Spline-Functions - Theory And Applications

This article analyses the theoretical background of different interpolation techniques. Polynomial interpolation is a classical approach. It has numerous advantages based on the simplicity of the method. However, on the basis of its disadvantages like dealing with polynomials of high degree we derive the theory and applicability of spline-interpolation to financial time-series. High degree polynomial interpolation could turn out to be useless on a practical point of view due to the resulting increased calculation efforts and large oscillations in the data sample. Solution to this problem is provided by piecewise polynomial interpolation, that produces pieces of polynomial curves, that automatically tie together smoothly. These piecewise polynomial curves are called spline functions. Presented are also the Lagrange polynomial, different spline curves, the essence of regression splines and Akima splines. The article ends up with a discussion of the applicability of natural cubic splines as an interpolation technique for financial time-series with missing values like those on a frontier stock market on the instance of the Bulgarian Stock Exchange.