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News Impact Curves for S&P 500 Returns using Coefficients from GARCH and GJR
Model Estimates: 33 GARCHin Mean • We expect a risk to be compensated by a higher return. So why not let
the return of a security be partly determined by its risk? • Engle, Lilien and Robins (1987) suggested the ARCHM specification.
A GARCHM model would be • δ can be interpreted as a sort of risk premium. • It is possible to combine all or some of these models together to get
more complex “hybrid” models  e.g. an ARMAEGARCH(1,1)M
model.
34 What Use Are GARCHtype Models? • GARCH can model the volatility clustering effect since the conditional
variance is autoregressive. Such models can be used to forecast volatility. • We could show that
Var (yt ⏐ yt1, yt2, ...) = Var (ut ⏐ ut1, ut2, ...) • So modelling σt2 will give us models and forecasts for yt as well. • Variance forecasts are additive over time. 35 Forecasting Variances using GARCH Models
• Producing conditional variance forecasts from GARCH models uses a
very similar approach to producing forecasts from ARMA models.
• It is again an exercise in iterating with the conditional expectations
operator.
• Consider the following GARCH(1,1) model:
, ut ∼ N(0,σt2),
• What is needed is to generate are forecasts of σT+12 ⏐ΩT, σT+22 ⏐ΩT, ...,
σT+s2 ⏐ΩT where ΩT denotes all information available up to and
including observation T.
• Adding one to each of the time subscripts of the above conditional
variance equation, and then two, and then three would yield the
following equations
σT+12 = α0 + α1 +βσT2 , σT+22 = α0 + α1 +βσT+12 , σT+32 = α0 + α1 +βσT+22 36 Forecasting Variances
using GARCH Models (Cont’d)
• • • • Let
be the one step ahead forecast for σ2 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 2step ahead forecast for σ2 made at time T,
calculated? Taking the conditional expectation of the second equation
at the bottom of slide 36:
= α0 + α1E( ⏐ ΩT) +β
where E(
⏐ ΩT) is the expectation, made at time T, of
, which is
the squared disturbance term.
37 Forecasting Variances
using GARCH Models (Cont’d)
•
• • • We can write
E(uT+12  Ωt) = σT+12
But σT+12 is not known at time T, so it is replaced with the forecast for
it,
, so that the 2step ahead forecast is given by
= α0 + α1
+β
= α0 + (α1+β)
By similar arguments, the 3step ahead forecast will be given by
= ET(α0 + α1 + βσT+22)
= α0 + (α1+β)
= α0 + (α1+β)[ α0 + (α1+β)
= α0 + α0(α1+β) + (α1+β)2
Any sstep ahead forecast (s ≥ 2) would be produced by 38 What Use Are Volatility Forecasts? 1. Option pricing
C = f(S, X, σ2, T, rf)
2. Conditional betas 3. Dynamic hedge ratios
The Hedge Ratio  the size of the futures position to the size of the
un...
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This note was uploaded on 09/20/2013 for the course FINA 5170 taught by Professor Janebargers during the Summer '13 term at Greenwich School of Management.
 Summer '13
 JaneBargers
 Financial Markets

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