We assume that we have been given n+1 data points, \( (t_0,f_0), (t_1,f_1), ..., (t_n,f_n)\), where \(t_i\) and \(f_i\) represent the abscissa and ordinate of each point, respectively. The objective of interpolation is to find the values of the ordinates for some t within the range \( t_0 \) to \(t_n\). The usual approach to the problem is to fit a continuous polynomial function of some degree, which we will call the interpolating polynomial or function, between the given data points, and then use the fitted function to get the interpolated value for any desired t (within the range of course!).
One can fit an n degree polynomial through the n+1 data points. This will make the solution easier; however, high degree polynomials will exhibit high oscillations, an undesirable property in the financial applications. Hence, the popular approaches in finance are to fit low degree polynomials between two consecutive points (or each interval), but impose some structure so that the polynomials blend together well.
We present the three most popular approaches: linear, Cubic (Bessel) Hermite, and Cubic Spline interpolations. We give a detailed and self-contained account of these interpolation methodologies, outlining the underlying assumptions and structural conditions, and detailing the step by step derivation of key results/formulae.