Estimating the volatility term structure
Antonio D ́ıaz, Francisco Jareno, and Eliseo Navarro ̃Abstract.In this paper, we proceed to estimate term structure of interest rate volatilities,
finding that these estimates depend significantly on the model used to estimate the term structure
(Nelson and Siegel or Vasicek and Fong) and the heteroscedasticity structure of errors (OLS
or GLS weighted by duration). We conclude in our empirical analysis that there are significant
differences between these volatilities in the short (less than one year) and long term (more than
ten years). Finally, we can detect that three principal components explain 90% of the changes
in volatility term structure. These components are related to level, slope and curvature.Key words:volatility term structure (VTS), term structure of interest rates (TSIR), GARCH,
principal components (PCs)1 Introduction
We define the term structure of volatilities as the relationship between the volatil-
ity of interest rates and their maturities. The importance of this concept has been
growing over recent decades, particularly as interest rate derivatives have developed
and interest rate volatility has become the key factor for the valuation of assets such
as caplets, caps, floors, swaptions, etc. Moreover, interest rate volatility is one of
the inputs needed to implement some term structure models such as those of Black,
Derman and Toy [4] or Hull and White [12], which are particularly popular among
practitioners.
However, one of the main problems concerning the estimation of the volatility
term structure (VTS) arises from the fact that zero coupon rates are unobservable.
So they must be previously estimated and this requires the adoption of a particular
methodology. The problem of the term structure estimation is an old question widely
analysed in the literature and several procedures have been suggested over the last
thirty years.
Among the most popular methods are those developed by Nelson and Siegel [14]
and Vasicek and Fong [17]. In Spain, these methods have been applied in Nu ́nez [15] ̃
and Contreras et al. [7] respectively.
A large body of literature focuses on the bond valuation ability of these alternative
models without analysing the impact of the term structure estimation method onM. Corazza et al. (eds.), Mathematical and Statistical Methodsfor Actuarial Sciencesand Finance
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