Approximating term structure of interest rates using cubic L1 splines

A simple and general approach to fitting the discount curve under no-arbitrage constraints

“…Therefore, for cubic and higher-degree splines, one needs to content oneself with spline approximations that achieve monotonicity only at a finite collection of points; see Tobler (1996) for such an approach to modeling discount factors. Alternatively, one requires more structural assumptions, such as local convexity, as in Chiu et al (2008), or additional and more sophisticated optimization techniques, as proposed in Turlach (2005) and Papp and Alizadeh (2014).…”

“…In the literature, there is consensus about the usefulness of constrained estimators, but less is known about the relative importance of the degrees of the spline and the loss functions used for constrained estimation. This is because a single constrained estimator is usually compared against unconstrained fits; see Barzanti and Corradi (1999); Ramponi (2003); Chiu et al (2008); Laurini and Moura (2010). The general estimation framework that we suggest allows us to obtain a more complete picture of the impact of the B-spline order, the penalty, and the loss function on estimation efficiency.…”

“…A linear B-spline approximation may therefore not be satisfactory for certain applications. Chiu et al (2008) use a cubic B-spline instead, but for monotonicity to hold, they impose a local convexity condition to hold piecewise between the spline segments. This assumption could be questioned, as there is no economic theory that would require the discount curve to be convex, either locally or globally.…”

Exact Smooth Term-Structure Estimation

“…Barzanti and Corradi (1998) use tension splines where the tension in the spline is increased manually until problematic behaviour is avoided. Chiu et al (2008), Laurini andMoura (2010), andFengler andHin (2015), among others, impose shape constraints on the B-splines used to represent the discount function. The discount curve produced by our method is not guaranteed to be positive or monotonic non-increasing, however we did not find this to be a problem in the numerical examples we have explored.…”

A Combinatorial Optimization Model of Interest Rate Term Structure Using Genetic Algorithms

“…For example, Vasicek and Fong [5] suggested exponential splines whereas Steely [6] preferred basis splines. Chiu et al [7] proposed cubic spline model. Laurini and Moura [8] applied the constrained smoothing-splines to interpolate and construct measures associated with the term structure of interest rates.…”