Problem
Suppose that you are presented with 2 models: (a) an EWMA model with parameter l=0.89; and (b) a GARCH(1,1) model with parameters a= 0.09, b= 0.91, and w = 0.000008.
1. What is the long-run average volatility in both models?
2. Assume that the most recent return un-1 is estimated at 1% and the most recent volatility sn-1 is estimated at 2%. Update the volatility estimate in both models. For all the remaining items, consider only the following GARCH(1,1) model with parameters a= 0.08, b= 0.91, and w = 0.000005.
3. If the current volatility is 1.5% per day, what is your estimate of the volatility in 50, 75, and 800 days.
4. What volatility should be used to price 50-, 75-, and 800-day options?
5. Suppose that there is an event that decreases the volatility from 1.5% per day to 0.5% per day. Estimate the effect on the volatility in 50, 75, and 800 days.
6. Estimate by how much the event increases the volatilities used to price 50-, 75-, and 800-day options.