13.4 Cubic spline interpolation


13.4 Cubic spline interpolation

Cubic spline interpolation is a type of piecewise polynomial approximation that uses cubic polynomials between successive pairs of nodes [ 13 ]. Additionally, the constructed cubic piecewise interpolant is required to be twice continuously differentiable. This condition differentiates cubic spline interpolation from other types of cubic piecewise interpolation techniques, such as Cubic Hermite and Cubic Bessel interpolation (see [ 20 ]).

At each of the nodes across which the cubic splines are fitted, the following hold:

  • The values of the fitted splines equal the values of the original function at the node points.

  • The first and second derivatives of the fitted splines are continuous.

See Appendix for a detailed formulation of cubic spline interpolation.

As shown in (13.9) and (13.10), we require the derivatives of the initial term structures of A (0, ·) and B (0, ·). Although the construction of the cubic spline does not ensure the derivatives of the interpolant agree with the derivatives of the initial function [ 13 ], it does provide "acceptable approximations to derivatives" [ 20 ]. For this reason the derivatives of the fitted cubic splines are used as approximations to the required derivatives i.e. and .

Now consider approximating the derivative at each node point x j , by first fitting a quadratic polynomial to points x j ˆ’ 1 , x j and x j +1 and then evaluating its derivative at x j . Comparing these derivatives to those produced by taking the derivatives of the fitted cubic splines shows that the cubic splines produce much more extreme derivative values. In equations (13.7) and (13.10) the integral of the square of inverse of the derivative of B (0, ·) is required. Here, the more extreme derivatives generated by the cubic splines give rise to inconsistencies and rather high integral values between certain node points. The effect is that of unreasonable future term structure shapes of ‚( t, ·) and unreasonably high volatility values for some of the sub-options constituting the coupon bond option. We investigate two ways of mitigating the magnitude of this effect:

13.4.1 Cubic Bessel interpolation

Cubic Bessel interpolation (see [ 20 ]) uses the same basic methodology as Cubic Spline Interpolation, but places an additional constraint on the derivatives at the node points. The derivative at each node point x j , is set equal to the derivative of the quadratic polynomial fitted to points x j ˆ’ 1 , x j and x j +1 . For details of this procedure see the Appendix.

By forcing the derivatives at the nodes of the cubic interpolant to be equal to those of the quadratic polynomials, the derivatives between the nodes become less extreme and the integral of the square of the inverse of the derivative of the cubic polynomials becomes smoother. However, the integral between some of the short-term nodes is still rather large, again leading to overestimation of the volatility.

13.4.2 Interpolating the derivatives

The derivatives at the node points are assumed to equal those of the quadratic polynomials fitted as above. Cubic spline interpolation is then applied to these derivatives. This results in a smoothly interpolated derivative curve and hence smooth integral values. The integral required to evaluate (13.10) now becomes the integral of the square of the inverse of the cubic spline. The evaluation of such an integral is detailed in the Appendix.




Interest Rate Modelling
Interest Rate Modelling (Finance and Capital Markets Series)
ISBN: 1403934703
EAN: 2147483647
Year: 2004
Pages: 132

flylib.com © 2008-2017.
If you may any questions please contact us: flylib@qtcs.net