Volatility Modelling | The Mountain Path Academy

Volatility measures how widely returns fluctuate around their mean. It drives Value-at-Risk, option pricing, portfolio limits and capital decisions. Because market volatility changes through time, risk managers need models that update as new information arrives.

1Volatility Foundations

Historical volatility

Standard deviation of realised past returns. It answers: how volatile was the asset?

Implied volatility

The volatility consistent with current option prices. It reflects the market’s forward-looking expectation.

Forecast volatility

An econometric estimate based on past returns, recent shocks and persistence.

Historical daily volatility is commonly annualised with the square-root-of-time approximation:

\[\sigma_{annual}=\sigma_{daily}\sqrt{252}\]

Worked example

Daily volatility of 1.25% corresponds to approximately \(1.25\%\sqrt{252}=19.84\%\) annualised. This scaling assumes independent returns and stable volatility, so treat it as an approximation.

2Characteristics of Volatility

Clustering

Large moves tend to follow large moves, while calm periods tend to persist.

Mean reversion

Volatility moves toward a long-run level rather than drifting without limit.

Persistence

The effect of a shock fades gradually across future forecasts.

Fat tails

Extreme returns occur more often than a normal distribution predicts.

Leverage effect

Negative equity returns often raise volatility more than equal positive returns.

Latent state

True volatility is unobserved and must be inferred from market data.

Diagnostic clue

Returns may show little serial correlation, while squared or absolute returns show substantial correlation. That pattern motivates conditional variance models.

3Exponentially Weighted Moving Average

EWMA gives more weight to recent observations and lets older information decay geometrically:

\[\sigma_t^2=\lambda\sigma_{t-1}^2+(1-\lambda)r_{t-1}^2\]

Decay factor	Forecast behaviour
Higher \(\lambda\)	Smoother forecast with a slower response to new shocks
Lower \(\lambda\)	Faster response with more weight on the latest return
RiskMetrics daily value	\(\lambda=0.94\), so 94% weights prior variance and 6% weights the latest squared return

Link to GARCH

EWMA is a restricted GARCH(1,1) with \(\omega=0\), \(\alpha=1-\lambda\), \(\beta=\lambda\), and \(\alpha+\beta=1\). It reacts efficiently, but has no separate long-run mean reversion.

4ARCH Model

Autoregressive means the model uses its own history. Conditional means variance depends on currently available information. Heteroskedasticity means variance changes through time.

\[\sigma_t^2=\omega+\sum_{i=1}^{q}\alpha_i\varepsilon_{t-i}^2\]

Squaring removes direction. A return of +2% and a return of −2% both contribute \(0.02^2=0.0004\). ARCH therefore models the magnitude of shocks.

ARCH(1) calculation

Suppose \(\omega=0.00002\), \(\alpha=0.30\), and yesterday’s return was 2%. Then \(\sigma_t^2=0.00002+0.30(0.02)^2=0.000140\). Daily forecast volatility is 1.183%, or about 18.78% annualised.

5GARCH(1,1)

Pure ARCH may need many lags to capture slowly fading volatility. GARCH adds yesterday’s forecast variance, which summarises the older shocks using one term:

\[\boxed{\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\beta\sigma_{t-1}^2}\]

Parameter	Interpretation
\(\omega\)	Constant that anchors the long-run variance
\(\alpha\)	Reaction to the latest squared shock
\(\beta\)	Persistence carried from the previous variance forecast

Persistence and mean reversion

Covariance stationarity requires \(\alpha+\beta<1\). Values near one imply slow decay.

\[\sigma_{LR}^2=\frac{\omega}{1-\alpha-\beta},\qquad h_{1/2}=\frac{\ln(0.5)}{\ln(\alpha+\beta)}\]

6GJR-GARCH and Asymmetry

Standard GARCH treats equal positive and negative shocks alike. GJR-GARCH adds an indicator that activates after a negative innovation:

\[\boxed{\sigma_t^2=\omega+\alpha\varepsilon_{t-1}^2+\gamma\varepsilon_{t-1}^2I(\varepsilon_{t-1}<0)+\beta\sigma_{t-1}^2}\]

Reading the leverage term

A positive shock has news impact \(\alpha\). A negative shock has news impact \(\alpha+\gamma\). When \(\gamma>0\), bad news raises the next variance forecast more strongly.

7Interactive Forecasting Lab

Change the parameters below. Previous daily volatility is fixed at 1%.

GARCH and GJR-GARCH forecast engine

\(\omega\) baseline 0.000002\(\alpha\) news response 0.08\(\beta\) memory 0.90Last shock −2.50%\(\gamma\) asymmetry 0.10

GARCH next-day vol

GJR next-day vol

Persistence

Shock half-life

Long-run daily vol

8Model Comparison

Model	Memory mechanism	Mean reversion	Primary use
Historical volatility	Fixed observation window	Not modelled	Describing realised risk
EWMA	Exponential weights	No	Fast operational forecasts
ARCH(q)	Past squared shocks	Yes	Foundation and diagnostics
GARCH(1,1)	Last shock and last variance	Yes	VaR and multi-period forecasting
GJR-GARCH	GARCH plus negative-shock indicator	Yes	Equities and asymmetric markets

Model validation

Test for remaining ARCH effects in standardised residuals. Check parameter constraints and persistence. Consider Student-t innovations when residual tails remain heavy. Compare out-of-sample forecasts and VaR backtests rather than relying only on in-sample fit.

9Practical Applications

Use these applications after the guide to fit, compare and interpret volatility models using market data.

ARCH, GARCH, GJR-GARCH and EGARCH

Compare symmetric and asymmetric volatility models across the same return series.

Open comparison app ↗

Volatility Forecasting

Move from market returns to an operational volatility forecast and examine model sensitivity.

Open forecasting app ↗

Prepared for The Mountain Path AcademySource material: Volatility Forecasting presentation