GARCH Model – The Standard
Developed by Tim Bollerslev (Engle's student) in 1986, GARCH (Generalized ARCH) is the upgraded version of ARCH and is the industry standard today.
Why GARCH?
Remember ARCH(q) needed many past lags (yesterday, day before, last week...) to predict volatility. GARCH solves this by saying: "Today's Variance depends on Yesterday's Shock AND Yesterday's Variance."
By including yesterday's variance (Sigma^2), it effectively remembers the entire history of the process with just a few parameters. This makes it "Generalized".
The GARCH(1,1) Formula
This is the most popular model in finance.
sigma_t^2 = omega + alpha * epsilon_(t-1)^2 + beta * sigma_(t-1)^2
Where:
- omega: Weighted long-term variance.
- alpha: Reaction to recent news (Yesterday's shock).
- beta: Persistence (Yesterday's variance).
Interpretation of Parameters (alpha + beta)
The sum alpha + beta tells you how "sticky" volatility is.
- If alpha + beta is close to 1 (e.g., 0.99): Shocks die out very slowly. Volatility is very persistent (e.g., after the 2008 crash, markets stayed volatile for months).
- If alpha + beta < 1: The process is stable and mean-reverting.
- If alpha + beta > 1: The model is unstable (explosive volatility) - usually indicates bad data or a crash.
Real World: For daily S&P 500 returns, typical values are $\alpha \approx 0.05$ and $\beta \approx 0.94$. This means 94% of today's volatility is just "carried over" from yesterday.
Applications
- VaR (Value at Risk): Banks use GARCH to predict potential losses for tomorrow.
- Option Pricing: Replacing the constant volatility in Black-Scholes with GARCH volatility.
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