# VaR of ARCH model

Consider the following:

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

$$u_t=\sigma_t\epsilon_t$$

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

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

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)}

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