Using GARCH to Forecast Volatility

🎯 GARCH is not just a descriptive model — its purpose is to forecast the volatility of future returns on invested capital.

In previous posts, we:

• Explained the GARCH model
• Estimated its parameters using the Maximum Likelihood Method (MLE)
• Validated the model by removing volatility clustering

Now we answer the key question:

How do we use the model once we are confident it works properly?

🔧 Forecasting Mechanism

The original model is defined as:$$sigma_{t+1}^2 = omega + alpha u_t^2 + beta sigma_t^2$$

However, when forecasting, future shocks are unknown.
Therefore, we replace them with their expected values, which gives:

$$E[sigma_{t+k}^2] = V_L + (alpha+beta)^k(sigma_t^2 – V_L)$$

This means that volatility evolves according to a simple logic:

👉 its current level gradually moves toward a long-term level — a kind of gravitational equilibrium

Where:$$V_L = frac{omega}{1 – alpha – beta}$$

📈 What This Looks Like in Excel

• We start from the current variance (σₜ²)
• We forecast forward (t = 1…30)
• We convert variance into volatility

The result:

👉 a smooth curve that converges toward equilibrium

🔍 What Determines the Shape of the Curve

Two key factors:

• Persistence (α + β)
  High → slow adjustment
  Low → fast adjustment
• Mean Reversion
  Volatility does not remain at extreme levels — it gradually returns to equilibrium

📎 Excel: Volatility Forecast – GARCH

Adapted from:

Options, Futures & Other Derivatives, John C. Hull