Estimation of Alternative Volatility Models

As pointed out above, we will analyze the Bund future options market. Our data set of daily closing prices for the Bund future starts at the beginning of 1988 and runs through August 1995. We follow standard procedures and take futures prices for those series only for which the time to maturity is larger than one month. This gives us a consistent series, that is, however, not stationary. Hence, instead of using the prices we employ returns series.

The daily returns are calculated as the differences in the logarithms of the prices and hence constitute continuously compounded rates. They exhibit much the same characteristics than stock returns. They are characterized by volatility clustering and fat tailed distributions as can be seen from the summary statistics reported in Table 8.1. In this table we present descriptive statistics not only for the returns based on closing prices but also for returns calculated from opening to closing and closing to opening prices. These statistics show the following results: the mean is very close to zero so that it can be neglected if necessary and all series have excess kurtosis.

As a final first step we analyze the return series for possible autocorrelations. Looking at the autocorrelation functions of the returns and the squared returns suggests that there are no significant linearities in the data but strong nonlinearities that can be the result of heteroscedastic-

Table 8.1. Descriptive statistics of daily returns

Close to Close

Opening to Close (Active Period)

Close to Opening (Passive Period)

Mean

-4.7E-5 %

-0.014 %

0.014 %

Standard Dev.

0.360 %

0.326 %

0.196 %

Skewness

-0.163

0.066

0.034

Kurtosis

7.137

7.081

12.734

ity. This again points to the use of a GARCH model. In what follows we estimate three alternative GARCH models. A standard GARCH(1,1), an EGARCH(1,1) and a GJR(1,1) specification. Tables 8.2, 8.3 and 8.4 report the estimation results.

Table 8.2. GARCH (1,1) estimates

Variable

Coefficient

t-Statistic

Probability

c

8.29E-5

1.17

0.241

u

1.4E-7

5.15

0.000

a

0.0646

8.74

0.000

ß

0.9249

136.13

0.000

Table 8.3. EGARCH (1,1) estimates

Variable Coefficient t-Statistic Probability

Table 8.3. EGARCH (1,1) estimates

Variable Coefficient t-Statistic Probability

c

4.62E-5

0.66

0.509

u

-0.1707

-5.42

0.000

a

0.1314

9.10

0.000

ß

0.9847

359.00

0.000

e

-0.0348

-4.63

0.000

These results imply the following. First, there is a strong persistence in conditional variances as expressed by values for 3 above 0.9 for all three specifications. Second, for both the GARCH(1,1) and for the GJR(1,1) model the sum of the estimated parameters is close to unity suggesting an integrated process for the conditional variance. Finally, returns show significant leverage effects. In order to select one of the models as the appropriate one, we performed several diagnostic checks which are reported in the following tables. Table 8.5 reports some de-

Table 8.4. GJR(1,1) estimates

Variable

Coefficient

t-Statistic

Probability

c

4.59E-5

0.64

0.524

u

1.56E-7

5.54

0.000

a

0.0318

4.09

0.000

3

0.9330

139.39

0.000

e

0.0441

4.42

0.000

scriptive statistics of the normalized residuals for the three models. From these we see that the kurtosis is substantially reduced which is in accord with international evidence on stock returns. Table 8.6 reports the test statistic of the Box-Ljung autocorrelation test, which indicates that both the residuals and the squared residuals are not autocorrelated. Finally, the BDS test reported in Table 8.7 supports consistency with i.i.d. residuals for all three specifications. Hence, to decide on one of the alternative models we make use of the estimated likelihood as shown in Table 8.8. On the basis of these results we will mainly use the GJR(1,1) model for the following analysis.

Table 8.5. Descriptive statistics of normalized residuals

Model

Mean

Std. Dev.

Skewness

Kurtosis

GARCH(1,1)

-0.0165

0.9981

-0.1435

4.653

EGARCH(1,1)

-0.0016

0.9984

-0.0341

4.801

GJR(1,1)

-0.0027

0.9980

-0.0906

4.667

Table 8.6. Autocorrelation test (Box-Ljung test with lag = 20)

Normalized Residuals

Squared Normalized Residuals

Model

X2-Statistic Probability

X2-Statistic

Probability

GARCH(1,1)

21.35 0.377

14.00

0.830

EGARCH(1,1)

23.79 0.252

15.83

0.727

GJR(1,1)

21.27 0.381

12.84

0.884

As pointed out in Section 8.2 the exponentially weighted moving average exhibits many characteristics of a GARCH model. The two important differences are that the EWMA does violate the strict stationarity

Table 8.7. BDS test of daily returns and normalized returns

Embedding Dimension = 2

Embedding Dimension = 5

Model

BDS-Statistic

Probability

BDS-Statistic

Probability

Daily Returns

8.08

0.000

13.12

0.000

GARCH(1,1)

-1.23

0.220

-1.41

0.160

EGARCH(1,1)

-1.36

0.173

-1.17

0.241

GJR(1,1)

-1.19

0.235

-1.01

0.312

Table 8.8. Estimated likelihood of GARCH models

Model

Number of Parameters

Log-Likelihood

GARCH(1,1)

4

9276.29

EGARCH(1,1)

5

9273.31

GJR(1,1)

5

9282.18

assumption and that it does not include any constant. Since the GARCH estimates demonstrate that the data are consistent with stationary variances the EWMA model does not seem to be a correct specification. Nevertheless we make use of this volatility model and estimate its parameters via least squares methods. The corresponding estimated dynamic equation is given by of+1 = 0, 0704mt + 0, 9296<rt2 (8.9)

with mt = (rt — c) and c as the sample mean and a0 as the sample variance. Since our primary objective is to use the five different models for volatility forecasts, we need to evaluate their relative forecasting performance. This is not a trivial exercise, since volatility is not directly observable. Hence, we need to identify some benchmark. Here we choose the sample standard deviation as a reference. Based on this measure, Table 8.9 presents the in-sample and out-of-sample forecasting performance evaluated along the lines of two different measures, the mean squared error (MSE) and the mean absolute error (MAE). Based on these results, again the GJR model dominates the others. Hence, for the remainder of the paper we mainly use this specification.

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