for months after markets had calmed, overstating the true current risk. This is the motivation for weighted approaches.

EWMA (Exponentially Weighted Moving Average)

The Exponentially Weighted Moving Average (EWMA) model addresses the equal-weighting problem by assigning exponentially declining weights to past observations. Recent returns matter more; distant returns fade away.

Where λ (lambda) is the decay factor, typically between 0.90 and 0.99. JPMorgan’s RiskMetrics system, introduced in 1994, popularized λ = 0.94 for daily data and λ = 0.97 for monthly data. The weight assigned to an observation k days ago is (1 – λ)λ^k, which declines geometrically — at λ = 0.94, data from 30 trading days ago receives only about 16% of the weight given to yesterday’s return.

To illustrate, consider Apple (AAPL) stock with yesterday’s EWMA variance estimate of σ_n-1² = 0.00025 (1.58% daily vol) and yesterday’s return of r_n-1 = -4.5% following an earnings miss. Using λ = 0.94: σ_n² = 0.94 × 0.00025 + 0.06 × 0.002025 = 0.000235 + 0.000122 = 0.000357, giving σ_n = 1.89% daily. The single large return immediately increased the volatility estimate from 1.58% to 1.89% — and under EWMA, this elevated level will persist in all future forecasts until offset by smaller returns.

The GARCH(1,1) Model — Formula and Parameters

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, introduced by Tim Bollerslev in 1986, extends EWMA by adding a crucial third component: a constant term ω that pulls volatility back toward a long-run level.

Where ω > 0, α ≥ 0, β ≥ 0, and α + β < 1 (the covariance stationarity condition).

Parameter interpretation:

• α (alpha) — shock sensitivity: how strongly yesterday’s squared return (new information) affects today’s variance estimate. Higher α means volatility reacts more sharply to market surprises.
• β (beta) — persistence: how much of yesterday’s conditional variance carries forward. Higher β means volatility decays slowly after a shock.
• ω (omega) — baseline contribution: ensures variance has a positive floor and drives mean reversion to the long-run level.

The sum α + β is called persistence — it determines how slowly volatility decays back toward V_L after a shock. The volatility half-life (trading days for a shock to decay by half) is approximately ln(0.5) / ln(α + β). This is the key advantage of GARCH over EWMA: mean reversion. Elevated volatility will gradually return to V_L, and depressed volatility will gradually rise back to V_L.

The “(1,1)” in GARCH(1,1) refers to 1 lag of the squared return and 1 lag of the conditional variance. This specification is sufficient for the vast majority of financial applications — higher-order models like GARCH(2,1) rarely provide meaningful improvement.

GARCH Model Example

Maximum Likelihood Estimation

GARCH parameters (ω, α, β) cannot be calculated with a simple formula — they are estimated using maximum likelihood estimation (MLE). The idea is to find the parameter values that make the observed return data most probable under the assumed model.

There is no closed-form solution — the log-likelihood must be maximized numerically using optimization algorithms (such as Nelder-Mead or BFGS). The process requires initial parameter guesses and convergence criteria. In practice, software handles this automatically: Python’s arch package, R’s rugarch, or MATLAB’s garch function.

A reliable GARCH fit typically requires 500 to 1,000+ daily observations (roughly 2-4 years of trading data). Fewer than 250 observations often produces unstable or unreliable parameter estimates. After fitting, always check standardized residuals (r_t / σ_t) for remaining ARCH effects — if the model is well-specified, these residuals should behave like independent draws. If standardized residuals show excess kurtosis (fat tails), a Student-t distribution may be more appropriate than the Gaussian assumption.

EWMA vs GARCH Model — Key Differences

Both EWMA and GARCH capture volatility clustering, but they differ in fundamental ways that affect when each model is appropriate.

For short-horizon applications like 1-day VaR, EWMA and GARCH often produce similar results. The difference becomes significant for multi-day forecasts — GARCH’s mean reversion produces a declining volatility term structure after a shock, while EWMA forecasts a constant level indefinitely. For risk managers building multi-day or multi-week VaR models, GARCH provides a more realistic volatility path.

Volatility Forecasting

One of the most practical applications of GARCH is multi-step-ahead volatility forecasting. The GARCH model’s mean-reversion property creates a natural term structure of volatility.

GARCH forecasting: The h-day-ahead forecast gradually converges toward the long-run variance V_L as the forecast horizon h increases. If volatility is currently elevated (as in our S&P 500 example), the model predicts it will decay back toward V_L. If volatility is currently depressed, the model predicts it will rise. At sufficiently long horizons, the GARCH forecast equals V_L regardless of the starting point.

EWMA forecasting: Because EWMA has no unconditional variance, its multi-day forecast stays flat at the current estimated level — it has no long-run anchor to converge toward. This is the IGARCH property: every shock has a permanent effect on the forecast.

This forecasting behavior directly affects risk models. Parametric VaR scales the volatility input across the holding period. Using EWMA for a 10-day VaR will produce a higher estimate (during stressed periods) than GARCH, because GARCH predicts the volatility will partially revert during those 10 days. The choice of model can meaningfully impact capital requirements and risk limits.

Common Mistakes with GARCH Models

Volatility estimation involves statistical complexity that creates several common pitfalls. Avoiding these mistakes is essential for reliable risk management.

1. Using simple historical volatility for risk management. Equally weighted variance ignores volatility clustering — the most important stylized fact of financial returns. After a market shock, simple historical volatility is slow to react (if the lookback window is long) or noisy (if the window is short). Conditional models like EWMA and GARCH are designed specifically for this problem.

2. Blindly using λ = 0.94 without validation. JPMorgan’s RiskMetrics default was calibrated for a broad set of assets in the 1990s. It may not be optimal for all asset classes, time periods, or market regimes. Validate the decay factor against your specific data or use GARCH to estimate the effective weighting from the data itself.

3. Ignoring mean reversion for long-horizon forecasts. Using an EWMA-based volatility estimate for a 30-day or 90-day risk horizon implicitly assumes volatility stays at its current level indefinitely. If volatility is elevated after a crisis, this overstates risk; if depressed during calm markets, it understates risk. GARCH’s mean reversion provides more realistic multi-day forecasts.

4. Estimating GARCH with too few data points. Reliable MLE estimation requires at least 500 daily observations (~2 years). With fewer data points, parameter estimates become unstable and can produce misleading results. The long-run variance V_L is especially sensitive to sample size.

5. Confusing conditional and implied volatility. GARCH estimates conditional volatility from historical return data — it is backward-looking. Implied volatility is extracted from option prices and reflects the market’s forward-looking expectation. These two measures serve different purposes and often diverge, especially during market stress.

6. Failing to validate model residuals. Fitting a GARCH model is not the final step. Always check standardized residuals for remaining ARCH effects (using Ljung-Box tests on squared residuals). If the model is well-specified, residuals should be approximately independent. Backtest any VaR model built on GARCH output against realized returns.

7. Assuming normally distributed errors when tails are heavy. Financial returns typically exhibit excess kurtosis — extreme events occur more frequently than a normal distribution predicts. Using Gaussian GARCH when returns have fat tails will underestimate tail risk. Consider Student-t or skewed-t innovations when standardized residuals show excess kurtosis.

GARCH Model Limitations

Leverage effect not captured. The asymmetric response of volatility to negative versus positive returns is well-documented in equity markets. After sharp declines, volatility tends to spike more than it would after an equally large rally. Extensions such as EGARCH (Nelson, 1991) and GJR-GARCH (Glosten, Jagannathan, and Runkle, 1993) add a leverage term to capture this asymmetry. For equity applications, these models often provide a better fit than standard GARCH(1,1).

Parameter sensitivity to sample period. GARCH parameters estimated over a calm period (e.g., 2017) can differ significantly from those estimated over a volatile period (e.g., 2008-2009). This sensitivity means the long-run variance V_L itself is an estimate, not a fixed constant. Practitioners should re-estimate parameters periodically and conduct sensitivity analysis.

Distributional assumptions affect tail risk. The standard Gaussian GARCH model underestimates the probability of extreme returns. Using a Student-t distribution for innovations improves tail fit but introduces an additional parameter (degrees of freedom) that must be estimated. The choice of distribution directly affects risk measures like Value at Risk and parametric VaR at high confidence levels.

Higher-order models add complexity with marginal benefit. GARCH(2,1) or GARCH(1,2) models add more lag terms, but empirical evidence consistently shows that GARCH(1,1) captures the vast majority of conditional variance dynamics. The additional parameters rarely justify the added estimation complexity.

Despite these limitations, GARCH(1,1) remains the workhorse of volatility estimation. Its combination of parsimony (three parameters), mean reversion, and strong empirical performance makes it the default choice for practitioners across risk management, derivatives pricing, and portfolio construction. Volatility estimation also plays a role beyond trading desks — real options analysis requires estimating project cash flow uncertainty, and employee stock option valuation under FASB ASC 718 depends on reliable volatility inputs.