Choosing Realized Variance or Volatility as a Forecast Target
Summary
The document distinguishes forecasts of realized variance from forecasts of realized volatility. Realized variance is based on the sum of squared returns, while realized volatility is its square root. Forecasting the two quantities separately can produce different outputs: averaging realized volatility does not generally yield the same result as forecasting variance and then taking a square root.
The answer favors realized variance when the forecast is intended for option pricing. It argues that variance is additive across periods and is the quantity accumulated in pricing models, so targeting it aligns the forecast with the modeled input and mean squared error objective. Forecasting volatility instead may require a further conversion or bias adjustment to obtain a variance estimate, adding parameter estimation and potential error. The response offers a reasoned recommendation rather than a comparative empirical study; the appropriate target still depends on the model, loss function, and intended use of the forecast.
Key ideas
- Realized volatility is the square root of realized variance, so forecasts of the two targets need not agree.
- A conditional average of realized volatility differs from the square root of a conditional variance forecast.
- Variance is additive across periods, which can make it a natural target for cumulative pricing applications.
- Converting a volatility forecast into variance may require bias adjustment and additional estimation.
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Full text
# Target Realized Volatility or Realized Variance in Forecasting # Target Realized Volatility or Realized Variance in Forecasting 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? ## Answer by Summer_More_More_Tea (score 1) https://quant.stackexchange.com/a/80666 My two cents to fish more inputs. I choose to forecast the realised variance with the reasons - 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; - 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; - 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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