GARCH Model for Volatility

Worked Solution

How to Think About It: If you look at any equity index return series, two things jump out immediately. First, quiet days cluster together and so do wild days -- volatility is not constant, it comes in regimes. Second, even though individual daily returns look roughly symmetric, the unconditional distribution has much fatter tails than a normal. A constant-$\sigma$ model misses both of these features entirely. GARCH is the simplest model that captures them: it lets today's variance depend on yesterday's shock and yesterday's variance, creating a self-reinforcing loop that produces clustering and fat tails from purely Gaussian innovations.

Key Insight: GARCH models volatility as a weighted average of a long-run level, yesterday's squared return (the "news" term), and yesterday's variance (the "persistence" term). This single recursion generates remarkably realistic return dynamics.

The Method:

GARCH(1,1) specification. Let $r_t$ be the return at time $t$. The model is:

$r_t = \sigma_t \, \epsilon_t, \quad \epsilon_t \sim N(0,1) \text{ i.i.d.}$

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

where $\omega > 0$, $\alpha \geq 0$, $\beta \geq 0$, and the stationarity condition is $\alpha + \beta < 1$.

The parameters have direct interpretations: - $\omega$ sets the floor -- it pulls variance toward a long-run level. - $\alpha$ is the reaction coefficient -- how much yesterday's surprise moves today's variance. - $\beta$ is the persistence coefficient -- how much of yesterday's variance carries forward.

Long-run (unconditional) variance. By taking the unconditional expectation of both sides:

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

This is well-defined only when $\alpha + \beta < 1$. Typical equity index estimates give $\alpha \approx 0.05\text{-}0.10$ and $\beta \approx 0.85\text{-}0.90$, so $\alpha + \beta \approx 0.95$, meaning shocks to variance decay slowly.

Uses in finance: - Volatility forecasting -- the multi-step forecast mean-reverts toward $\bar{\sigma}^2$ at rate $(\alpha + \beta)^h$ per step $h$, giving a term structure of expected volatility. - Value-at-Risk and risk management -- time-varying $\sigma_t$ produces much more accurate tail risk estimates than a fixed-$\sigma$ model. - Option pricing -- GARCH-based option models produce implied volatility smiles from a single model, unlike Black-Scholes. - Dynamic hedging -- hedge ratios depend on current volatility, and GARCH provides a principled estimate of it.

Phenomena GARCH captures that constant-$\sigma$ models miss:

1. Volatility clustering. Large returns (of either sign) tend to be followed by large returns. The $\alpha r_{t-1}^2$ term injects yesterday's shock into today's variance, so a big move raises tomorrow's expected volatility.

1. Mean reversion of volatility. After a spike, variance gradually decays back to $\bar{\sigma}^2$. The speed is controlled by $\alpha + \beta$ -- closer to 1 means slower reversion. This matches the empirical observation that VIX spikes are temporary.

1. Fat tails (excess kurtosis). Even though the innovations $\epsilon_t$ are Gaussian, the unconditional distribution of $r_t$ has heavier tails than normal. The excess kurtosis of a stationary GARCH(1,1) is:

$\kappa = \frac{6\alpha^2}{1 - 2\alpha^2 - (\alpha + \beta)^2}$

(provided the denominator is positive). With typical parameters ($\alpha = 0.08$, $\beta = 0.90$), this gives $\kappa \approx 0.5\text{-}2.0$, consistent with empirical return distributions.

1. Time-varying conditional distributions. At any point in time, the model gives you a full conditional distribution $r_t \mid \mathcal{F}_{t-1} \sim N(0, \sigma_t^2)$, not just a static one. This is essential for any risk measure that depends on the current state of the market.

Practical Considerations:

• What GARCH(1,1) does NOT capture: It treats positive and negative shocks symmetrically -- a 3% drop raises variance by the same amount as a 3% rally. In practice, negative returns increase volatility more (the "leverage effect"). Extensions like EGARCH and GJR-GARCH add an asymmetry term.
• Parameter estimation uses maximum likelihood. The log-likelihood is straightforward since the conditional distribution is Gaussian (or Student-$t$ if you use fat-tailed innovations).
• The IGARCH boundary: When $\alpha + \beta = 1$, shocks to variance never decay -- this is Integrated GARCH. Equity indices often estimate close to this boundary.

Answer: GARCH(1,1) models the conditional variance as $\sigma_t^2 = \omega + \alpha r_{t-1}^2 + \beta \sigma_{t-1}^2$, with $\alpha + \beta < 1$ for stationarity. It captures volatility clustering, mean-reverting variance, and fat-tailed unconditional returns -- three features that constant-volatility models miss entirely. It is the workhorse model for volatility forecasting, VaR computation, and dynamic hedging in practice.