Explanation:

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

where:

$\alpha =$ weighting on the previous period's return
$\beta =$ weighting on the previous volatility estimate
$\omega =$ weighted long-run variance

\sigma_n^2 = 0.000005 + 0.04(0.02^2) + 0.92(0.01^2) = 0.000113

\sigma_n = \sqrt{0.000113} = 0.01063 = 1.063\%

Long-run average variance $= \frac{\omega}{1 - \alpha - \beta} = \frac{0.000005}{1 - 0.04 - 0.92} = 0.000125$

Implied long-run volatility $= \sqrt{0.000125} = 0.01118 = 1.118\%$

Q.36 An analyst wishes to estimate the volatility of a stock. He uses a generalized autoregressive conditional heteroskedastic [GARCH (1,1)] model with the following parameters: $\omega = 0.000005$ , $\alpha = 0.04$ , $\beta = 0.92$ . If the stock's return today is 2% and the daily volatility is estimated to be 1%, calculate the new estimate of volatility and the implied long-run volatility level.

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RRohit

Last updated: June 22, 2026 at 14:25

New volatility = 1.063%; long-run volatility level = 1.118%.

New volatility = 0.0113%; long-run volatility level = 0.000125.

New volatility = 1.225%; long-run volatility level = 2.5%.

New volatility = 0.0125%; long-run volatility level = 3.2%.

Financial Risk Manager Part 1