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There are many academic paper doing volatility forecasts using realised variance and realised volatility interchangeably -- both targeting the proxy estimation of sum of squared returns (realized variance restrictedly). The two are different actually as using the realized volatility (root of the sum of squared returns) as forecast target can also yield different numbers.

Mathematically, the realized volatility forecasts are taking the (conditional) arithmetic average of the realised volatility in a regression model and the realized variance forecasts are taking the (conditional) squared average of the realised volatility. The realised variance forecast outputs can be proven restrictedly larger than the realised volatlity forecast in reality. Which would be a more meaningful target under the option pricing's model?

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  • $\begingroup$ Why don't you backtest it yourself? It is quite straightforward to test the accuracy of a ex-ante vol forecast by "pulling backward" realized volatility from the future. You just to make sure the relevant horizons are the same. $\endgroup$
    – KaiSqDist
    Commented Jul 30 at 1:17
  • $\begingroup$ @KaiSqDist I believe you didn't check/trade the volatility data before. Targeting volatility tends to forecast the volatility low and if you trade on it a large market shock will wipe out most of the cumulative profits though the error metrics better. $\endgroup$
    – Summer_More_More_Tea
    Commented Jul 30 at 2:45
  • $\begingroup$ I'm not asking you to trade on it, all I'm doing is asking for you backtest the volatility forecast (whether it is the ex-ante vol or var method forecast) to the actual realized volatility so you can see how accurate it is. Hopefully that would help inform you as to what is a more meaningful target under the option pricing model or whatever is your end goal. $\endgroup$
    – KaiSqDist
    Commented Jul 30 at 3:15
  • $\begingroup$ @KaiSqDist ok, in this regard, when comparing the performance, do I need to compare with realised volatility or realised variance? And if I choose to target realised variance, the error on realised variance should be smaller, and vice versa. $\endgroup$
    – Summer_More_More_Tea
    Commented Jul 30 at 5:53
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    $\begingroup$ Thanks! In light of your response, does my edit of your post make sense? $\endgroup$
    – Richard Hardy
    Commented Aug 1 at 8:47

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My two cents to fish more inputs. I choose to forecast the realised variance with the reasons

  1. realized variance is additive than realized volatility, which makes optimizing to sample mean more economically meaningful. And it is what aimed at when minizing the RMSE;
  2. in pricing model, it is the cumulative realized variance matters rather than realised volatility, so the forecast should target to be unbiased towards realised variance;
  3. if target realize volatility, there needs to be a bias-adjustment from volatility to variance when used in pricing. This involves an extra parameter estimation with more errors introduced than end-to-end variance target.
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