# Can we model Implied volatility using GARCH?

Can I use Implied volatility as a dependent variable in a GARCH model? I believe my IV data shows ARCH effects and hence can I use it to model volatility of the volatility? I know literature has used logged price differences in GARCH model, So I am a little confused If IV can be used or not (Since GARCH models historical volatility whereas IV is a forward looking measure)?

## 1 Answer

I am not sure where you're going, but GARCH models usually have two equations: (1) an equation describing the conditional expectation of the dependent variable and (2) an equation describing the conditional variance of the error term in equation (1) as a process that is perfectly anticipated 1 period ahead.

When you use returns in equation (1), equation (2) is then used to filter out the condition volatility process from returns. If you say you want to model IV or its growth rate using GARCH, you're trying to model the volatility of IV.

Is this what you want to do?

• Yes, I want to allow for the volatility of implied volatility. So, can I use GARCH in that context? One concern I have is GARCH models describe conditional variance (P-measure), while IV is risk-neutral (Q-measure) – Raghav Goyal Nov 6 '20 at 5:05
• GARCH model describes conditional P-variance when applied to P processes. If you take IV to be Q-volatility, you are modeling Q-vol-of-vol with GARCH on IV. It doesn't miraculously become a P dynamics because you apply GARCH on it. We could quibble over how these measures will be polluted, but taken as it is you would be doing what you want to do. – Stéphane Nov 6 '20 at 18:59
• So, just a follow-up question: Will the F-tests and T-tests still be valid if I use IV as the dependent variable. (modeling its volatility) – Raghav Goyal Nov 7 '20 at 7:36
• In a time series context, you can often obtain an asymptotic distribution for those tests statistics. Usually, that would be a chi square and normal law, respectively, because your estimator of the covariance matrix is going to converge (i.e., it's like it's known). In small samples, your scaled down Wald statistic is Fisher because your covariance matrix estimator is going to be random in finite sample -- and likewise for the t statistic and the Student distribution. The validity of those tests depend on whether you can reasonably make the assumptions you use when deriving their laws. – Stéphane Nov 8 '20 at 16:23
• Maybe you can take your condescending attitude elsewhere. IV at heart is just a dimension reduction gimmick: it's a convenient way to say cheap or expensive for options. I don't know where the hell you get "forward movement." VIX used to aggregate IV's. The newer formulation based on pricing variance swaps behaves very similarly and it gives you an estimate of Q volatility polluted by higher moments (Martin, 2017). So, my comment about vol of vol under Q do make sense. – Stéphane Nov 12 '20 at 4:19