Yield Curve Dynamics Research Articles

Modeling the Term Structure of Interest Rates: Where Do We Stand?

No-arbitrage term structure models are becoming increasingly important to policy makers and practitioners alike. Several factors justify this trend. First, modeling progress has been tremendous over the last years, allowing a much better fit of actual yield curve dynamics and increased model realism (see Dai and Singleton (2002a,b)). Second, increases in computing power allow the efficient panel estimation of term structure models. Given that term structure models have implications for both the cross-section and time series dimension of yields, panel estimation techniques are to be preferred over either crosssection or time series techniques. Third, term structure models have recently been extended in ways that are of direct interest to policy makers. Example given, Dewachter and Maes (2001) model the international term structure of interest rates, taking into account the role of the exchange rate in a no-arbitrage economy, while amongst others Hordahl et al. (2002) and Dewachter et al. (2002) jointly model the term structure of interest rates with the dynamics of macroeconomic variables. The latter approach allows to study (i) the driving factors behind the term structure and the risk premia in terms of clearly interpretable macroeconomic variables and their determinants, and (ii) the effects of monetary policy on the term structure of interest rates and macroeconomic variables within a consistent no-arbitrage framework.

Interest rates and information geometry

Interest rates and information geometry

A family of humped volatility models

Recent empirical studies on interest rate derivatives have shown that the volatility structure of interest rates is frequently humped. Several researchers have modelled interest rate dynamics in such a way that humped volatility structures are possible and yet analytical formulas for European options on discount bonds are derived. However, these models are Gaussian, and hence interest rates may become negative. Here, a family of interest rate models is proposed where (i) humped volatility structures are possible; (ii) the interest rate volatility may depend on the level of the interest rates themselves; and (iii) the valuation of interest rate derivative securities can be accomplished through recombining lattices. The second item implies that a number of probability distributions are possible for the yield curve dynamics, and some of them ensure that interest rates remain positive. Proportional models of the Ritchken and Sankarasubramanian type and the Black and Karasinski model are proposed. To ensure computational tractability the embedding of all models in this paper in either the Ritchken and Sankarasubramanian or the Hull and White class of models is demonstrated.

A family of humped volatility models

Recent empirical studies on interest rate derivatives have shown that the volatility structure of interest rates is frequently humped. Several researchers have modelled interest rate dynamics in such a way that humped volatility structures are possible and yet analytical formulas for European options on discount bonds are derived. However, these models are Gaussian, and hence interest rates may become negative. Here, a family of interest rate models is proposed where (i) humped volatility structures are possible; (ii) the interest rate volatility may depend on the level of the interest rates themselves; and (iii) the valuation of interest rate derivative securities can be accomplished through recombining lattices. The second item implies that a number of probability distributions are possible for the yield curve dynamics, and some of them ensure that interest rates remain positive. Proportional models of the Ritchken and Sankarasubramanian type and the Black and Karasinski model are proposed. To ensure computational tractability the embedding of all models in this paper in either the Ritchken and Sankarasubramanian or the Hull and White class of models is demonstrated.

Optimal stopping in Hilbert spaces and pricing of American options

We consider an optimal stopping problem for a Hilbert-space valued diffusion. We prove that the value function of the problem is the unique viscosity solution of an obstacle problem for the associated parabolic partial differential equation in the Hilbert space. The results are applied to investigate the pricing of American interest rate options in the lognormal Heath-Jarrow-Morton model of yield curve dynamics.

Modeling Term Structure Dynamics: An Infinite Dimensional Approach

Motivated by stylized statistical properties of interest rates, we propose a modeling approach in which the forward rate curve is described as a stochastic process in a space of curves. After decomposing the movements of the term structure into the variations of the short rate, the long rate and the deformation of the curve around its average shape, this deformation is described as the solution of a stochastic evolution equation in an infinite dimensional space of curves. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates, the structure of principal components of forward rates and their variances. In particular we show that a flat, constant volatility structures already captures many of the observed properties. Finally, we discuss parameter estimation issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.