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I would like to know what is the most reasonable way to measure volatility in a sample of past observations. Aside from standard deviation, are more complex models like GARCH used for (historical) volatility measurement if one is not interested in forecasting future volatility?

For context, as mentioned in a comment below, I need a measure of past monthly volatility to study the relationship between (monthly) mutual fund alphas and (monthly) market volatility over a past period of time.

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    $\begingroup$ Do you want some time specific measure of volatility $\sigma^2_t$? The simplest possible thing to do is estimate based upon some rolling window. Volatility models such as GARCH can also be used to back out some estimated $\sigma^2_t$ process from the data. $\endgroup$ – Matthew Gunn May 18 at 17:24
  • $\begingroup$ What would be the benefits of using a GARCH model instead of a simple moving average standard deviation in the context of in-sample estimation, if any? $\endgroup$ – Gianluca May 18 at 20:28
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    $\begingroup$ When you compute std deviation for a subperiod $[T_1,T_2]$ you get a single number $\sigma_{T_1,T_2}$ which you can think of as an average over all days in $T_1,T_2$ of something called the "latent volatility" or "instantaneous volatility" on each day. This instantaneous volatility is not observable but you might estimate it with a model such as GARCH. This is a possible benefit of a model. But my question for you is: what do you need volatility for? $\endgroup$ – noob2 May 18 at 21:26
  • $\begingroup$ I need a measure of past monthly volatility to study the relationship between (monthly) mutual fund alphas and (monthly) market volatility over a past period of time. Based on what you say, it seems to me that using GARCH would only make sense if I wanted to estimate daily volatilities. I am trying to understand whether measures other than the monthly standard deviation could be of any use when measuring monthly volatility. Would estimating daily volatilities with GARCH and then converting them into monthly volatilities have any benefit in this context? $\endgroup$ – Gianluca May 19 at 12:50
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    $\begingroup$ I believe you do not need GARCH for your application $\endgroup$ – noob2 May 20 at 19:56
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Why would you need to model volatility to test an hypothesis. Just use the historical realised volatility and if you want to test the hypothesis how funds relate in the near future, then use the VIX index, it's a forward looking measure. Or you use some volatility tracking fund as a proxy, why use some model to estimate relationships when obviously some modelling errors will creep in.

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  • $\begingroup$ By "historical realised volatility" do you mean the historical standard deviation? I am asking because I am aware there are also RV models using intra-day data, to which I do not have access. $\endgroup$ – Gianluca May 21 at 18:18
  • $\begingroup$ Yes, historical stdev $\endgroup$ – Dhruv Mahajan May 22 at 16:42
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    $\begingroup$ because, for instance, implied vol is known to be consistently higher than associated realized vol. in short, because you think you can do better than something quick and dirty. $\endgroup$ – Chris May 22 at 22:12

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