Volatility Forecasting: Why Shocks Linger and Downside Moves Matter

Avoid solving a weak fit by repeatedly adding model terms until residual tests become favourable. The search itself changes the strength of subsequent inference. A simpler model with transparent limitations can be preferable to an elaborate specification whose apparent success depends on one recent event. Model complexity should earn its place through an improvement in the intended out-of-sample forecast or risk decision.

Risk changes even when the expected direction of a market is unclear. Large moves can be followed by large moves of either sign, leaving return forecasts weak while forecasts of the size of the next move remain useful. Conditional-volatility models formalise that distinction by updating the variance forecast from the information available at each point in time.

This reference report consolidates the introductory volatility-model lessons and advanced teaching material into a coherent model-selection and validation process. The aim is a defensible forecast input for a risk system. It does not turn an estimated variance into a prediction that a particular market will rise or fall.

The distinction between a measurement and a conditional forecast

A rolling standard deviation describes a selected historical window. A conditional variance is an expectation about uncertainty given an information set. Both are estimates, and neither automatically represents the full distribution of future losses.

Write a return as $r_t=\mu_t+\epsilon_t$, where $\mu_t$ is its conditional mean and $\epsilon_t$ the innovation. Express the innovation as $\epsilon_t=\sigma_t z_t$, with $z_t$ standardised to mean zero and variance one. The model then specifies how $\sigma_t^2$, the conditional variance, evolves.

This formulation allows conditional variance to change while a stationary unconditional variance exists. There is no contradiction: the long-run distribution can be stable even though today’s forecast depends on yesterday’s shock. Conversely, using log returns does not guarantee stationarity. Structural breaks, changing market microstructure and evolving regimes still need examination.

The basic teaching example converts a daily standard deviation of 0.76% into an annualised estimate of approximately $0.76\sqrt{252}=12.06\%$. The calculation is correct under that convention. It does not establish that every next-day move falls in a normal-distribution band or that the estimate captures a scheduled event not represented in past returns.

ARCH models make yesterday’s shock matter

An ARCH model makes conditional variance depend on lagged squared innovations:

$$\sigma_t^2=\omega+\sum_{j=1}^{q}\alpha_j\epsilon_{t-j}^2.$$

The positive constant $\omega$ is a baseline term; $\alpha_j$ controls the weight on the innovation $j$ periods ago. Non-negative lag weights help preserve positive variance. In the familiar finite-variance setting, a sum of lag coefficients below one provides a mean-reverting second-moment structure.

ARCH captures clustering because a large positive or negative innovation produces a large squared term. The sign is discarded. If persistence lasts for many days, however, an ARCH specification may need many lags and therefore many parameters. This motivates a more economical recursion.

GARCH separates shock response from persistence

The GARCH(1,1) recursion is:

$$\sigma_t^2=\omega+\alpha\epsilon_{t-1}^2+\beta\sigma_{t-1}^2.$$

In words: Today’s variance combines a baseline, the previous squared surprise and the previous variance, each with its stated weight.

The coefficient $\alpha$ measures immediate response to the last shock, while $\beta$ carries forward the previous conditional variance. With $\omega>0$, non-negative coefficients and $\alpha+\beta<1$, the standard finite unconditional variance is:

$$\bar\sigma^2=\frac{\omega}{1-\alpha-\beta}.$$

The constant $\omega$ is therefore not itself the long-run variance. Calling it that hides the effect of persistence. Two models with the same $\omega$ can imply very different long-run levels when their coefficient sums differ.

A retained numerical example has $\alpha=0.3$, $\beta=0.5$ and $\omega=0.02$. Its coefficient sum is 0.8 and its unconditional variance is 0.10 in the input’s variance units. Before translating that into volatility or a currency loss, establish whether the returns were supplied as decimals or percentage points.

If percentage returns are used, a variance of 1 means a daily standard deviation of one percentage point. If decimal returns are used, a variance of 1 means 100% daily standard deviation. Recording the scale in the model metadata prevents a factor-of-10,000 variance error.

Persistence determines how slowly a shock fades

For the usual GARCH expectation recursion, deviations of forecast variance from its long-run level decay geometrically at a rate related to $\alpha+\beta$. The corresponding variance-shock half-life is:

$$H=\frac{\ln(0.5)}{\ln(\alpha+\beta)}.$$

Variance-shock half-lives under the retained coefficient sums

These values preserve the source’s numerical examples. The periods are trading observations if the model is fitted to daily trading returns. The half-life describes expected decay under a fixed fitted model, not a deadline by which the market must calm down.

When the coefficient sum approaches one, the inferred long-run variance becomes highly sensitive to estimation error. A seemingly small parameter revision can materially change a long-horizon forecast. A model with a stable-looking fitted volatility line may consequently have an unstable long-run interpretation.

Exponential weighting is a useful restricted model

An exponentially weighted moving-average recursion is:

$$\sigma_t^2=(1-\lambda)\epsilon_{t-1}^2+\lambda\sigma_{t-1}^2.$$

In words: Today’s variance averages the previous squared surprise and the previous variance using complementary weights.

The teaching material uses $\lambda=0.94$ as a conventional daily setting. It requires no likelihood optimisation and updates easily as a new observation arrives. That makes it a practical baseline and a useful fallback during a model-service failure.

The recursion has no positive intercept anchoring a finite long-run mean in the same way as a stationary GARCH model. Its forward conditional-variance forecast is flat under the restricted specification. That limitation does not make every use beyond one day “meaningless”; it means the model does not itself generate an additional mean-reversion story. Its suitability must be assessed for the intended horizon.

Equity-like asymmetry requires an additional mechanism

Plain GARCH gives equal immediate variance effects to positive and negative shocks of equal magnitude. A threshold variant introduces an extra response for negative innovations:

$$\sigma_t^2=\omega+\alpha\epsilon_{t-1}^2 +\gamma I_{t-1}\epsilon_{t-1}^2+\beta\sigma_{t-1}^2,$$

In words: A negative surprise receives an additional variance weight alongside the baseline and the previous variance.

where $I_{t-1}=1$ when the innovation is negative and zero otherwise. A positive $\gamma$ increases the negative-shock response from $\alpha$ to $\alpha+\gamma$.

The retained illustrative coefficients $\alpha=0.05$, $\gamma=0.12$ and $\beta=0.85$ make the negative-shock coefficient 0.17, compared with 0.05 for a positive shock. Under symmetric standardised innovations, a relevant second-moment persistence quantity is $\alpha+\beta+\gamma/2=0.96$, not simply $\alpha+\beta$. Using the plain-GARCH stationarity check on this asymmetric specification would miss part of the persistence.

An exponential GARCH model instead evolves log variance. A common form uses the magnitude and sign of the previous standardised residual separately. Negative sign coefficients can then produce a larger response to downside shocks. Working in log variance guarantees a positive variance after exponentiation, but does not automatically guarantee a well-behaved fit or forecast.

Innovation distributions and variance dynamics are different choices

A heavy-tailed unconditional return distribution can arise from changing conditional variance even with normal conditional innovations. There can also be heavy tails left in the standardised innovations after volatility has been modelled. These are distinct issues.

The source recommends comparing normal and Student-t innovations and gives degrees of freedom around five to seven as a representative range. Retain this as a model candidate, not an assertion that a particular asset must have that distribution. Estimation, tail diagnostics and out-of-sample evaluation should determine whether the additional shape parameter is useful.

Replacing normal innovations with Student-t does not automatically “get the tail right.” Tail estimates remain sensitive to sample size, breaks, serial dependence and the selected distribution. Likewise, higher volatility widens a distribution but does not necessarily increase its standardised tail thickness. Variance level and kurtosis should not be conflated.

Fit, diagnose and evaluate in the correct order

Begin with clean returns and a credible conditional-mean specification. Inspect autocorrelation in returns and squared residuals. An ARCH-LM test can assess whether the selected lagged squared residuals carry joint explanatory power. Its p-value is evidence under a specified null, not an instruction that one model must always be accepted or rejected.

After fitting, examine standardised residuals and their squares. Residual autocorrelation may reveal a deficient mean model; dependence in squared standardised residuals may reveal missing variance dynamics. Distributional diagnostics then ask whether the remaining innovations match the chosen error distribution.

Information criteria help compare fitted candidates but do not establish trading usefulness. Compare forecast losses out of sample, using identical dates and targets. For a risk system, also examine loss-quantile exceedances, clustering of breaches and behaviour during stress periods. A model that improves average squared error can still be unsuitable for a particular tail-risk use.

Refit policy matters. Re-estimating parameters daily and updating only the variance state daily are different systems. Frequent refits may adapt faster but introduce parameter noise; a long fixed window may be stable but slow to recognise a break. Log both the fitted parameters and the operational decision about when to update them.

Translate forecast changes into a bounded risk input

The teaching example uses capital of 10,000,000 and a target annual volatility of 12%. With a 10% forecast, the simple scaling rule gives 1.20 times base notional, or 12,000,000. With a 22% forecast, it gives approximately 0.5455 times base notional, or 5,454,545.

These are arithmetic consequences of $\text{notional}=\frac{\text{capital}\times\text{target}}{\text{forecast}}$, not assurances that portfolio volatility will actually equal 12%. Forecast error, basis risk, non-linear payoffs and exposure constraints all intervene. Leverage caps, minimum forecast floors and executable rebalance rules should be separate controls rather than hidden adjustments to the estimated variance.

A crisis can arrive between observations, so model-driven sizing cannot guarantee that exposure is reduced before the loss. Nor should a recent calm regime justify unlimited leverage. The model’s role is to organise the information available, not to eliminate the possibility of a new shock.

Match a multi-period risk estimate to the forecast recursion

A next-day variance forecast and a multi-day holding-period variance are different outputs. Under the basic GARCH assumptions, forecast variance for later days moves toward the fitted long-run level. The expected variance over a holding interval is constructed from the relevant sequence of conditional daily variances, with additional covariance terms if the return model requires them.

It is therefore not always appropriate to take today’s elevated conditional standard deviation and multiply by the square root of a long horizon. That shortcut assumes a constant variance path and suitable return dependence. A mean-reverting forecast can give a different result after a shock, while a persistent or regime-shifting process can invalidate an optimistic normalisation assumption.

Distinguish expected variance from expected volatility as well. Taking the square root of an expected variance gives a useful risk scale, but it is not generally identical to the expectation of the random future standard deviation. This distinction becomes more important when the forecast is used for a volatility-linked payoff rather than a linear exposure’s variance budget.

The same discipline applies to asymmetric-model half-lives. A log-variance persistence coefficient describes a different state from the plain model’s variance deviation. Report which state is decaying and under which recursion instead of comparing all quoted half-lives as though they measured exactly the same object.

Parameter uncertainty belongs in the interpretation

A single fitted parameter set conceals how much the estimate would change with another plausible sample. Near a persistence boundary, this uncertainty can be economically important even when the numerical optimiser reports convergence. Examine parameter stability across nearby windows and compare the forecast dispersion, especially after a small number of extreme observations.

A conditional-volatility service should return its forecast horizon and units, latest data timestamp, parameter set, estimation window, convergence status and principal validation diagnostics. A fallback should be visible when used. This makes disagreement with another estimator explainable and prevents a stale or failed model from silently influencing position limits.

The framework’s enduring contribution is the separation of shock response, persistence, asymmetry and tail shape. Treating those as distinct modelling choices produces a more useful risk input than asking one volatility number to explain every market condition.