Estimating the volatility term structure 1252Data
The database we use in this research contains daily volume-weighted averages of all
the spot transaction prices and yields of all Spanish Treasury bills and bonds traded
and registered in the dealer market or Bank of Spain’s book entry system. They are
obtained from annual files available at the “Banco de Espana” website. ̃^2 We f o cu s o n
27 different maturities between 1 day and 15 years. Our sample runs from January
1994 to December2006.
First of all, in order to refine our data, we have eliminated from the sample those
assets with a trading volume less than 3 million euros (500 million pesetas) in a single
day and bonds with term to maturity less than 15 days or larger than 15 years. Besides,
in order to obtain a good adjustment in the short end of the yield curve, we always
include in the sample the one-week interest rate from the repo market.
From the price (which must coincide with the quotient between effective volume
and nominal volume of the transaction) provided by market, we obtain the yield to
maturity on the settlement day. Sometimes this yield diverges from the yield reported
by the market. Controlling for these conventions, we recalculate the yield using com-
pound interest and the year basis ACT/ACT for both markets.^3
We estimate the zero coupon bond yield curve using two alternative methods. The
first one we use fits Nelson and Siegel’s [14] exponential model for the estimation
of the yield curve.^4 The second methodology is developed in Contreras et al. [7]
where the Vasicek and Fong [17] term structure estimation method (VFO) is adapted
to the Spanish Treasury market.VFOuses a non-parametric methodology based on
exponential splines to estimate the discount function. A unique variable knot, which is
located to minimise the sum of squared residuals, is used to adjust exponential splines.
With respect to the estimation methodology we apply both OLS and GLS. In the
second case we adjust the bond price errors by the inverse of the bond Macaulay
duration in order to avoid penalisation of more interest rate errors in the short end of
the term structure.
In Figure 1 we illustrate the resulting estimations of the term structure in a single
day depending on the weighting scheme applied to the error terms. It can be seen how
assuming OLS or GLS affects mainly the estimates in the short and long ends of the
TSIR even though in both cases we use the Nelson and Siegel model.^5
(^2) http://www.bde.es/banota/series.htm. Information reported is only about traded issues. It
contains the following daily information for each reference: number of transactions, settle-
ment day, nominal and effective trading volumes, maximum, minimum and average prices
and yields.
(^3) These divergences are due to simple or compound interest and a 360-day or 365-day year
basis depending on the security term to maturity. http://www.bde.es/banota/actuesp.pdf
(^4) See, for example, D ́ıaz and Skinner [10], D ́ıaz et al. [8] and D ́ıaz and Navarro [9] for a more
detailed explanation. Also, a number of authors have proposed extensions to the NS model
that enhance flexibility [16].
(^5) When using the Vasicek and Fong model, these differences are mainly shown in the short
term. We observe differences depending on the model employed (VF or NS) even when the
same error weighting scheme is used.