Building Econometric Models

Forecasting Variances using GARCH Models (Cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Let be the one step ahead forecast for 2 made at time T. This is easy to calculate since, at time T, the values of all the terms on the RHS are known.
• would be obtained by taking the conditional expectation of the first equation at the bottom of slide 36:
• Given, how is , the 2-step ahead forecast for 2 made at time T, calculated? Taking the conditional expectation of the second equation at the bottom of slide 36:
• = 0 + 1E(  T) +
• where E(  T) is the expectation, made at time T, of , which is the squared disturbance term.

Forecasting Variances using GARCH Models (Cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• We can write
• E(uT+12  t) = T+12
• But T+12 is not known at time T, so it is replaced with the forecast for it, , so that the 2-step ahead forecast is given by
• = 0 + 1 +
• = 0 + (1+)
• By similar arguments, the 3-step ahead forecast will be given by
• = ET(0 + 1uT+22 + T+22)
• = 0 + (1+)
• = 0 + (1+)[ 0 + (1+) ]
• = 0 + 0(1+) + (1+)2
• Any s-step ahead forecast (s  2) would be produced by

What Use Are Volatility Forecasts?

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• 1. Option pricing
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• C = f(S, X, 2, T, rf)
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• 2. Conditional betas
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• 3. Dynamic hedge ratios
• The Hedge Ratio - the size of the futures position to the size of the underlying exposure, i.e. the number of futures contracts to buy or sell per unit of the spot good.
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