Suppose I have a forecasted daily volatility for K days.

How can I get the forecasted monthly volatility from the daily ?

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#### Best Answer

If you assume the underlying time series $x_t$ is not autocorrelated (which is a reasonable assumption for daily financial returns), then $$ text{Var}( x_{t+1} + dotsc + x_{t+K} ) = text{Var}(x_{t+1}) + dotsc + text{Var}(x_{t+K}) $$ and you can substitute forecasts for the theoretical quantities: $$ widehat{text{Var}}( x_{t+1} + dotsc + x_{t+K} ) = widehat{text{Var}}(x_{t+1}) + dotsc + widehat{text{Var}}(x_{t+K}). $$ You have your daily foreasts $widehat{text{Var}}(x_{t+1}), dotsc, widehat{text{Var}}(x_{t+K})$ with $K$ around 22 for working days or 30 for calender days; this allows you to obtain the monthly forecast $widehat{text{Var}}( x_{t+1} + dotsc + x_{t+K} )$.

Meanwhile, in presence of autocorrelation you would have $$ text{Var}( x_{t+1} + dotsc + x_{t+K} ) = text{Var}(x_{t+1}) + dotsc + text{Var}(x_{t+K}) + sum_{i=1}^K sum_{j=1}^K text{Cov}(x_{t+i},x_{t+j}) $$ and a corresponding expression for forecasts in places of theoretical quantities.

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