To estimate the updated volatility of AAA Holdings using the GARCH(1,1) model, we first define the components of the model:

• $\omega$ (omega) - This is the constant term in the GARCH model, representing the baseline variance when no previous information is considered.
• $\alpha$ (alpha) - The coefficient for the squared residual from the previous day, reflecting the immediate impact of past shocks on today's volatility.
• $\beta$ (beta) - The coefficient for the previous day's variance, indicating how much of yesterday's volatility persists into today.

The GARCH(1,1) model formula to calculate today's variance ($\sigma_n^2$) is expressed as:

$$\sigma_n^2 = \omega + \alpha \times u_{n-1}^2 + \beta \times \sigma_{n-1}^2$$

For the calculations:

• $u_{n-1}^2$ - The squared residual from the previous period. Given a 5% drop in prices, $u_{n-1} = 0.05$, thus $u_{n-1}^2 = 0.05^2 = 0.0025$.
• $\sigma_{n-1}^2$ - The variance of the previous day, based on a volatility of 2.7%, so $\sigma_{n-1} = 0.027$ and $\sigma_{n-1}^2 = 0.027^2 = 0.000729$.

Substituting these values and the given parameters ($\omega = 0.000032$, $\alpha = 0.12$, $\beta = 0.8$) into the formula, we calculate the updated variance:

$$\sigma_n^2 = 0.000032 + 0.12 \times 0.0025 + 0.8 \times 0.000729$$ $$\sigma_n^2 = 0.000032 + 0.0003 + 0.0005832 = 0.0009152$$

Finally, we find the updated daily volatility by taking the square root of the variance:

$$\sigma_n = \sqrt{0.0009152} \approx 0.030252 \text{ or } 3.0\%$$