# (solution) Consider the standard GARCH(1,1) model, as described in exercise 7.2 with α + β 1. A

Consider the standard GARCH(1,1) model, as described in exercise 7.2 with α + β 1. A different way to model volatility is to use an exponentially weighted moving average. That is, one estimates/forecasts volatility by means of weighted average of squared returns: ∞ h t = (1 − λ ) λ j − 1 ( y t − j =1 j − y ¯) 2 where 0 λ 1. For daily data λ = 0 . 94 gives JP Morgan’s Risk Metrics model. Show that (7.84) can also be written as h t = λ h t − 1 + (1 − λ )( y t − 1 − y ¯) 2 What is the intuition behind this representation? Derive the optimal 1-step, 2-step and 3-step ahead point forecasts of h t in the GARCH(1,1) model (that is, derive expressions for h ˆ t + k | t = E[ h t + k | Y t ] for k = 3 , 4 and 5, where Y t denotes the information set available at t .) What is the optimal 100-step ahead point forecast? Derive the optimal 1-step, 2-step and 3-step ahead point forecasts of h t in the RiskMetrics model. What is the crucial difference between forecasts of the RiskMetrics model and the GARCH(1,1) model?