Compare and contrast the concepts of unconditional and conditional volatility.

The simplest volatility model is moving average. Why is it so simple and what are the main disadvantages?

What is the EWMA model? What are its strengths and weaknesses?

What are the conditional and unconditional volatilities of the EWMA model?

If you simulate the EWMA model, what will the volatilities eventually become?

What is the ARCH(\(L_1\)) model?

What is the kurtosis of the conditionally normal ARCH(1) model and what does that say about fat tails?

How can we capture the relationship between volatility and expected returns?

What is the leverage effect and how can is be modelled?

What is the memory and half life of a GARCH model?

What are the parameter restrictions we always have to impose in the estimation of GARCH(1,1) models and which are only needed in special cases?

What is the log likelihood function for the GARCH(1,1) model? What issues arise with at \(t=1\)?

What is variance targeting and why may it be beneficial?

What is a likelihood ratio test and why does it only work for nested models?

What is residual analysis?

What is VIX and how is it different from the conditional volatility models?

What is realized volatility and how is it different from the conditional volatility models?

Suppose two different estimation procedures are used to estimate a GARCH model for an asset, giving the following two estimation results, called model a and model b; \[\begin{aligned} \text{model a} && \sigma^2_{t+1}= 0.003+0.2 y_t^2 + 0.5 \sigma_{t}^2\\ \text{model b} && \sigma^2_{t+1}= 0.003+0.3 y_t^2 + 0.9 \sigma_{t}^2\end{aligned}\] where \(\sigma_t^2\) is \(1\%\) for both models. Suppose we observe the following future returns: \[\begin{aligned} y_{t}=&0.2\\ y_{t+1}=&0.1\\ y_{t+2}=&-0.2\\ y_{t+3}=&-0.1\\ y_{t+4}=&0.1\\\end{aligned}\]

What is your volatility forecast on day \(t+5\) for models \(a\) and \(b\)?

Suppose the outcome in the last question was problematic in your specific circumstances. What steps could you take to overcome the problem?

Suppose you observe returns for the next thousand days. What would you expect the volatility for the two models to be on day \(t+1000\)?

Considering the two results in the previous question, is there any reason to believe that either model is more incorrect than the other?