13.16.1 Cubic Spline of Yields
One approach to smoothing yield curves is the use of cubic splines. A cubic spline is a series of third-degree polynomials that have the form:
Figure 13-14 Cubic Spline of Yields Formula
Description of formula to calculate the Cubic Spline of Yields follows:
Where:
x = years to maturity (independent variable)
y = yield (dependent variable)
These polynomials are used to connect-the-dots formed by observable data. For example, a US Treasury yield curve might consist of interest rates observable at 1, 2, 3, 5, 7, and 10 years. To value a fixed-income option, we need a smooth yield curve that can provide yields for all possible yields to maturity between zero and 10 years. A cubic spline fits a different third-degree polynomial to each interval between data points (0 to 1 year, 1 to 2 years, 2 to 3 years, and so on). In the case of a spline fitted to swap yields, the variable x (independent variable) is years to maturity and the variable y (dependent variable) is yield. The polynomials are constrained so they fit together smoothly at each knot point (the observable data point); that is, the slope and the rate of change in the slope to time to maturity have to be equal for each polynomial at the knot point where they join. If this is not true, there will be a kink in the yield curve (that is, continuous but not differentiable).
However, two more constraints are needed to make the cubic spline curve unique. The first constraint restricts the zero-maturity yield to equal the 1-day interest rate (for example, the federal funds rate in the U.S. market). At the long end of the maturity spectrum, several alternatives exist. The most common one restricts the yield curve at the longest maturity to be either straight (y"=0) or flat (y'=0). There are other alternatives if the cubic spline is fitted to zero-coupon bond prices instead of yields.
Our function will also extrapolate the original yield curve outside its domain of definition. The resulting smoothed yield curve will be constant and equal to:
- The first term yield for T ≤ first term
- The last term yield for T ≥ last term