D.1 Making Vol Observable

• Name and explain 4 different methods for determining the last trading day's return volatility of the S&P 500

1. calculate the mean and then add up the squared deviation and divide by the $n-1$ to get the variance.
2. take the log of returns and do 1. to get the log variance
3. calculate the VIX of the returns
4. Fit an ARMA model and calculate an expected return, take the squared deviation from that to calculate the volatility.

I'm not sure what the question is particularly asking for, therefore I wouldn't consider my solution here reliable...

ALT:

• compute 'realized variance' (sum of squared returns)
• estimate parametric model, e.g. ARCH-GARCH
• short cuts: e.g. daily high log price minus its daily low log price.
• option-implied variance

D.2 Mean vs Variance Equation

Take any non-trivial ARMA(p,q) model to explain the concept of 'mean-' and 'variance' equation

$$ r_{t} \equiv \mu_{t-1} + \epsilon_{t} $$

Where $\mu _{t-1}$ is the mean part of the return decomposition, and $\epsilon _{t}$ is the variance part of the return decomposition.

The task of the mean equation is to capture auto correlation in the realized returns. (The vast majority of returns weakly auto correlated). An ARMA(p,q) parametrization can remove the correlation.

Hence we get the following mean equation: (k-factor risk premiums are optional)