Polynomial Interpolation

A trading desk observes market rates at only a handful of maturities — say 1, 5, and 10 years — but needs a rate for every maturity in between, like 3 years, to price an odd-dated instrument. There's no market quote for exactly 3 years, so the rate has to be interpolated: constructed from the known points in a way that's smooth and internally consistent. Polynomial interpolation is the most basic tool for this: find the (essentially unique) polynomial curve that passes exactly through every known point, and read off its value anywhere in between.

An analogy: connecting dots with the smoothest possible curve

Think of connecting a handful of dots on a page not with straight lines (which would have visible sharp corners at each dot) but with a single smooth, wavy curve that threads through every dot exactly. With enough dots, a sufficiently flexible curve can be drawn through all of them — that curve, if it's a polynomial, is unique for a given set of dots: there is exactly one polynomial of the right degree that passes through $n$ points and no other of that degree does.

The math, one piece at a time

Given $n+1$ known data points $(x_0, y_0), \ldots, (x_n, y_n)$ with distinct $x$-values, there exists exactly one polynomial of degree at most $n$ that passes through every point exactly. Lagrange's form writes this polynomial explicitly:

$$P(x) = \sum_{i=0}^{n} y_i \, L_i(x), \qquad L_i(x) = \prod_{j \neq i} \frac{x - x_j}{x_i - x_j} .$$

In words: $P(x)$ is a weighted sum of the known $y$-values, where each weight $L_i(x)$ is a specially constructed function that equals exactly 1 at its own point $x_i$ and exactly 0 at every other known point $x_j$ — so at any known point, only that point's term survives and $P$ reproduces the correct $y$-value exactly, while in between, all the weights blend smoothly to interpolate. The construction of $L_i(x)$ guarantees this: the product runs over every other point $j$, and each factor $\frac{x-x_j}{x_i-x_j}$ is built to vanish at $x=x_j$ and equal 1 at $x=x_i$.

The catch is that with more points, the interpolating polynomial's degree rises, and high-degree polynomials tend to swing wildly between the known points — a phenomenon called Runge's phenomenon — especially near the edges of the data range, even though the curve is forced to pass exactly through every point. More data doesn't always mean a better-behaved curve.

Worked example 1: interpolating a rate by hand

Given 1-year rate 3.0%, 5-year rate 4.0%, and 10-year rate 4.5% — three points, so a degree-2 (quadratic) polynomial fits exactly. Using Lagrange's formula to estimate the 3-year rate ($x=3$):

$$L_0(3) = \frac{(3-5)(3-10)}{(1-5)(1-10)} = \frac{(-2)(-7)}{(-4)(-9)} = \frac{14}{36} \approx 0.389,$$ $$L_1(3) = \frac{(3-1)(3-10)}{(5-1)(5-10)} = \frac{(2)(-7)}{(4)(-5)} = \frac{-14}{-20} = 0.700,$$ $$L_2(3) = \frac{(3-1)(3-5)}{(10-1)(10-5)} = \frac{(2)(-2)}{(9)(5)} = \frac{-4}{45} \approx -0.089 .$$

Estimate: $P(3) = 3.0(0.389) + 4.0(0.700) + 4.5(-0.089) = 3.567\%$. Note the weight on the 10-year point is negative — the quadratic curve overshoots slightly to bend through all three points, a mild early warning sign of Runge-type behavior.

Worked example 2: watching the wobble with more points

Adding a fourth point (a 30-year rate of 4.2%, lower than the 10-year rate because of curve inversion) forces a degree-3 polynomial through all four points. Because 4.2% at 30 years sits below 4.5% at 10 years while all the shorter maturities were rising, the cubic polynomial connecting them can overshoot well above 4.5% or dip well below 3.0% somewhere between the plotted points to accommodate that direction change — producing implied rates for, say, a 7-year point that look implausible even though every actual quoted point is matched exactly. This overshoot, worsening as more points and a higher polynomial degree are added, is exactly why practitioners rarely use a single high-degree polynomial across a full yield curve.

What this means in practice

Polynomial interpolation underlies filling in a curve — a yield curve, a volatility smile, a dividend schedule — from sparse market quotes, and it's the conceptual foundation for numerical integration and root-finding methods that implicitly fit a local polynomial. In practice, though, quants rarely fit one high-degree polynomial across an entire dataset; instead they use lower-degree polynomials on small local pieces stitched together smoothly (see cubic splines), precisely to avoid the overshoot that a single global polynomial produces once the data has more than a handful of points.