Interpolation Methods for Curve Construction

Abstract

This paper surveys a wide selection of the interpolation algorithms that are in use in financial markets for construction of curves such as forward curves, basis curves, and most importantly, yield curves. In the case of yield curves the issue of bootstrapping is reviewed and how the interpolation algorithm should be intimately connected to the bootstrap itself is discussed. The criterion for inclusion in this survey is that the method has been implemented by a software vendor (or indeed an inhouse developer) as a viable option for yield curve interpolation. As will be seen, many of these methods suffer from problems: they posit unreasonable expections, or are not even necessarily arbitrage free. Moreover, many methods lead one to derive hedging strategies that are not intuitively reasonable. In the last sections, two new interpolation methods (the monotone convex method and the minimal method) are introduced, which it is believed overcome many of the problems highlighted with the other methods discussed in the earlier sections.

Keywords:

• Yield curve
• interpolation
• bootstrap

Acknowledgement

We wish to thank an anonymous referee for several useful suggestions.

Notes

1. By this we mean the intervals between the fixed payments of the swap, such as three or six months.

2. For example, it is not reasonable to expect a bootstrapper to differentiate bonds according to tax status, rather, some value adjustment should be made a priori to the prices one set of bonds or the other, so that what is input can be considered to have the same tax status.

3. To paraphrase the nomenclature of de Boor (1978, 2001): a cubic spline where the first derivative is known (in addition to the function values) is called a Hermite spline. The Bessel method is an intermediate method, where the derivatives are estimated from the function values, and then the Hermite method is applied.

4. A matrix in a‐1‐b bandwidth form means that the entry in the ith row and jth column in this representation actually lies in the ith row and i + j−a−1th column in the canonical matrix representation. Note that the entries in the top left and bottom right of the bandwidth matrix are redundant, they can be set to anything. This redundancy is denoted by a ×.

5. Why not f(r) = a_i  + b_i (r−r_i ) + c_i (r−r_i )² + d_i (r−r_i )³ + e_i (r−r_i )⁴? We would prefer on principle this format. However, the bandwidth matrices which result are very unwieldy.

6. This observation is consistent with Adams (2001) where, after interpolating the forward curves, one additional piece of information is needed to recover the interpolatory function on the yields i.e. the method of Adams (2001) has one remaining degree of freedom.

7. If g_{i −1} = 0 = g_i , then the curve is flat, and g = 0.

8. It is advisable to ensure no problems with code execution (‘division by zero’) to trap the cases x(r) = 0 and x(r) = 1 up front: in these cases, g(r) = g_{i −1} and g(r) = g_i respectively.

9. The bound function is given by bound(a, x, b) = min(max(a, x), b).

10. The same trapping as before has .