198 E. Luciano and P. Semeraro
- find the numerical copulaCˆtof the process over the prespecified grid;
- compute the distance between the numerical and Gaussian copula.^1 Please note
that, since the linear correlationρX(t)in (10) is independent of time, the Gaussian
copula remains the same too: (Ct=Ct′in (11)).
3 Empirical investigation
3.1 Data
The procedure outlined above has been applied to a sample of seven major stock
indices: S&P 500, Nasdaq, CAC 40, FTSE 100, Nikkei 225, Dax and Hang Seng. For
each index we estimated the marginal VG parametersunder the risk neutral measure,
using our knowledge of the (marginal) characteristic function, namely (4). From the
characteristic function, call option theoretical prices were obtained using the Frac-
tional Fast Fourier Transform (FRFT) in Chourdakis [2], which is more efficient than
the standard Fast Fourier Transform (FFT). The data for the corresponding observed
prices are Bloomberg quotes of the corresponding options with three months to ex-
piry. For each index, six strikes (the closest to the initial price) were selected, and the
corresponding option prices were monitored over a one-hundred-day window, from
7/14/06 to 11/30/06.
3.2 Selection of theα-VG parameters
We estimated the marginal parameters as follows: using the six quotes of the first
day only, we obtained the parameter values which minimised the mean square error
between theoretical and observed prices, the theoretical ones being obtained by FRFT.
We used the results as guess values for the second day, the second day results as guess
values for the third day, and so on. The marginal parameters used here are the average
of the estimates over the entire period. The previous procedure is intended to provide
marginal parameters which are actually not dependent on an initial arbitrary guess
and are representative of the corresponding stock index price, under the assumption
that the latter is stationary over the whole time window. The marginal values for the
VG processes are reported in Table 1.
We performed our analysis using the marginal parameters reported above and the
maximal correlation allowed by the model. The idea is indeed that positive and large
dependence must be well described. For each pair of assets, Table 2 gives the maximal
possible value ofa, namelya=min{α^11 ,α^12 }(lower entry) and the corresponding
correlation coefficientρ(upper entry), obtained using (10) in correspondence to the
maximala.
(^1) Since we have the empirical copula only on a grid we use the discrete version of the previous
distance.