(R2 is above 90 percent). These interpolated rates range from 1 percent to 1.5 percent (annual).
Unlike S&P 500, S&P 100 index futures are not traded at organized exchanges. Therefore, an independent
source of data on the dividend rate is not available. Standard & Poor’s routinely collects data related to the
performance of its indices and publishes dividend rates once a month. However, these rates are the ex-post
ones, which are apt to diverge from the daily market’s assessment of the future dividend streams. Therefore,
I employ the data on the Europeamstyle S&P 100 index options to infer dividend rates from the European
put-call parity relationship, Ct — Pt = Ste λπ' t — e l'π" t'> X. To obtain S, I always take XEO calls and puts
with high trading volume and open interest at a strike price closest to the underlying’s price. Interpolated
dividend rates range from 1 percent to 4 percent (annual).
Since XEO options are used to calibrate the parameters only, I converted XEO calls into puts according
to the parity relationship. A summary of the option data is presented in Tables 1 and 2.6
6.2. Parameter Calibration
In principle, there are several ways to obtain the parameters of the model. A direct and analytically appealing
approach is to use the underlying’s price data only (together with “known” r and S) and estimate the
parameters by maximum likelihood.
However, for several reasons, I do not dare to use MLE. First, even if vt were observable for every t
corresponding to the sample time-series data points, sample likelihood may be computationally prohibitive
to evaluate just for one trial set of parameters. In present context, the sample likelihood function is the
joint density of a series of random vectors (st,vt)'. A well-known technique from the time-series analysis is
to represent this density as a product of conditional densities. Each of these would have the form specified
by equation (9), as it is sufficient to condition on the most recent previous values of the state variables.
Therefore, in analytical terms, sample likelihood presents no difficulty. Nevertheless, numerical integration
in (9) is computationally demanding and must be performed as many times as there are data points in the
sample. Further parameter search would require this whole task to be repeated iteratively, until the maximum
of the sample likelihood is achieved.
Second, in reality the state variable vt is not observed. Still, it is a necessary component in the option
pricing formula. The combined task of simultaneously estimating vt series and maximizing the sample
likelihood is, indeed, daunting.
An approach to recovering vt series has been suggested by Chernov and Ghysels (2000). Their filter-
ing method is based on the reprojection procedure introduced by Gallant and Tauchen (1998).7 Chernov
6S&P U.S. 100 index closing prices were: 553.87 (June 30), 549.01, 547.17, 543.33, 544.25, 540.21, and 542.63 (July 9).
7 “Reprojecting partially observable systems with application to interest rate diffusions,” Journal of American Statistical
Association 93, 10-24.
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