How can I compare 30 day implied volatility forecasts with GARCH forecasts?

Question

I'm trying to understand whether there is a good way to compare forecasts for volatility from different sources i.e., implied volatility and GARCH. I'll outline a few statements that I believe and if anyone could verify if they are correct or explain why I'm wrong I would be grateful.

$\textbf{1.}$ The 30 day implied volatility is the average implied vol for an option with an expiry 30 days from now. The value is annualized and so (roughly) represents a measure of the standard deviation of the $\textit{prices}$ of the stock over the next year. In order to get a daily value for the 30 day implied volatility we use $$\sigma_{\text{day}}=\frac{\sigma_{\text{annualized}}}{\sqrt{365}}$$

$\textbf{2.}$ GARCH models should always be applied to the returns or the log returns rather than the prices, as often we work under assumptions of normality and we believe returns follow a normal distribution a lot more than prices do. The $\textit{volatility}$ output from GARCH models is the conditional variance, $\text{Var}[y_t|y_{t-1},...]$ which I believe is the cond. variance of the returns used to model the GARCH? I believe this since modelling the returns and log returns give different variances which would not be consistent if GARCH outputted the cond. variance of the underlying stock price.

Hence my main questions are,

Given that implied volatility represents a measure of changes in the underlying price of the stock, and GARCH outputs the conditional variance of the returns, how would one go about comparing the two? Is there a way to change the GARCH forecasts so that we talk about the variance of the prices?

Given that I have forecasts for the implied volatility, and GARCH forecasts (and can perform some transformation to get them both in terms of prices or returns, see previous question), how can I compare these out of sample forecasts to the subsequent realised volatility? Would this be done by a Mincer-Zarowitz regression, stating a relevant error measure?

One last question, if I use a stochastic volatility model to give the conditional variance, such as Taylor's (1986) (implemented in the $\texttt{stochvol}$ package) can I perform the same sort of transformation used on the GARCH forecasts to get the volatility of prices rather than returns?

As you can see, I'm relatively confused about the many different ways one can quote/model/forecast volatility. If anyone can answer my questions, please do :) Thanks

Answer 1

This is a partial answer to your 2. statement. The main points are,

• the conditional (on information up to time $t-1$) variance of the price $P_t$ is the same as the conditional variance of the "return" $P_{t}-P_{t-1}$;
• the conditional variance of $\log P_t$ is the same as the conditional variance of $\log P_t - \log P_{t-1}$;
• the conditional variance of $P_t$ is not the same as the conditional variance of $\log P_t$ (and similarly for $P_{t}-P_{t-1}$ vs $\log P_{t} - \log P_{t-1}$).

The following is therefore incorrect:

GARCH models should always be applied to the returns or the log returns rather than the prices

Suppose $P_t$ is the sum of two components:

• a deterministic $\mu_t=g(I_{t-1})$, where $g(\cdot)$ is some function and $I_{t-1}$ is information up to time $t-1$, and
• a stochastic $\varepsilon_t$.

The only component unknown as of time $t-1$ is $\varepsilon_t$, the conditional variance of which is the conditional variance of $P_t$ (conditional on $I_{t-1}$).

Meanwhile, the "return" $P_{t}-P_{t-1}=g(I_{t-1})+\varepsilon_t-g(I_{t-2})-\varepsilon_{t-1}$. Here once again the only component that is unknown at time $t-1$ is $\varepsilon_t$, so the conditional variance of $P_{t}-P_{t-1}$ is the conditional variance of $\varepsilon_t$ -- the same as for $P_t$.

All the same logic would also hold of you decomposed the price multiplicatively rather than additively.

However, things change if you consider logarithms instead of levels. If you assume $\log P_t=g(I_{t-1})+\varepsilon_t$, you will get a different conditional variance than if you had assumed $P_t=g(I_{t-1})+\varepsilon_t$. This can be proven by a counterexample (any would do).