Daily vs. Monthly Vol

GammaSkimmer · November 28th, 2007, 9:41 pm

I am trying to come up with a scenario where a set of daily realized vols is less than monthly vols. I have tried assuming 250 days/year and including 238 days with the same price change, ie zero daily vol and including one day per month of a random, very large move. This, however, is overpowered by the annualization factor (sqrt(250) vs. sqrt(12)) and daily is at best just slightly larger.Is there any mathematical way for daily to be higher than monthly?

Jim · November 28th, 2007, 9:54 pm

Monthly vol will be greater than daily vol if the series exhibits trends; monthly vol will be less than daily vol if the series exhibits mean reverting behavior.

GammaSkimmer · November 28th, 2007, 11:18 pm

Sorry, I should have clarified. Everything I was doing was based on percentage changes thus the order of the events should not matter I believe.

Jim · November 29th, 2007, 1:34 pm

Whether you have an additive or multiplicative process is irrelevant. What matters for trending/mean reverting is whether the stochastic shocks exhibit positive or negative serial correlation.

Athletico · November 29th, 2007, 1:48 pm

This is a very common interview question; I believe it's in Crack's book. Jim is right of course. To the original q:> Is there any mathematical way for daily to be higher than monthly? If the log-price process were pure white noise, the answer is no (outside of sampling noise). But if you see this as a persistent feature in your time-series, then trending or mean-reversion exists, as Jim said.

eredhuin · November 29th, 2007, 6:06 pm

It's also helpful to remember that sampling vols are chi-squared. I.e., the confidence interval is large.E.g. try your monthly exercise using end of month, beginning of month and middle of month.