8 using garch to forecast future volatility 2 2 2 1 n

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8. Using GARCH to forecast future volatility. 2 2 2 1 ( ) ( n t L n t L n t L V u V V σ α β σ + + − + − = + 1 ) 2 ) , since the expected value of is , hence: 2 1 n t u + − 2 1 n t σ + − 2 2 1 [ ] ( ) [ ] ( ) ( t n t L n t L n L E V E V σ α β σ α β σ + + − = + = + V 9. Volatility term structures – the relationship between the implied volatilities of the options and their maturities. - Define 2 1 ( ) ( ) and a=ln + n t V t E σ α β + = Æ - 0 1 1 ( ) [ (0) ] at T L L e V t dt V V V T aT = + , or in per year Æ - 2 1 ( ) 252 [ (0) ] at L e T V V aT σ L V = + Æ can be used to estimate a volatility term structure based on the GARCH (1,1) model. 10. Impact of volatility changes. - 2 2 1 (0) ( ) 252 [ ] 252 at L L e T V aT σ σ = + V . - 6 -
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