Interpolation Methods in Interest Rate Applications
13.1 INTRODUCTION AND OBJECTIVES
In this chapter we discuss interpolation which is an important tool in many financial applications.
We are interested in the use of interpolation applied in the construction of the yield curve and of the forward curve; x values are times while y values can be zero rates, forward rates, discount factors or some functions of these.
The choice of interpolation method determines some characteristics of its output. In particular, we require that the output of the interpolation satisfies some quality criteria related to the nature of the problem, for example preserving arbitrage-free conditions, behaviour under perturbation of only one input, smoothness, positivity and stability of forward rates. We address these requirements and we propose several interpolation methods that satisfy them. We deal with the problem of preserving the shape of data in the interpolant y(x), for example non-negativity, monotonicity and convexity. We may also take geometric properties into consideration, for example preventing spurious behaviour near points where rapid changes take place. This may be even more important than ensuring the asymptotic accuracy of the interpolation method.
With these requirements in mind, starting from popular interpolation methods such as linear and cubic spline we then focus on some other schemes: