GARCH(1,1) prediction in R - Basic Questions

Question

Background to question:

Hi, I was trying to fit a GARCH(1,1) model to the variance of log returns of a series, and ARMA(0,0) for the mean. I was using the fGarch package to do this. The aim of the modeling is to generate a predicted volatility number to feed into the Black-Scholes model to an generate option price and therefore option deltas. I plan to backtest the delta from GARCH volatility to hedge my option positions (as opposed to deltas derived from implied vol prices).

Questions:

A) This might be a very noob question: I used the 'predict' function in the package to generate a 'n-day ahead' volatility forecast. As I understand GARCH, these numbers are annualized standard deviation numbers. To hedge a 1 month option I want to forecast 30 day volatility. I can simply put 'n-days ahead = 30 to get the numbers, but how do I combine those 30 numbers to get a annualized vol number?

B) Could anyone also please explain how to use the nroll argument in the package? Basically I want rolling GARCH estimates of volatility. For example, at day 10, I want to use the past 10 days of data to get a vol prediction for day 11, at day 50 I want to use 50 days of data for vol prediction of day 51 etc.

Any help would be greatly appreciated.

Answer 1

Annualized volatility is not calculated generally by forecasting the volatility n days ahead. what is done is that the next period volatility is calculated and then it is multiplied by square root of n where n is the number of the periods contained in the year as the scaling factor. so if you calculate daily volatility and the number of trading days is 250 then the annual volatility is the next period standard deviation multiplied by square root of 250.

GARCH already uses all the past days data to predict the next period volatility. what you can do is to assign more weightage to the recent days values if you want 11 days or give more importance to longer term values if you want 50 values.