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:

$\begin{aligned} \sigma_{n}^{2} & =\omega+\alpha \times u_{n-1}^{2}+\beta \times \sigma_{n-1}^{2} \\ & =0.000032+0.12 \times 0.0025+0.8 \times 0.000729 \\ & =0.000032+0.0003+0.0005832 \\ & =0.0009152 \end{aligned}$

The square root of the updated variance yields the updated standard deviation (volatility):

$\sigma_{n}=\sqrt{0.0009152} \approx 0.03$ or $3%$

This calculation shows that the updated volatility for AAA Holdings, factoring in the recent 5% price drop through the GARCH(1,1) model parameters, is approximately 3%.

Chapter: Measuring and Monitoring Volatility

Learning objectives:

• Apply the exponentially weighted moving average (EWMA) approach and the GARCH (1,1) model to estimate volatility.
• Explain and apply approaches to estimate long-horizon volatility/VaR and describe the process of mean reversion according to a GARCH (1,1) model.