Suppose I have a forecasted daily volatility for K days.
How can I get the forecasted monthly volatility from the daily ?
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.
Similar Posts:
- Solved – How to measure the true underlying daily volatility from daily data
- Solved – Volatility of x and y variables in linear regression
- Solved – Disaggregate monthly forecasts into daily data
- Solved – the proper procedure for forecasting a VAR estimated in differences
- Solved – Realized volatility vs close to close return and open to close return