Interpolation

Paper Accepted for Publication in the Wilmott Magazine

Posted on by Michele Maio

We are happy to announce that our paper Curiosities on the Monotone Preserving Cubic Splines  has been accepted for publication in the Wilmott Magazine.

The content of the paper is nicely summarised by the starting paragraph of the referee report: "The authors have two results. The first is a worthwhile observation that the Hyman constraint is automatically satisfied, and can thus be omitted. The second is a sensitivity calculation. These results deserve to be published ..".

Needless to say, we are happy for such an achievement and proud to get recognition for the quality of our research from an international refereed journal. This is for us a confirmation of our capacities and an additional boost of motivation.

A working version of the paper can be found in our library page. The final publication can be obtained directly from the Technical Papers section of the Wilmott Magazine journal here.

Curiosities on the Monotone Preserving Cubic Spline

Posted on by Michele Maio

In this paper we describe some new features of the monotone-preserving cubic splines and the Hyman’s monotonicity constraint, that is implemented into various spline interpolation methods to ensure monotonicity. We find that, while the Hyman constraint is in general useful to enforce monotonicity, it can be safely omitted when the monotone-preserving cubic spline is considered. We also find that, when computing sensitivities, consistency requires making some specific assumptions about how to deal with non-differentiable locations, that become relevant for special values of the parameter space.

Keywords: Yield curve, fixed-income, interpolation, Hyman, monotone preserving cubic splines.

Click here for the full paper.

On the non-differentiability of Hyman Monotonicity constraint

Posted on by Michele Maio

In this paper we describe some new features of Hyman’s monotonicity constraint, which is implemented into various cubic spline interpolation methods. We consider the problem of understanding how sensitive such methods are to small changes of the input y-values and, in particular, how relevant Hyman’s constraint is with respect to such changes. We find that many things cancel out and that eventually Hyman’s constraint can be safely omitted when the monotone-preserving cubic spline is used. We also find that consistency requires including some specific boundary conditions that become relevant for special values of the parameter space.

Keywords: Yield curve, interpolation, monotone preserving cubic splines.

Click here for the full paper.

Interpolation methods and the Hagan-West paper

Posted on by Michele Maio

Interpolation is a very useful technique for extracting data when the available information does not come in a continuous form.

From a non-technical point of view, any inference or decision process (sometimes subconsciously) is based on a kind of interpolation or best fitting or regression of the available informations. We as people are normally quite good at generalising (often too fast) from the little amount of information that we have about other people, situations, or even numerical data. This is possible because our brain can recognise patterns and see trends in any kind of data. However, technically speaking, interpolation is more that just finding a trend.

Technically, we are often given a discrete set of data corresponding to a certain function which is known at specific points, or nodes (for example, we have made an experiment for specific input values and measured the outputs  corresponding to that input), and is otherwise unknown.  In principle this is a multi-dimensional problem, and the interpolating hyper-surface will give an idea of the missing information. In fact, even if it is true that such a hyper-surface can always be numerically constructed,  however the uniqueness issue remains. Given the same input data, many different constructions can be engineered, all satisfying to various -more or less realistic- criteria, and all passing through the same input points. Continue reading →