Cubic Spline vs Piecewise Linear Curve Fitting

A curve builder knows zero rates at a handful of maturities — say 1, 2, 5, 10, and 30 years — from observable market instruments. But pricing a bond that pays a cash flow in year 7 requires a rate at exactly year 7, a point the market never quoted directly. Something has to fill in the gaps, and how it fills them in changes the price, the hedge ratios, and even whether the resulting curve looks financially sensible.

Piecewise linear

Between two known points $(t_1, z_1)$ and $(t_2, z_2)$, linear interpolation is just:

$$z(t) = z_1 + (z_2 - z_1) \times \frac{t - t_1}{t_2 - t_1}$$

In words: move a fraction of the way from $z_1$ to $z_2$ equal to how far $t$ has moved between $t_1$ and $t_2$. It's the simplest possible rule, entirely local — changing one input point only affects the curve in the segments touching that point — but the slope of the curve jumps abruptly at every input knot, which means the forward rate curve derived from it looks like a staircase of sudden jumps rather than something smooth.

Cubic spline

A cubic spline instead fits a separate cubic polynomial between each pair of knots, chosen so that the pieces join up smoothly — not just matching in level at each knot, but also matching in slope and curvature. This produces a visually smooth curve with continuous forward rates, which is usually closer to how real markets actually price nearby maturities relative to each other.

Worked example

Known zero rates: 2-year at 3.00%, 5-year at 3.60%. Interpolate the 3-year rate.

1. Piecewise linear: $z(3) = 3.00 + (3.60 - 3.00) \times \frac{3-2}{5-2} = 3.00 + 0.60 \times 0.333 = 3.20\%$.
2. A cubic spline using these two points plus additional neighboring knots (say 1-year at 2.70% and 10-year at 3.90%) would fit a smooth curve through all four, typically landing close to but not exactly at the linear answer — often slightly higher or lower depending on the curvature implied by the neighboring points, precisely because the spline "looks beyond" just the two adjacent knots.
3. Forward-rate check: the 2-to-5-year linear forward rate is a single constant number across that whole segment (a flat piece of the staircase), while the spline's forward rate over the same stretch continuously varies — closer to how forward rates observed in the market usually behave.

What this means in practice

Piecewise linear interpolation (often on log-discount-factors rather than rates directly) remains popular in production systems because it's numerically stable, fast, and entirely local — a data error at one maturity can't distort the curve far away. Cubic splines are preferred where smooth, continuous forward rates genuinely matter, such as pricing instruments sensitive to the curve's shape between quoted points, but they can overshoot ("wiggle") when input points are sparse or unevenly spaced, occasionally producing implied forward rates that go briefly negative or unrealistically high.