Consider the following:

$r_t = \theta r_{t-1}+u_t$


$\sigma^2_t=\omega+\alpha u^2_{t-1}$

$-1<\theta<1,\omega>0,\alpha \in(0,1)$

What is the 99% 2-day VaR of a long position at time $t$?

First, the problem given does not explicitly say it, but I am assuming the question only has a "nice" answer if we assume $\epsilon_t \sim N(0,1)$ and are iid. Fine. But after taking a recursive approach, I get (ASSUMING WE START AT DAY $t-2$)

$P(\theta^2r_{t-2}+\theta \sqrt{\omega+\alpha \sigma_{t-2}^2\epsilon_{t-2}^2}\epsilon_{t-1}+\epsilon_t \sqrt{\omega+\alpha \epsilon_{t-1}^2(\omega+\alpha \sigma_{t-2}^2 \epsilon_{t-2}^2)}<x)=.01$

But this has no obvious solution to me. Any help will be MUCH appreciated.


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