Empirical Properties of Financial Data and GARCH Volatility Models

The loss operator links portfolio value to underlying risk factors via

V_t = f(t, Z_t)

. Computing VaR and ES requires a model for the distribution of risk-factor changes

X_{t+1} = Z_{t+1} - Z_t

. The analytic formulas derived in Topic 2 assumed

X

was normally distributed; this topic shows that assumption is systematically wrong, describes the empirical alternatives, and introduces the time-series models — GARCH and its extensions — that are the industry standard for capturing the dynamics of financial returns.

The central fact is that daily log-returns are not iid and not normal: they exhibit volatility clustering (large moves cluster in time), heavy tails (kurtosis far above 3), gain/loss asymmetry, and a tendency toward tail dependence across series in multivariate data. These deviations are not rare exceptions — they appear in all liquid markets and at all horizons below a month. Standard normal or lognormal models therefore systematically underestimate tail risk.

GARCH models resolve these issues by treating volatility as a latent dynamic process: $r_t = \mu + \sigma_t \varepsilon_t$ where $\sigma_t$ evolves as a function of past squared returns and past variances. Even with normally distributed innovations $\varepsilon_t$ , the unconditional distribution of $r_t$ is heavy-tailed; with Student- $t$ innovations it matches empirical tails closely.

By the end of this topic you should be able to:

name and interpret the six univariate and four multivariate stylised facts of financial returns;
compute sample skewness and kurtosis and apply the Jarque–Bera test;
define ARMA $(p, q)$ stationarity, ACF, PACF, and the Ljung–Box test;
write the ARCH $(q)$ and GARCH $(1,1)$ variance equations and interpret their coefficients;
state GJR-GARCH and EGARCH and explain why they are needed;
explain filtered historical simulation (FHS) and why it outperforms plain historical simulation;
compute the implied multi-period VaR from a GARCH model and state the limitations of the square-root-of-time scaling rule.

In this lesson

Contents
The big picture

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