Volatility, Tails, and Financial Risk

Why finance changes the second moment

Daily returns often have weak linear autocorrelation but strong dependence in squared or absolute returns. A useful decomposition is

The conditional mean $\mu_t$ and conditional variance $h_t$ are separate modelling problems. An unconditional ARMA model can describe $E(r_t\mid\mathcal F_{t-1})$ while missing the risk clustering in $h_t$.

From ARCH to GARCH

Engle's ARCH model lets recent squared shocks drive variance. Bollerslev's GARCH model adds lagged variance:

If $\alpha+\beta<1$, the unconditional variance is

Interpretation:

• $\alpha$ is the immediate variance response to news;
• $\beta$ is persistence in the variance state;
• $\alpha+\beta$ controls the decay of a volatility shock;
• $\omega$ sets the long-run scale jointly with persistence.

When $\alpha+\beta$ is close to one, volatility forecasts revert slowly. That is not the same as a unit root in the return level.

Conditional covariance as a matrix

Given past information, a block of future shocks has covariance

only when future standardised innovations are conditionally uncorrelated. In a multivariate model, $H_t$ is a full positive-definite matrix and must simultaneously control variances and correlations.

The one-step GARCH forecast is

For $k\ge2$,

Gaussian quasi-likelihood

Ignoring constants, the conditional Gaussian objective is

It is a quasi-likelihood when $z_t$ is not Gaussian. Under suitable conditions it can still estimate the variance recursion consistently, but Gaussian tail probabilities and conventional standard errors need not be valid.

Always inspect standardised residuals

Remaining dependence in $\hat z_t^2$ signals an inadequate variance recursion; heavy tails in $\hat z_t$ signal an inadequate innovation distribution.

R laboratory: estimate GARCH by recursion

The constraints are built into the parameter transformation. This is safer than accepting negative variances after optimisation. The simulated returns are in percentage-like units, so the reported VaR and ES use the same units.

Heavy tails and leverage

Two extensions answer different diagnostics:

• Student-$t$ innovations: large standardised shocks occur more often than under a Normal distribution.
• Asymmetric variance dynamics: negative and positive shocks of equal magnitude can affect future volatility differently.

For example, a GJR-style recursion is

$\gamma>0$ represents a larger variance response after negative news. Do not use asymmetry and heavy tails as interchangeable explanations.

From volatility to a risk decision

For lower-tail probability $p$, define the return quantile $q_{p,t}$ by

If losses are $L_t=-r_t$, then $\operatorname{VaR}_{1-p,t}=-q_{p,t}$. Expected shortfall is the expected loss beyond VaR. State the sign convention; otherwise a “negative VaR” can mean either a gain or a loss.

A volatility model is not validated merely because its likelihood is high:

Finance versus economics

GARCH is central in finance because the conditional scale of returns is itself a decision input. In macroeconomics it is useful only when time-varying uncertainty is part of the question—for example inflation uncertainty or exchange-rate volatility. For GDP nowcasting, publication timing and latent-state estimation are usually more central than a daily GARCH recursion.

Practice

1. If $\alpha+\beta=0.98$, what fraction of a variance deviation remains after ten forecast steps?
2. Why can uncorrelated returns still be dependent?
3. A model has correct average VaR coverage but all exceptions occur in one crisis month. Is it calibrated?

Next: Unit Roots, Cointegration, and Error Correction.