Two-Day Value at Risk for an AR(1) ARCH Return Model

Summary

The document poses a two-day value-at-risk problem for a long position whose returns follow an autoregressive conditional heteroskedastic model. The conditional variance depends on the previous shock, while the return also depends on the previous return. The question asks for a 99% VaR and points out that the innovation distribution is not specified; the author considers independent standard normal innovations as a possible assumption.

Expanding the recursion produces a two-period return expression in which later shocks and conditional variances depend on earlier shocks. The post does not give a quantile or a numerical VaR, and it notes that the resulting probability equation has no obvious closed-form solution. It therefore highlights the need to specify the innovation distribution and initial conditions, and suggests that evaluating the multi-period tail probability may require a numerical approach. No answer or empirical evidence is included.

Key ideas

• The return model combines autoregressive returns with shock-dependent conditional variance.
• A multi-day VaR requires the distribution of the aggregated return, not only a one-day conditional variance.
• The post assumes independent standard normal innovations only as a possible added assumption.
• The recursive probability expression is presented without a closed-form solution or numerical VaR.

Tags

# VaR of ARCH model


# 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 $t-2$)

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

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

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.