Unformatted text preview: , GARCH or implied volatility? •
Weekly closing prices (Wednesday to Wednesday, and Friday to Friday)
for the S&P100 Index option and the underlying 11 March 83 - 31 Dec. 89 • Implied volatility is calculated using a non-linear iterative procedure.
46 The Models
• The “Base” Models
For the conditional mean
And for the variance (2) or (3) where
RMt denotes the return on the market portfolio
RFt denotes the risk-free rate
ht denotes the conditional variance from the GARCH-type models while
σt2 denotes the implied variance from option prices.
47 The Models (cont’d) • Add in a lagged value of the implied volatility parameter to equations (2)
and (3) becomes
• We are interested in testing H0 : δ = 0 in (4) or (5).
Also, we want to test H0 : α1 = 0 and β1 = 0 in (4),
and H0 : α1 = 0 and β1 = 0 and θ = 0 and γ = 0 in (5).
48 The Models (cont’d)
• If this second set of restrictions holds, then (4) & (5) collapse to
(4’) • and (3) becomes
(5’) • We can test all of these restrictions using a likelihood ratio test. 49 In-sample Likelihood Ratio Test Results:
GARCH Versus Implied Volatility 50 In-sample Likelihood Ratio Test Results:
EGARCH Versus Implied Volatility 51 Conclusions for In-sample Model Comparisons &
• IV has extra incremental power for modelling stock volatility beyond
GARCH. • But the models do not represent a true test of the predictive ability of
IV. • So the authors conduct an out of sample forecasting test. • There are 729 data points. They use the first 410 to estimate the
models, and then make a 1-step ahead forecast of the following week’s
volatility. • Then they roll the sample forward one observation at a time,
constructing a new one step ahead forecast at each step.
52 Out-of-Sample Forecast Evaluation
• They evaluate the forecasts in two ways:
The first is by regressing the realised volatility series on the forecasts plus
is the “actual” value of volatility, and
is the value forecasted
for it during period t.
• Perfectly accurate forecasts imply b0 = 0 and b1 = 1.
• But what is the “true” value of volatility at time t ?
Day & Lewis use 2 measures
1. The square of the weekly return on the index, which they call SR.
2. The variance of the week’s daily returns multiplied by the number
of trading days in that week.
53 Out-of Sample Model Comparisons 54 Encompassing Test Results: Do the IV Forecasts Encompass those
of the GARCH Models? 55 Conclusions of Paper • Within sample results suggest that IV contains extra information not
contained in the GARCH / EGARCH specifications. • Out of sample results suggest that nothing can accurately predict
volatility! 56 Multivariate GARCH Models • • • Multivariate GARCH models are used to estimate and to forecast
covariances and correlations. The basic formulation is similar to that of the
GARCH model, but where the...
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